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Lakhmir Singh Manjit Kaur Chemistry 2019 Solutions for Class 10 Science Chapter 1 Chemical Reactions And Equations are provided here with simple step-by-step explanations. These solutions for Chemical Reactions And Equations are extremely popular among Class 10 students for Science Chemical Reactions And Equations Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Lakhmir Singh Manjit Kaur Chemistry 2019 Book of Class 10 Science Chapter 1 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Lakhmir Singh Manjit Kaur Chemistry 2019 Solutions. All Lakhmir Singh Manjit Kaur Chemistry 2019 Solutions for class Class 10 Science are prepared by experts and are 100% accurate.

#### Question 1:

Why is respiration considered an exothermic process?

In respiration, glucose undergoes slow combustion by combining with oxygen to form carbon dioxide, water and energy. Apart from other functions, this energy maintains our body heat.

C6H12O6 (aq) + 6O(g) $\to$ 6CO2 (g) + 6H2O (l) + Energy
Due to this production of energy, respiration is considered an exothermic process.

#### Question 2:

On what basis is a chemical equation balanced?

A chemical equation is balanced on the basis of law of conservation of mass; it states that "matter can neither be created nor be destroyed". If numbers of atoms of various elements are different in reactants' and products' side, it means they are either destroyed or created, which violate the law of conservation of mass.

#### Question 3:

What happens chemically when quicklime is added to water filled in a bucket?

When quicklime (calcium oxide) is added to water filled in a bucket, they react chemically to form slaked lime (calcium hydroxide) and produce heat energy. So, water in the bucket becomes hot.
CaO (s)       +       H2O (l)   $\to$    Ca(OH)2 (aq)      +      Heat
Calcium oxide           Water          Calcium hydroxide

#### Question 4:

Why should magnesium ribbon be cleaned before burning in air?

Magnesium ribbon should be cleaned before burning in air to remove the basic magnesium carbonate coating on it. This layer should be removed so that the ribbon may readily combine with oxygen while burning in air. Basic magnesium carbonate layer consists of magnesium carbonate and magnesium hydroxide.

#### Question 5:

State whether the following statement is true or false:
A chemical equation can be balanced easily by altering the formula of a reactant or product.

False
Formula of a reactant or product cannot be altered, but chemical equation is balanced by multiplying the reactant or product with numbers like 2,3,4,5 and so on to balance the number of atoms in the chemical equation.

#### Question 6:

What is wrong with the following chemical equation?

In this reaction, oxygen is written as O, but it is a diatomic molecule and its formula is O2.

The correct balanced chemical equation read as follows:

2Mg (s) + O2 (g) $\to$ 2MgO (s)

#### Question 7:

What does the symbol (aq) represent in a chemical equation?

The symbol (aq) stands for aqueous solution. When it is written after a formula in a chemical equation, it represents that the element or compound is a solution made in water.

#### Question 8:

Why is photosynthesis considered an endothermic reaction?

Photosynthesis is considered an endothermic reaction because in this process light energy from sunlight is absorbed to produce oxygen and glucose from carbon dioxide and water in the presence of chlorophyll.

#### Question 9:

How will you indicate the following effects in a chemical equation?
(a) A solution made in water
(b) Exothermic reaction
(c) Endothermic reaction

(a) A solution made in water is indicated by writing (aq) after its formula in a chemical equation.
Example: 6CO2 (g) + 6H2O (l) $\underset{\mathrm{chlorophyll}}{\overset{\mathrm{sunlight}}{\to }}$ C6H12O6 (aq) + 6O2 (g)

(b) Exothermic reaction is indicated by writing “heat” or “heat energy” or “energy” on the products’ side of a chemical reaction.
Example:  CH4 + 2O2 → CO2 + 2H2O + Heat

(c) Endothermic reaction is indicated by writing “heat” or “heat energy” or “energy” on the reactants’ side of a chemical reaction.
Example: N2 + O2 + Heat → 2NO

#### Question 10:

Translate the following statements into chemical equations and then balance the equations:
(a) Hydrogen sulphide gas burns in air to give water and sulphur dioxide.
(b) Phosphorus burns in oxygen to give phosphorus pentoxide.
(c) Carbon disulphide burns in air to give carbon dioxide and sulphur dioxide.
(d) Aluminium metal replaces iron from ferric oxide, Fe2O3, giving aluminium oxide and iron.
(e) Barium chloride reacts with zinc sulphate to give zinc chloride and barium sulphate.

(a) 2H2S (g) + 3O2 (g) $\to$ 2H2O (g) + 2SO2 (g)

(b) P4 + 5O2 $\to$ 2P2O5

(c) CS2 (g) + 3O2 (g) $\to$ CO2 (g) + 2SO2 (g)

(d) 2Al (s) + Fe2O3 (s) $\to$ Al2O3 (s) + 2Fe (s)

(e) BaCl2 + ZnSO4 $\to$ ZnCl2 + BaSO4

#### Question 11:

Write the balanced chemical equations for the following reactions:
(a)
(b)

(a) Ca(OH)2 + CO2 $\to$ Ca(CO3) + H2O

(b) 2Al + 3CuCl2 $\to$ 2AlCl3 + 3Cu

#### Question 12:

Complete and balance the following equations:
(a)
(b)

(a) 2NaOH + H2SO4 $\to$ Na2SO4 + 2H2O

(b) Ca(OH)2 + CO2 $\to$ CaCO3 + H2O

#### Question 13:

Correct and balance the following equations:
(i)
(ii)

The corrected and balanced equations read as follows:

(i) Ca + 2H2O $\to$ Ca(OH)2 + H2

(ii) N2 + 3H2 $\to$ 2NH3

#### Question 14:

Write complete balanced equations for the following reactions:
(a)
(b)

(a) Ca (s) + 2H2O (l) $\to$ Ca(OH)2 (aq) + H2 (g)

(b) 2SO2 (g) + O2 (g) $\to$ 2SO3 (g)

#### Question 15:

Balance the following equations:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)

(i) 4Na + O2 $\to$ 2Na2O

(ii) 2H2O2 $\to$ 2H2O + O2

(iii) Mg(OH)2 + 2HCl $\to$ MgCl2 + 2 H2O

(iv) 4Fe + 3O2 $\to$ 2Fe2O3

(v) 2Al(OH)3 $\to$ Al2O3 + 3H2O

(vi) 2NH3 + 3CuO $\to$3Cu + N2 + 3H2O

(vii) Al2(SO4)3 + 6NaOH $\to$ 2Al(OH)3 + 3Na2SO4

(viii) 2HNO3 + Ca(OH)2 $\to$ Ca(NO3)2 + 2H2O

(ix) 2NaOH + H2SO4 $\to$ Na2SO4 + 2H2O

(x) BaCl2 + H2SO4 $\to$ BaSO4 + 2HCl

#### Question 16:

Fill in the following blanks with suitable words:
(a) Chemical equations are balanced to satisfy the law of ............
(b) A solution made in water is known as an ........... solution and indicated by the symbol ...........

(a) Chemical equations are balanced to satisfy the law of conservation of mass.

(b) A solution made in water is known as an aqueous solution and indicated by the symbol (aq).

#### Question 17:

(a) Give one example of a chemical reaction.
(b) State two characteristics of the chemical reaction which takes place when dilute sulphuric acid is poured over zinc granules.
(c) Give two characteristics of the chemical reaction which occurs on adding potassium iodide solution to lead nitrate solution.

(a) Burning of magnesium ribbon in air to give white powder of magnesium oxide is an example of chemical reaction.
2Mg (s) + O2(g) $\stackrel{HEAT}{\to }$ 2MgO (s)

(b) When dilute sulphuric acid is poured over zinc granules, hydrogen gas is evolved and the mixture becomes hot. Hence, "evolution of a gas" and "change in temperature" are two characteristics of this chemical reaction.
Zn (s) + H2SO4 (aq) $\to$ ZnSO4 (aq) + H2 (g) + HEAT

(c) On adding potassium iodide solution to lead nitrate solution, yellow precipitates of lead iodide are formed and the colour of the solution changes from colourless to yellow. Hence, "formation of a precipitate" and "change in colour" are the characteristics of this chemical reaction.
2KI (aq) + Pb(NO3)(aq)$\to$ PbI(s) + 2KNO(aq)

#### Question 18:

(a) What is a chemical equation? Explain with the help of an example.
(b) Giving examples, state the difference between balanced and unbalanced chemical equations.
(c) Balance the following chemical equations:
(i)
(ii)

(a) The method of representing a chemical reaction with the help of symbols and formulae of the substances involved in it is known as chemical equation.
For example, zinc metal reacts with dilute sulphuric acid to form zinc sulphate and hydrogen gas.
This can be represented by chemical equation as follows:
Zn (s) + H2SO4 (aq) $\to$ ZnSO4 (aq) + H2 (g)

(b)

 Balanced Chemical Equation Unbalanced Chemical Equation A balanced chemical equation has an equal numbers of atoms of different elements in the side of reactants and products. An unbalanced chemical equation has an unequal number of atoms of one or more elements in the reaction. In a balanced chemical equation the sum of masses of the reactants is equal to the sum of the masses of the products. In an unbalanced chemical equation the sum of masses of the reactants is  not equal to the sum of the masses of the products. It obeys the law of conservation of mass It does not obey the law of conservation of mass Example:   Zn (s) + H2SO4 (aq) $\to$ ZnSO4 (aq) + H2 (g)  Here, number of Zn, H, S and O are same in both reactants' and products' side. Example:   H2 + O2 $\to$ H2O Here, one oxygen atom is more in reactant's side than in product's side; it means one atom of oxygen is being destroyed, which is a violation of law of conservation of mass.

(c)  (i) 2NH3 $\to$ N2 + 3H2

(ii) C + CO2 $\to$ 2CO

#### Question 19:

When hydrogen is passed over copper oxide, copper and steam are formed. Write a balanced equation for this reaction and state which of the chemicals are:
(i) elements
(ii) compounds
(iii) reactants
(iv) products
(v) metals
(vi) non-metals

The balanced equation for the reaction of hydrogen gas and copper oxide is written as follows:
H2 (g) + CuO (s) $\to$ Cu (s) + H2O (g)

(i) Elements: Hydrogen (H2) in reactants' side and copper (Cu) in products' side are elements.
(ii) Compounds: Copper oxide (CuO) in reactants' side and water (H2O) in products' side are the compounds.
(iii) Reactants: H2 and CuO are the reactants.
(iv) Products: Cu and H2O are the products.
(v) Metals: Cu is a metal.
(vi) Non-metals: H2 is a the non-metal.

#### Question 20:

(b) Write balanced chemical equation from the following information:
An aqueous calcium hydroxide solution (lime water) reacts with carbon dioxide gas to produce a solid calcium carbonate precipitate and water.

• Indicating physical states of reactants and products.
• Indicating heat change in the reaction.
• Indicating the reaction condition.
Example:
2KClO3 (s) + Heat 2KCl (s) + 3O2 (g)

where,
(s) and (g) signifies solid and gaseous state of compounds, respectively.
Heat written on reactants' side signifies that the reaction consumes heat.
MnO2 written above right-handed arrow signifies reaction takes place in the presence of MnO2.

(b) The chemical equation for the given reaction reads as follows:
Ca(OH)2 (aq) + CO2 (g) $\to$ CaCO3 (s) + H2O (l)

#### Question 21:

(a) What is a balanced chemical equation? Why should chemical equations be balanced?
(b) Aluminium burns in chlorine to form aluminium chloride (AlCl3). Write a balanced chemical equation for this reaction.
(c) Potassium metal reacts with water to give potassium hydroxide and hydrogen gas. Write a balanced chemical equation for this reaction.

(a) A balanced chemical equation has an equal number of atoms of different elements in the reactants' and products' sides.
Chemical equation should be balanced because if the number of atoms in reactants' or products' side is different, it means that the atoms are either created or destroyed, which is a clear violation of the law of conservation of mass.

(b) 2Al (s) + 3Cl2 (l) $\to$ 2AlCl3 (s)

(c) 2K (s) + 2H2O (l) $\to$ 2KOH (aq) + H2 (g)

#### Question 22:

(a) Explain, with example, how the physical states of the reactants and products can be shown in a chemical equation.
(b) Balance the following equation and add state symbols:

(c) Convey the following information in the form of a balanced chemical equation:
"An aqueous solution of ferrous sulphate reacts with an aqueous solution of sodium hydroxide to form a precipitate of ferrous hydroxide and sodium sulphate remains in solution."

(a) Physical state of reactants and products can be shown by writing (s) for solid, (l) for liquid, (aq) for aqueous solution and (g) for gaseous substance after its formula or symbol in chemical reaction.
Example:
C (s) + O2 (g) $\to$ CO2 (g) + HEAT
Here, Carbon is taken in solid state and carbon dioxide and oxygen are in gaseous state.

(b) Zn (s) + 2HCl (aq) $\to$ ZnCl2 (aq) + H2 (g)

(c) FeSO4 (aq) + 2NaOH (aq) $\to$ Fe(OH)2 (s) + Na2SO4 (aq)

#### Question 23:

Write any two observations in an activity which may suggest that a chemical reaction has taken place. Give an example in support of your answer.

Two observations that may suggest that a chemical reaction has taken place are
(i) evolution of a gas and
(ii) change in temperature.

For example, zinc granules react with dilute sulphuric acid to evolve hydrogen gas, which is easily observed by the bubbles of hydrogen gas appearing around zinc granules. Also, the flask becomes hot, which shows heat is evolved during the reaction—that is, change in temperature.
Zn + H2SO4 → ZnSO4 + H2↑ + Heat

#### Question 24:

(a) Aluminium hydroxide reacts with sulphuric acid to form aluminium sulphate and water. Write a balanced equation for this reaction.
(b) Balance the following chemical equation:

(a) 2Al(OH)3 (aq) + 3H2SO4 (aq) $\to$ Al2(SO4)3 (aq) + 6H2O (l)

(b) MnO2 + 4HCl $\to$ MnCl2 + Cl2 + 2H2O

#### Question 25:

Write the balanced equations for the following reactions, and add the state symbols:
(a) Magnesium carbonate reacts with hydrochloric acid to produce magnesium chloride, carbon dioxide and water.
(b) Sodium hydroxide reacts with sulphuric acid to produce sodium sulphate and water.

(a) MgCO3 (s) + 2HCl (aq) $\to$ MgCl2 (aq) + CO2 (g) + H2(l)

(b) 2NaOH (aq) + H2SO4 (aq) $\to$  Na2SO4 (aq) + 2H2(l)

#### Question 26:

Carbon monoxide reacts with hydrogen under certain conditions to form methanol (CH3OH). Write a balanced chemical equation for this reaction indicating the physical states of reactants and product as well as the conditions under which this reaction takes place.

Carbon monoxide reacts with hydrogen under the following conditions:
(i) 300 atm. pressure
(ii) 300oC temperature
(iii) in the presence of ZnO and CrO3 as catalysts

The balanced chemical equation for the given reaction is as follows:
CO (g) + 2H2 (g)  CH3OH (l)

#### Question 27:

(a) Potassium chlorate (KClO3) on heating forms potassium chloride and oxygen. Write a balanced equation for this reaction and indicate the evolution of gas.
(b) Rewrite the following information in the form of a balanced chemical equation:
Magnesium burns in carbon dioxide to form magnesium oxide and carbon.

(a) 2KClO3 (s) 2KCl (s) + 3O2 (g)$↑$

(b) 2Mg (s) + CO2 (g) $\to$ 2MgO (s) + C (s)

#### Question 28:

(a) Substitute formulae for names and balance the following equation:
Calcium carbonate reacts with hydrochloric acid to produce calcium chloride, water and carbon dioxide gas.
(b) Write balanced chemical equation with state symbols for the following reaction:
Sodium hydroxide solution reacts with hydrochloric acid solution to produce sodium chloride solution and water.

(a) Ca(CO3) (s) + 2HCl (aq) $\to$ CaCl2 (aq) + H2O (l) + CO2 (g)

(b) NaOH (aq) + HCl (aq) $\to$ NaCl (aq) + H2O (l)

#### Question 29:

Ammonia reacts with oxygen to form nitrogen and water. Write a balanced chemical equation for this reaction. Add the state symbols for all the reactants and products.

4NH3 (g) + 3O2 (g) $\to$ 2N2 (g) + 6H2O (l)

#### Question 30:

Write a balanced chemical equation for the process of photosynthesis giving the physical states of all the substances involved and the conditions of the reaction.

6CO2 (g) + 6H2O (l) $\underset{\mathrm{Chlorophyll}}{\overset{\mathrm{Sunlight}}{\to }}$ C6H12O6 (aq) + 6O2 (g)

#### Question 31:

Translate the following statement into chemical equation and then balance it:
Barium chloride solution reacts with aluminium sulphate solution to form a precipitate of barium sulphate and aluminium chloride solution.

3BaCl2 (aq) + Al2(SO4)3 (aq) $\to$ 3BaSO4 (s) + 2AlCl3 (aq)

#### Question 32:

When potassium nitrate is heated, it decomposes into potassium nitrite and oxygen. Write a balanced equation for this reaction and add the state symbols of the reactants and products.

2KNO3 (s)  2KNO2 (s) + O2 (g)

#### Question 33:

(a) What is meant by chemical reaction? Explain with the help of an example.
(b) Give one example each of a chemical reaction characterised by:
(i) evolution of a gas
(ii) change in colour
(iii) formation of a precipitate
(iv) change in temperature
(v) change in state.

(a) Chemical reaction is the process in which atoms of some elements rearrange—that is, undergo a chemical change—to form a new product with different properties. This involves breaking of old bonds between the reacting atoms and making of new chemical bonds between the rearranged atoms.

For example, the burning of magnesium ribbon in air. In this reaction, magnesium ribbon, on heating, combines with oxygen to form a white powder of magnesium oxide, which is totally different in its properties from magnesium and oxygen.

(b)
(i) Evolution of gas: When zinc granules react with dilute sulphuric acid in a flask, hydrogen gas evolves and zinc sulphate solution is formed.

${\mathrm{Zn+ H}}_{2}{\mathrm{SO}}_{4}{\to ZnSO}_{4}{+ H}_{2}↑$

(ii) Change in colour: When sulphur dioxide gas is passed through acidified potassium dichromate solution, orange colour of potassium dichromate solution changes to green colour.

(iii) Formation of a precipitate: When potassium iodide is added to lead nitrate solution (colourless and clear), yellow precipitate of lead iodide solution is formed.

(iv) Change in temperature: When quicklime reacts with water, slaked lime is formed with a release of large amount of heat, causing change in temperature in the reaction mixture.

(v) Change in state: The combustion reaction of candle wax (solid) melts initially to produce molten wax (liquid) and then produces carbon dioxide (gas) and water vapour (liquid). So, in this reaction, state of matter gets changed from solid to liquid and gas.

#### Question 34:

(a) State the various characteristics of chemical reactions.
(b) State one characteristic each of the chemical reaction which takes place when:
(i) dilute hydrochloric acid is added to sodium carbonate
(iii) dilute sulphuric acid is added to barium chloride solution
(iv) quicklime is treated with water
(v) wax is burned in the form of a candle

Following are the various characteristics of a chemical reaction:

1. Evolution of gas: Zinc and dilute sulphuric acid react to form zinc sulphate with the evolution of hydrogen gas.
2. Formation of precipitate: Sulphuric acid and barium chloride react to form white precipitates of barium sulphate.
3. Change in colour: Purple colour of potassium permanganate solution on addition of citric acid changes to colourless solution.
4. Change in temperature: Barium hydroxide and ammonium chloride react to form barium chloride, ammonia and water. This is an endothermic reaction in which heat is absorbed, causing the temperature to fall.
5. Change in state: Combustion of candle wax (solid) leads to the formation of water (liquid) and carbon dioxide (gas).
(b)
(i) When sodium carbonate is added to dilute hydrochloric acid, carbon dioxide gas evolves. Hence, this reaction is characterised by evolution of gas.

(ii) Lemon juice (contains citric acid) changes the purple colour of potassium permanganate solution to a colourless solution. Hence, this reaction is characterised by change in colour.

(iii) When dilute sulphuric acid is added to barium chloride solution, it forms white precipitates of barium sulphate. Hence, this reaction is characterised by formation of precipitate.

(iv) When quicklime is treated with water, slaked lime is formed with the evolution of large amount of heat. Hence, this reaction is characterised by change in temperature.

(v) When wax (solid) is burnt in the form of candle, it produces water (liquid) and carbon dioxide (gas). Hence, this reaction is characterised by change in state.

#### Question 35:

(a) What do you understand by exothermic and endothermic reactions?
(b) Give one example of an exothermic reaction and one of an endothermic reaction.
(c) Which of the following are endothermic reactions and which are exothermic reactions?
(i) Burning of natural gas
(ii) Photosynthesis
(iii) Electrolysis of water
(iv) Respiration
(v) Decomposition of calcium carbonate

(a) Exothermic reaction is the chemical reaction that produces heat along with products.
Example:

Endothermic reaction is the chemical reaction that consumes heat to form products.
Example:

(b) Example of exothermic reaction: Zinc granules on reacting with dilute sulphuric acid produce zinc sulphate along with heat energy.

Example of endothermic reaction: Barium hydroxide on reacting with ammonium chloride produces ammonia, water and barium chloride. It also results in decrease in temperature, as it consumes heat energy.

(c)
(i) Burning of natural gas: Exothermic reaction
Burning of natural gas releases a huge amount of heat energy.

(ii) Photosynthesis: Endothermic reaction
Photosynthesis occurs by absorbing light energy from sunlight.

(iii) Electrolysis of water: Endothermic reaction
Electrolysis of water needs energy to break water into hydrogen and oxygen.

(iv) Respiration: Exothermic reaction
Respiration is an exothermic reaction in which glucose undergoes combustion to produce carbon dioxide, water and energy.

(v) Decomposition of calcium carbonate: Endothermic reaction
Decomposition of calcium carbonate is an endothermic reaction because it needs energy to decompose into carbon dioxide and calcium oxide.

#### Question 36:

One of the following does not happen during a chemical reaction. This is:
(a) Breaking of old chemical bonds and formation of new chemical bonds
(b) Formation of new substances with entirely different properties
(c) Atoms of one element change into those of another element to form new products.
(d) A rearrangement of atoms takes place to form new products.

(c) atoms of one element change into those of another element to form new products

It is not atoms but the bonds between these atoms break and form during chemical reactions. Atoms of elements can rearrange but cannot change into other element.

#### Question 37:

Which of the following does not involve a chemical reaction?
(a) digestion of food in our body
(b) process of respiration
(c) burning of candle wax when heated
(d) melting of candle wax on heating

(d) melting of candle wax on heating

On heating, wax changes only its physical state (solid to liquid), not its properties. Hence, only physical change takes place.

#### Question 38:

You are given the solution of lead nitrate. In order to obtain a yellow precipitate you should mix with it a solution of:
(a) potassium chloride
(b) potassium nitride
(c) potassium sulphide
(d) potassium iodide

(d) potassium iodide
Potassium iodide on reacting with lead nitrate gives yellow precipitate of lead iodide.

#### Question 39:

An acid which can decolourise purple coloured potassium permanganate solution is:
(a) sulphuric acid
(b) citric acid
(c) carbonic acid
(d) hydrochloric acid

(b) citric acid
Reaction of purple-coloured potassium permanganate with citric acid is characterised by change in colour from purple to colourless.

#### Question 40:

The chemical reaction between two substances is characterised by a change in colour from orange to green. These two substances are most likely to be:
(a) potassium dichromate solution and sulphur dioxide
(b) potassium permanganate solution and sulphur dioxide
(c) potassium permanganate solution  and lemon juice
(d) potassium dichromate solution and carbon dioxide.

(a) potassium dichromate solution and sulphur dioxide
Reaction between these two are characterised by change in colour. The orange colour of acidified potassium dichromate changes to green on passing sulphur dioxide.

#### Question 41:

The chemical reaction between quicklime and water is characterised by:
(a) evolution of hydrogen gas
(b) formation of slaked lime precipitate
(c) change in temperature of mixture
(d) change in colour of the product

(c) change in temperature of mixture
Reaction of quicklime and water is highly exothermic, i.e., it evolves heat energy and changes the temperature of mixture.

#### Question 42:

One of the following is an endothermic reaction. This is:
(a) combination of carbon and oxygen to form carbon monoxide
(b) combination of nitrogen and oxygen to form nitrogen monoxide
(c) combination of glucose and oxygen to form carbon dioxide and water
(d) combination of zinc and hydrochloric acid to form zinc chloride and hydrogen

(b) combination of nitrogen and oxygen to form nitrogen monoxide
When nitrogen and oxygen are heated at very high temperature (3,000oC approx), they form nitrogen monoxide. Hence, it is an endothermic reaction.

#### Question 43:

Which of the following is not an endothermic reaction?
(a)
(b)
(c)
(d)

(d) C6H12O6 + 6O2 $\to$ 6CO2 + 6H2O
This is the reaction of respiration process in which glucose burns in oxygen to produce heat energy needed by our body. So, this reaction is an exothermic reaction.

#### Question 44:

One of the following is an exothermic reaction. This is:
(a) electrolysis of water
(b) conversion of limestone into quicklime
(c) process of respiration
(d) process of photosynthesis

(c) process of respiration
In the process of respiration, heat energy that maintains our body temperature is released. Therefore, this process is an exothermic reaction.

#### Question 45:

The chemical equations are balanced to satisfy one of the following laws in chemical reactions. This law is known as:
(a) law of conservation of momentum
(b) law of conservation of mass
(c) law of conservation of motion
(d) law of conservation of magnetism

(b) law of conservation of mass
Equations are balanced when the mass of atoms of different elements in  reactants' side equals the atoms in products' side. If it is not so, it means that matter is either created or destroyed in reactions, which is impossible.

#### Question 46:

When the solution of substance X is added to a solution of potassium iodide, then a yellow solid separates out from the solution.
(a) What do you think substance X is likely to be?
(b) Name the substance which the yellow solid consists of.
(c) Which characteristic of chemical reaction is illustrated by this example?
(d) Write a balanced chemical equation for the reaction which takes place. Mention the physical states of all the reactants and products involved in the chemical equation.

(a) Substance X is likely to be lead nitrate, because on reacting with potassium iodide it forms a yellow precipitate.

(b) The yellow solid or precipitate consists of lead iodide, which is one of the products in this reaction.

(c) This chemical reaction is characterised by the formation of precipitate.

(d) 2KI (s) + Pb(NO3)2 (aq) $\to$ PbI2 (s) + 2KNO3 (aq)

#### Question 47:

When water is added gradually to a white solid X, a hissing sound is heard and a lot of heat is produced forming a product Y. A suspension of Y in water is applied to the walls of a house during white washing. A clear solution of Y is also used for testing carbon dioxide gas in the laboratory.
(a) What could be solid X? Write its chemical formula.
(b) What could be product Y? Write its chemical formula.
(c) What is the common name of the solution of Y which is used for testing carbon dioxide gas?
(d) Write chemical equation of the reaction which takes place on adding water to slid X.
(e) Which characteristic of chemical reactions is illustrated by this example?

(a) Solid X is calcium oxide, commonly known as quicklime. Its chemical formula is CaO.

(b) Product Y is calcium hydroxide, commonly known as slaked lime. Its chemical formula is Ca(OH)2.

(c) Common name of the solution of Y is "lime water", which is used to test carbon dioxide gas.

(d) CaO (s) + H2O (l) $\to$ Ca(OH)2 (aq) + HEAT

(e) This chemical reaction is characterised by change in temperature because a huge amount of heat is evolved in this reaction, causing rise in temperature of the mixture.

#### Question 48:

When metal X is treated with a dilute acid Y, then a gas Z is evolved which burns readily by making a little explosion.
(a) Name any two metals which can behave like metal X.
(b) Name any two acids which can behave like acid Y.
(c) Name the gas Z.
(d) Is the gas Z lighter than or heavier than air?
(e) Is the reaction between metal X and dilute acid Y, exothermic or endothermic?
(f) By taking a specific example of metal X and dilute acid Y, write a balanced chemical equation for the reaction which takes place. Also indicate physical states of all the reactants and products.

(a) Zinc (Zn) and magnesium (Mg) metals can behave like metal X.

(b) Sulphuric acid (H2SO4) and hydrochloric acid (HCl) can behave like acid Y.

(c) Gas Z is hydrogen gas (H2).

(d) Gas Z (i.e., hydrogen) is lighter than air because it is the lightest element in the periodic table.

(e) The reaction between metal X and dilute acid Y is exothermic because it produces huge amount of heat.

(f) If X is zinc and Y is sulphuric acid, then the equation can be written as follows:
Zn (s) + H2SO4 (aq) $\to$ ZnSO4 (aq) + H2 (g)

#### Question 49:

A solid substance P which is very hard is used in the construction of many buildings, especially flooring. When substance P is heated strongly, it decomposes to form another solid Q and a gas R is given out. Solid Q reacts with water with the release of a lot of heat to form a substance S. When gas R is passed into a clear solution of substance S, then a white precipitate of substance T is formed. The substance T has the same chemical composition as starting substance P.
(a) What is substance P? Write its common name as well as chemical formula.
(b) What is substance Q?
(c) What is gas R?
(d) What is substance S? What is its clear solution known as?
(e) What is substance T? Name any two natural forms in which substance T occurs in nature.

(a) Substance P is calcium carbonate. Its common name is limestone and its chemical formula is CaCO3.

(b) Substance Q is calcium oxide (CaO).

(c) Gas R is carbon dioxide gas (CO2).

(d) Substance S is calcium hydroxide [Ca(OH)2]. Its clear solution is known as lime water.

(e) Substance T is calcium carbonate (CaCO3). Two natural forms of calcium carbonate are calcite and mollusc shell.

#### Question 50:

A silvery-white metal X taken in the form of ribbon, when ignited, burns in air with a dazzling white flame to form a white powder Y. When water is added to powder Y, it dissolves partially to form another substance Z.
(a) What could metal X be?
(b) What is powder Y?
(c) With which substance metal X combines to form powder Y?
(d) What is substance Z? Name one domestic use of substance Z.
(e) Write a balanced chemical equation of the reaction which takes place when metal X burns in air to form powder Y.

(a) Metal X could be magnesium metal.

(b) Powder Y is magnesium oxide (MgO).

(c) Magnesium metal (X) combines with oxygen gas to form powdery magnesium oxide (Y).

(d) Substance Z is magnesium hydroxide [Mg(OH)2]. Suspension of magnesium hydroxide (Z) is used as an antacid.

(e) 2Mg (s) + O2 (g) $\to$ 2MgO (s)

#### Question 51:

A metal X forms a salt XSO4. The salt XSO4 forms a clear solution in water which reacts with sodium hydroxide solution to form a blue precipitate Y. Metal X is used in making electric wire and alloys like brass.
(a) What do you think metal X could be?
(b) Write the name, formula and colour of salt XSO4.
(c) What is the blue precipitate Y?
(d) Write a chemical equation of the reaction which takes place when salt XSO4 reacts with sodium hydroxide solution. Give the state symbols of all the reactants and products which occur in the above equation.

(a) Metal X could be copper (Cu).

(b) The salt XSO4 is copper sulphate; its formula is CuSO4 and it is blue in colour.

(c) Copper hydroxide [Cu(OH)2] is the blue precipitate.

(d) CuSO4 (aq) + 2NaOH (aq) $\to$ Cu(OH)2 (s) + Na2SO4 (aq)

#### Question 52:

The metal M reacts vigorously with water to form a solution S and a gas G. The solution S turns red litmus to blue whereas gas G, which is lighter than air, burns with a pop sound. Metal M has a low melting point and it is used as a coolant in nuclear reactors.
(a) What is metal M?
(b) What is solution S? Is it acidic or alkaline?
(c) What is gas G?
(d) Write a balanced chemical equation for the reaction which takes place when metal M reacts with water.
(e) Is this reaction exothermic or endothermic?

(a) Metal M is sodium (Na), which is used as a coolant in nuclear reactors.

(b) Solution S is sodium hydroxide solution (NaOH). It is alkaline, it turns red litmus to blue.

(c) Gas G is hydrogen gas, which is lighter than air.

(d) 2Na (s) + 2H2O (l) $\to$ 2NaOH (aq) + H2 (g) + HEAT

(e) This reaction is exothermic because it releases excessive heat.

#### Question 53:

When a mixture of gases X and Y is compressed to 300 atm pressure and then passed over a catalyst consisting of a mixture of zinc oxide and chromium oxide (heated to a temperature of 300°C), then an organic compounds Z having the molecular formula CH4O is formed. X is a highly poisonous gas which is formed in appreciable amounts when a fuel burns in a limited supply of air; Y is a gas which can be made by the action of a dilute acid on an active metal; and Z is a liquid organic compound which can react with sodium metal to produce hydrogen gas.
(a) What are X, Y and Z?
(b) Write a balanced chemical equation of the reaction which takes place when X and Y combine to form Z. Indicate the conditions under which the reaction occurs.

(a) X is carbon monoxide gas (CO); it is formed when a fuel burns in limited supply of air. Y is hydrogen gas (H2); it is formed by the action of dilute acid on active metal. Z is methanol (CH3OH); it is a liquid organic compound that can react with sodium metal to produce hydrogen gas.

(b) CO (g) + 2H2 (g)  CH3OH (l)

#### Question 54:

The white solid compound A decomposes quite rapidly on heating in the presence of a black substance X to form a solid compound B and a gas C. When an aqueous solution of compound B is reacted with silver nitrate solution, then a white precipitate of silver chloride is obtained along with potassium nitrate solution. Gas C does not burn itself but helps burn other things.
(a) What is compound A?
(b) What is compound B?
(c) What is gas C?
(d) What do you think is the black substance X? What is its function?
(e) What is the general name of substances like X?

(a) Compound A is potassium chlorate (KClO3).

(b) Compound B is potassium chloride (KCl).

(c) Gas C is oxygen gas (O2), which does not burn itself but helps burn other things.

(d)  Black substance X is manganese oxide (MnO2). Its function is to catalyse the reaction.

(e)  General name of substance like X (MnO2) is catalyst.

#### Question 55:

Gas A, which is the major cause of global warming, combines with hydrogen oxide B in nature in the presence of an environmental factor C and a green material D to form a six carbon organic compounds E and a gas F. The gas F is necessary for breathing.
(a) What is gas A?
(b) What is the common name of B?
(c) What do you think could be C?
(d) What is material D? Where is it found?
(e) Name the organic compound E.
(f) What is gas F? Name the natural process during which it is released.

(a) Gas A is carbon dioxide (CO2), which is the main cause of global warming.

(b) Common name of B is water (H2O).

(c) C is the sunlight, an environmental factor that helps in photosynthesis process.

(d) Material D is chlorophyll. It is a green material found in leaves of plants.

(e) Organic compound E is glucose (C6H12O6).

(f) Gas F is oxygen (O2) gas, which is necessary for breathing. It is released naturally during photosynthesis.

#### Question 1:

What type of reaction is represented by the digestion of food in our body?

Digestion of food in our body represents decomposition reaction. In this reaction, glucose undergoes slow combustion and combines with oxygen to produce carbon dioxide and water with evolution of heat energy.

C6H12O6 (aq) + 6O2 (g) $\to$ 6CO2 (g) + 6H2O (l) + HEAT
Glucose           Oxygen        Carbon       Water
dioxide

#### Question 2:

Name the various types of chemical reactions.

The various types of chemical reactions are as follows:

1. Combination reaction
2. Decomposition reaction
3. Displacement reaction
4. Double displacement reaction
5. Oxidation and reduction reaction

#### Question 3:

Why does the colour of copper sulphate solution change when an iron nail is kept immersed in it?

The colour of copper sulphate solution changes when iron nail is kept immersed in it because displacement reaction occurs between copper sulphate and iron to produce ferrous sulphate solution, which is green in colour.

CuSO4 (aq)        +          Fe (s)        $\to$          FeSO4 (aq)         +        Cu (s)
Copper sulphate                 Iron                   Ferrous sulphate                 Copper
(Blue solution)                 (Nail)               (Greenish solution)      (Reddish brown nail)

#### Question 4:

Write the balanced chemical equation for the following reaction:

The balanced chemical equation reads as follows:
Zn (s) + 2AgNO3 (aq) $\to$ Zn(NO3)2 (aq) + 2Ag (s)

#### Question 5:

Which term is used to indicate the development of unpleasant smell and taste in fat and oil containing foods due to aerial oxidation (when they are kept exposed for a considerable time)?

"Rancidity" is the term used to indicate the development of unpleasant smell and taste in fat and oil containing foods due to aerial oxidation (when they are kept exposed for a considerable time).

#### Question 6:

What is the general name of the chemicals which are added to fat and oil containing foods to prevent the development of rancidity?

Antioxidants is the general name of the chemicals that are added to fat and oil containing foods to prevent the development of rancidity. Antioxidants are reducing agents that prevent food from getting oxidised easily.

#### Question 7:

State an important use of decomposition reactions.

An important use of decomposition reaction is digestion of food in our body. It is because the carbohydrates and proteins in the food we eat decompose to simpler sugars like glucose and amino acids, respectively. These further break down to provide us energy to do work.

#### Question 8:

What are anti-oxidants? Why are they added to fat and oil containing foods?

Antioxidants are reducing agents that prevent oxidation of fats and oils. They are added to fatty and oily foods to prevent them from turning rancid (i.e., change in taste and smell of food) due to oxidation. Example: Butylated hydroxyanisole (BHA).

#### Question 9:

Explain why, food products containing fats and oils (like potato chips) are packaged in nitrogen.

Food products containing fats and oils are packaged in nitrogen gas because it is unreactive; also, no oxidation occurs because of the absence of oxygen. Hence, there is no rancidity (unpleasant smell).

#### Question 10:

Give one example of a decomposition reaction which is carried out:
(a) with electricity
(b) by applying heat

Examples of the decomposition reaction are as follows:

(a) with electricity:
2H2(l) → 2H2 (g) + O2 (g) (electrolysis of water)

(b) by applying heat:
CaCO3 (s) → CaO(s) + CO2 (g)
(Limestone)   (Quicklime)

#### Question 11:

What type of chemical reaction is used to extract metals from their naturally occurring compounds like oxides or chlorides?

Decomposition reaction, which is carried out by electricity, is used to extract metals from their naturally occurring compounds like oxides or chlorides.

#### Question 12:

Name two anti-oxidants which are usually added to fat and oil containing foods to prevent rancidity.

The two main antioxidants that are usually added to fatty and oily foods to prevent rancidity are:
1. Butylated hydroxyanisole (BHA)
2. Butylated hydroxytoluene (BHT)

#### Question 13:

Write one equation each for the decomposition reactions where energy is supplied in the form of (a) heat, (b) light, and (c) electricity.

(a) Decomposition reaction where energy is supplied in the form of heat:
CaCO3 (s) → CaO (s) + CO2 (g)

(b) Decomposition reaction where energy is supplied in the form of light:
2AgCl (s) → 2Ag (s) + Cl2 (g)

(c) Decomposition reaction where energy is supplied in the form of electricity:
2H2(l) → 2H2 (g) + O2 (g)

#### Question 14:

In the refining of silver, the recovery of silver from silver nitrate solution involved displacement by copper metal. Write down the chemical equation of the reaction involved.

Here, the more electropositive Cu displaces lesser electropositive Ag.

Cu (s) + 2AgNO3 (aq) → 2Ag (s)↓ + Cu(NO3)(aq)

#### Question 15:

What type of reactions are represented by the following equations?
(i)
(ii)
(iii)
(iv)
(v)

(i) Decomposition reaction by the action of heat

(ii) Combination reaction

(iii) Decomposition reaction by the action of heat

(iv) Decomposition reaction by the action of heat

(v) Combination reaction

#### Question 16:

What type of chemical reactions take place when:
(a) a magnesium wire is burnt in air?
(b) lime-stone is heated?
(c) silver bromide is exposed to sunlight?
(d) electricity is passed through water?
(e) ammonia and hydrogen chloride are mixed?

(a) Combination reaction

(b) Decomposition reaction by the action of heat

(c) Decomposition reaction by the action of light

(d) Decomposition reaction by the action of electricity

(e) Combination reaction

#### Question 17:

What type of chemical reactions are represented by the following equations?
(i)
(ii)
(iii)
(iv)
(v)

(i) Displacement reaction

(ii) Combination reaction

(iii) Decomposition reaction

(iv) Double displacement reaction

(v) Displacement reaction

#### Question 18:

Balance the following chemical equations:
(a)
(b)

(a) 2FeSO4 $\stackrel{\mathrm{Heat}}{\to }$ Fe2O3 + SO2 +SO3

(b) 2Pb(NO3)(s) $\stackrel{\mathrm{Heat}}{\to }$ 2PbO (s) + 4NO2 (g) + O2 (g)

#### Question 19:

Which of the following is a combination and which is a displacement reaction?
(a)
(b)

(a) Displacement reaction

(b) Combination reaction

#### Question 20:

What type of reactions are represented by the following equations?
(a)
(b)
(c)
(d)
(e)

(a) Combination reaction

(b) Displacement reaction

(c) Displacement reaction

(d) Decomposition reaction

(e) Double displacement reaction

#### Question 21:

In the following reaction between lead sulphide and hydrogen peroxide:

(a) which substance is reduced?
(b) Which substance is oxidised?

(a) Hydrogen peroxide (H2O2) is reduced.

(b) Lead sulphide (PbS) is oxidised.
This is because lead sulphide gains oxygen, while hydrogen peroxide loses oxygen.

#### Question 22:

Identify the component oxidised in the following reaction:

In the above reaction, H2S is oxidised to S, as hydrogen is being removed.

#### Question 23:

When SO2 gas is passed through saturated solution of H2S, the following reaction occurs:

In this reaction, which substance is oxidised and which one is reduced?

In the above reaction, SO2 is reduced to S as it loses oxygen. On the other hand, H2S, which gains oxygen, gets oxidised to H2O.

#### Question 24:

Fill in the following blanks with suitable words:
(a) The addition of oxygen to a substance is called ......... whereas removal of oxygen is called ........
(b) The addition of hydrogen to a substance is called ........ whereas removal of hydrogen is called .......
(c) Anti-oxidants are often added to fat containing foods to prevent .............. due to oxidation.

(a) The addition of oxygen to a substance is called oxidation, whereas removal of oxygen is called reduction.

(b) The addition of hydrogen to a substance is called reduction, whereas removal of hydrogen is called oxidation.

(c) Anti-oxidants are often added to fat-containing foods to prevent rancidity due to oxidation.

#### Question 25:

What is an oxidation reaction? Identify in the following reaction (i) the substance oxidised, and (ii) the substance reduced:

Oxidation reaction is a reaction that involves addition of oxygen to a substance and removal of hydrogen from a substance.

(i) C is oxidised to CO because oxygen is added.

(ii) ZnO is reduced to Zn because oxygen is removed.

#### Question 26:

(a) What is a redox reaction? Explain with an example.
(b) When a magnesium ribbon burns in air with a dazzling flame and forms a white ash, is magnesium oxidised or reduced? Why?
(c) In the reaction represented by the equation:

(i) name the substance oxidised.
(ii) name the oxidising agent.
(iii) name the substance reduced.
(iv) name the reducing agent.

(a) The reaction in which one substance gets oxidised and other gets reduced is known as redox reaction.

Example:

ZnO + C → Zn + CO

Here, C is oxidised to CO because oxygen is being added and ZnO is reduced to Zn because oxygen is being removed. Therefore, it is a redox reaction.

(b) Magnesium gets oxidised because oxygen is added to the substance (Mg). The chemical reaction is given as follows:

2Mg + O2 → 2MgO

(c) (i) HCl is oxidised to Cl2.

(ii) MnO2 is the oxidising agent.

(iii) MnO2 is reduced to MnCl2.

(iv) HCl is a reducing agent.

#### Question 27:

(a) Define a combination reaction.
(b) Give one example of a combination reaction which is also exothermic.
(c) Give one example of a combination reaction which is also endothermic.

(a) The reaction in which a single product is formed from two or more reactants is called a combination reaction.

(b) Carbon burns in air to form carbon dioxide.

C (s) + O2 (g) → CO2 (g)

Heat is released during this reaction.

(c) Nitrogen burns in air to form nitric oxide.

N2 (g) + O2 (g) → 2NO (g)

Heat is absorbed during this reaction.

#### Question 28:

(a) Give an example of an oxidation reaction.
(b) Is oxidation an exothermic or an endothermic reaction?
(c) Explain, by giving an example, how oxidation and reduction proceed side by side.

(a) Burning of magnesium ribbon in a dazzling flame of air (i.e., oxygen) is an oxidation reaction.

The chemical reaction is given below:

2Mg + O2 → 2MgO

(b) Most of the oxidation reactions are exothermic reactions because energy is released during oxidation.

Example: C (s) + O2 (g) → CO2 (g)

(c) It can be explained by the following example:

CuO + H2 → Cu + H2O

In this reaction, CuO gets reduced to Cu by losing oxygen and H2 gets oxidised to H2O by gaining oxygen. Hence, oxidation and reduction proceed side by side.

#### Question 29:

(a) What is the colour of ferrous sulphate crystals? How does this colour change after heating?
(b) Name the product formed on strongly heating ferrous sulphate crystals. What type of chemical reaction occurs in this change?

(a) The colour of ferrous sulphate crystal is green. After heating, the ferrous sulphate crystal loses water molecules and forms anhydrous ferrous sulphate, which is white in colour. Subsequently, it decomposes to give ferric oxide, which is brown in colour, sulphur dioxide and sulphur trioxide.

(b) The products formed are ferric oxide, sulphur dioxide and sulphur trioxide. During this change, decomposition reaction occurs.

The chemical reaction is given as follows:

2FeSO4 (s) → Fe2O3 (s) + SO2 (g) + SO3 (g)

#### Question 30:

What is a decomposition reaction? Give an example of a decomposition reaction. Describe an activity to illustrate such a reaction by heating.

Decomposition reaction is a chemical reaction in which a single compound breaks down to simpler products. It is the opposite of combination reaction.

Example: Decomposition of ferrous sulphate crystals.

2FeSO4 (s) → Fe2O3 (s) + SO2 (g) + SO3 (g)

Heat the ferrous sulphate crystals; the colour of ferrous sulphate crystals is green. On heating, the ferrous sulphate crystals lose their water molecules and therefore the colour of the crystals changes. It is a decomposition reaction because they decompose to simpler molecules.

#### Question 31:

Zinc oxide reacts with carbon, on heating, to form zinc metal and carbon monoxide. Write a balanced chemical equation for this reaction. Name (i) oxidising agent, and (ii) reducing agent, in this reaction.

The balanced chemical reaction is as follows:

ZnO + C → Zn + CO

(i) Oxidising agent: ZnO
It oxidises C to CO.

(ii) Reducing agent: C
It reduces ZnO to Zn.

#### Question 32:

Give one example of an oxidation-reduction reaction which is also:
(a) a combination reaction
(b) a displacement reaction

(a) 2Cu + O2 → 2CuO

It is a combination reaction, as well as a redox reaction, because two reactants combine to form a single compound.

(b) CuO + H2 → Cu + H2O

In this reaction, Cu is being displaced from the reaction, so it is a displacement reaction. It is also a redox reaction because CuO is reduced to Cu and H2 is oxidised to H2O.

#### Question 33:

(a) What is the difference between displacement and double displacement reactions? Write equations for these reactions.
(b) What do you mean by a precipitation reaction? Explain giving an example.

(a)

 Displacement Reaction Double Displacement Reaction Displacement reaction is a chemical reaction in which displacement of a metal from its solution takes place by the other metal that is more reactive than the metal being displaced. Double displacement reaction is a chemical reaction in which exchange of ions between both the reactants takes place. Example: Zn + CuSO4 → ZnSO4 + Cu Here, Zinc (Zn), which is more reactive than Cu, displaces copper from its solution. Example: Na2SO4 + BaCl2 → BaSO4 + 2NaCl Here, both Na2SO4 and BaCl2 exchange their ions.

(b) Precipitation reaction: It is a chemical reaction in which an insoluble solid (precipitate) is formed that is separate from the solution.

Example: Na2SO4 + BaCl2 → BaSO4 + 2NaCl
Here, white precipitate of barium sulphate is formed.

#### Question 34:

(a) Explain the following in terms of gain or loss of oxygen with one example each:
(i) oxidation (ii) reduction
(b) When copper powder is heated strongly in air, it forms copper oxide. Write a balanced chemical equation for this reaction. Name (i) substance oxidised, and (ii) substance reduced.

(a) (i) Oxidation is the gain of oxygen.

Example:
2Mg + O2 → 2MgO
Here, Mg is oxidised to MgO by accepting oxygen.

(ii) Reduction is the loss of oxygen.

Example:
ZnO + C → Zn + CO
Here, ZnO is reduced to Zn by losing oxygen.

(b) Balanced chemical reaction is as follows:
2Cu + O2 → 2CuO

(i) The substance oxidised is copper.
(ii) The substance reduced is oxygen.

#### Question 35:

(a) Define the following in terms of gain or loss of hydrogen with one example each:
(i) oxidation (ii) reduction
(b) When a magnesium ribbon is heated, it burns in air to form magnesium oxide. Write a balanced chemical equation for this reaction. Name (i) substance oxidised, and (ii) substance reduced.

(i) Oxidation is the loss of hydrogen.

Example:
MnO2+ 4HCl → MnCl2 + 2H2O + Cl2
Here, HCl oxidises to Cl2 by losing hydrogen.

(ii) Reduction is the gain of hydrogen.

Example:
CuO + H2 → Cu + H2O
Here, CuO reduces to Cu by gaining hydrogen.

(b) Balanced chemical reaction is given as follows:
2Mg + O2 → 2MgO

(i) The substance oxidised is magnesium.
(ii) The substance reduced is oxygen.

#### Question 36:

What is meant by (a) displacement reaction, and (b) double displacement reaction? Explain with help of one example each.

(a) Displacement reaction is a chemical reaction in which displacement of a metal from its solution takes place by the other metal that is more reactive than the metal being displaced.

Example:
Zn + CuSO4 → ZnSO4 + Cu

Here, Zn, being more reactive than Cu, displaces copper from its solution.

(b)Double displacement reaction is a chemical reaction in which both the reactants exchange their ions to form two new compounds.

Example:
Na2SO+ BaCl2 → BaSO4 + 2NaCl

#### Question 37:

(a) Why are decomposition reactions called the opposite of combination reactions? Explain with equations of these reactions.
(b) Express the following facts in the form of a balanced chemical equation:
"When a strip of copper metal is placed in a solution of silver nitrate, metallic silver is precipitated and a solution containing copper nitrate is formed".

(a) In decomposition reaction, a compound breaks down to two or more simple products, whereas in combination reaction, two or more reactants combine to form a single product. Therefore, both the reactions are opposite to each other.

For example, in a combination reaction, simple molecules of hydrogen and oxygen combine to form a water molecule.
2H2 (g) + O2 (g) → 2H2(l)

While in decomposition reaction, reverse process takes place—i.e., water decomposes to hydrogen and oxygen.
2H2(l) → 2H2 (g) + O2 (g)

(b) The balanced chemical equation is given as follows:
Cu (s) + 2AgNO3 (aq) → Cu(NO3)(aq) + 2Ag (s)

#### Question 38:

(a) What happens when a piece of iron metal is placed in copper sulphate solution? Name the type of reaction involved.
(b) Write balanced chemical equation with state symbols for the following reaction:
Barium chloride solution reacts with sodium sulphate solution to give insoluble barium sulphate and a solution of sodium chloride.

(a) When a piece of iron is placed in copper sulphate solution, it becomes brown in colour and the blue colour of the copper sulphate solution fades. This is because iron, being more reactive metal than copper, displaces copper from its solution.
Fe (s) + CuSO4 (aq) → FeSO4(aq) + Cu(s)
Thus, the type of reaction involved is displacement reaction.

(b) The balance chemical reaction is given as follows:
BaCl2 (aq) + 2NaSO4 (aq) → BaSO4 (s) + 2NaCl (aq)

#### Question 39:

In the reaction represented by the following equation:

(a) name the substance oxidised
(b) name the substance reduced
(c) name the oxidising agent
(d) name the reducing agent

(a) H2 gets oxidised to H2O.

(b) CuO gets reduced to Cu.

(c) CuO is the oxidising agent.

(d) H2 is the reducing agent.

#### Question 40:

What happens when silver nitrate solution is added to sodium chloride solution?
(a) Write the equation for the reaction which takes place.
(b) Name the type of reaction involved.

(a)The following reaction takes place:
AgNO3 + NaCl → NaNO3 + AgCl

(b) Exchange of ions takes place in the given reaction. Hence, it is a double displacement reaction.

#### Question 41:

What happens when silver chloride is exposed to sunlight? Write a chemical equation for this reaction. Also give one use of such a reaction.

When silver chloride is exposed to sunlight, decomposition of silver chloride takes place.

The chemical reaction is given as follows:
2AgCl (s) →  2Ag (s) + Cl2 (g)
This type of reaction is used in black and white photography.

#### Question 42:

What happens when a zinc strip is dipped into a copper sulphate solution?
(a) Write the equation for the reaction that takes place.
(b) Name the type of reaction involved.

When zinc strip is dipped into copper sulphate solution, it displaces copper from its solution because zinc is more reactive than copper.

(a) Zn (s) + CuSO4 (aq) → ZnSO4 (aq) + Cu (s)

(b) It is a displacement reaction.

#### Question 43:

(a) Explain the term "corrosion" with an example. Write a chemical equation to show the process of corrosion of iron.
(b) What special name is given to the corrosion of iron?
(c) What type of chemical reaction is involved in the corrosion of iron?
(d) Name any three objects (or structures) which are gradually damaged by the corrosion of iron and steel.

(a) Corrosion is the process by which a metal corrodes when it comes in contact with its surroundings such as moisture, air and acids, for example, powdered (brown) coating on iron objects such as iron railings.
In corrosion, oxidation of iron takes place.
Fe (s) → Fe2(aq) + 2e

Iron supplies electrons at the edge of the droplet to reduce oxygen from the air.
O(g) + 2H2(l) + 4e → 4OH (aq)

Within the droplet, the hydroxide ions react with the iron(II) ions and iron(II) hydroxide is precipitated.

Fe2(aq) + 2OH (aq) → Fe(OH)(s)

Rust is then quickly produced by the oxidation of the precipitate.

4Fe(OH)(s) + O2 (g) → 2Fe2O3 • H2(s) + 2H2(l)

(b) Corrosion of iron is called rust.

(c) Redox reaction is involved in the corrosion of iron.

(d) Following are the three objects that are damaged by corrosion:
(i) Iron railings
(ii) Ships
(iii) Bridges

#### Question 44:

(a) Explain the term "rancidity". What damage is caused by rancidity?
(b) What type of chemical reaction is responsible for causing rancidity?
(c) State and explain the various methods for preventing or retarding rancidity of food.

(a) Rancidity is the process of oxidation of fats and oils that spoil the food in such a way that it becomes unpleasant to taste and smell.
Rancidity damages food items due to the development of unpleasant smell, generally in fatty and oily foods.

(b) Oxidation is responsible for causing rancidity.

(c) For preventing or retarding rancidity of food, following methods can be applied:

(1) Use of antioxidants:
Antioxidants are those substances that prevent oxidation.

(2) Use of nitrogen gas:
Nitrogen gas is added during the packaging of the potato chips to prevent oxidation of fat and oil because it is an unreactive gas.

(3) Use of airtight containers:
Preserving food in airtight containers reduces its exposure to oxygen and thus slows down the oxidation process.

(4) Use of refrigerator:
Preserving food in the refrigerator slows down rancidity.

#### Question 45:

(a) What happens when an aqueous solution of sodium sulphate reacts with an aqueous solution of barium chloride?
(b) Write the balanced chemical equation for the reaction which takes place.
(c) State the physical conditions of reactants in which the reaction will not take place.
(d) Name the type of chemical reaction which occurs.
(e) Give one example of another reaction which is of the same type as the above reaction.

(a) Exchange of ions between both the reactants takes place; therefore, it is an example of double displacement reaction. Since, precipitate of barium sulphate forms, it is also a precipitation reaction.

(b) The balanced chemical reaction is given as follows:
Na2SO4 (aq) + BaCl2 (aq) → BaSO4 (s) + 2NaCl (aq)

(c) Reaction will not take place if the reactants are in solid state.

(d) Double displacement reaction takes place.

(e) Another reaction of same type is given as follows:
AgNO3 (aq) + NaCl (aq) →  AgCl (s) + NaNO(aq)

#### Question 46:

The removal of oxygen from a substance is called:
(a) oxidation
(b) corrosion
(c) reduction
(d) rancidity

(c) reduction
The removal of oxygen from the substance is called reduction.

Example: ZnO + C → Zn + CO
In this reaction, zinc loses oxygen and hence gets reduced. On the other hand, carbon gains oxygen and gets oxidised.

#### Question 47:

In the context of redox reactions, the removal of hydrogen from a substance is known as:
(a) oxidation
(b) dehydration
(c) reduction
(d) dehydrogenation

(a) oxidation
Removal of hydrogen from the substance is known as oxidation.

#### Question 48:

The chemical reaction involved in the corrosion of iron metal is that of:
(a) oxidation as well as displacement
(b) reduction as well as combination
(c) oxidation as well as combination
(d) reduction as well as displacement

(c) oxidation as well as combination

In the corrosion of iron metal, iron metal reacts with water and oxygen and forms hydrated oxide called rust.Addition of oxygen to the metal is a oxidation reaction.
4Fe + 3O2 + 2xH2O → 2Fe2O3.xH2O
Since iron metal combines with water and oxygen to corrode, it is also a combination reaction, in which iron metal, water and oxygen combine to form a single compound.

#### Question 49:

The term used to indicate the development of unpleasant smell and taste in fat and oil containing foods due to aerial oxidation is:
(a) acidity
(c) rabidity
(d) rancidity

(d) rancidity
The term used to indicate the development of unpleasant smell and taste in fat and oil containing foods due to aerial oxidation is rancidity.

#### Question 50:

In order to prevent the spoilage of potato chips, they are packed in plastic bags in an atmosphere of:
(a) Cl2
(b) H2
(c) N2
(d) O2

(c) N2
Nitrogen is an unreactive gas that prevents the spoilage of potato chips by preventing the oxidation of fat and oil present in the potato chips.

#### Question 51:

A white precipitate can be obtained by adding dilute sulphuric acid to:
(a) CuSO4 solution
(b) NaCl solution
(c) BaCl2 solution
(d) Na2SO4 solution

(c) BaCl2 solution

BaCl2    +    H2SO4    →     BaSO4    +     2HCl
White ppt.

#### Question 52:

A white precipitate will be formed if we add common salt solution to:
(a) Ba(NO3)2 solution
(b) KNO3 solution
(c) AgNO3 solution
(d) Mg(NO3)2 solution

(c) AgNO3 solution

On mixing aqueous solutions of silver nitrate and sodium chloride, a white curdy precipitate of silver chloride is formed. The chemical reaction is given below:
AgNO3 + NaCl → AgCl + NaNO3

#### Question 53:

Consider the following equation of the chemical reaction of a metal M:

This equation represents:
(a) combination reaction as well as reduction reaction
(b) decomposition reaction as well as oxidation reaction
(c) oxidation reaction as well as displacement reaction
(d) combination reaction as well as oxidation reaction

(d) combination reaction as well as oxidation reaction

In the above reaction, metal combines with the oxygen to form a single compound, so it is a combination reaction. Moreover, addition of oxygen takes place, so it is also a oxidation reaction.

#### Question 54:

The process of respiration is:
(a)  an oxidation reaction which is endothermic
(b) a reduction reaction which is exothermic
(c) a combination reaction which is endothermic
(d) an oxidation reaction which is exothermic

(d) an oxidation reaction that is exothermic

The food that we eat breaks down into simple compounds during digestion. For example, potatoes, bread and rice, which contain carbohydrates, breaks down to give glucose. Glucose combines with oxygen to give energy, so it is a oxidation reaction because addition of oxygen takes place; since energy is released in this process, it is also an exothermic reaction.

The chemical reaction is given as follows:
C6H12O6 + 6O2 → 6CO2 + 6H2O + Energy

#### Question 55:

Which of the following can be decomposed by the action of light?
(a) NaCl
(b) KCl
(c) AgCl
(d) CuCl

(c) AgCl

The chemical reaction is given as follows:
2AgCl (s) → 2Ag (s) + Cl2 (g)

#### Question 56:

Consider the reaction:

This is an example of:
(a) decomposition reaction
(b) combination reaction
(c) double displacement reaction
(d) displacement reaction

(c) double displacement reaction

In the given reaction, exchange of the ions takes place between both the reactants, so it is a double displacement reaction.

#### Question 57:

You are given the following chemical equation:

This equation represents:
(a) decomposition reaction as well as displacement reaction
(b) combination reaction as well as double displacement reaction
(c) redox reaction as well as displacement reaction
(d) double displacement reaction as well as redox reaction

(c) redox reaction as well as displacement reaction

In the given reaction, magnesium takes oxygen and gets oxidised, whereas copper loses oxygen and gets reduced, so it is a redox reaction. Also magnesium, being more reactive metal, displaces copper from the copper solution and hence it is a displacement reaction.

#### Question 58:

When a green iron salt is heated strongly, its colour finally changes to brown and odour of burning sulphur is given out.
(a) Name the iron salt.
(b) Name the type of reaction that takes place during the heating of iron salt.
(c) Write a chemical equation for the reaction involved.

(a) The iron salt is ferrous sulphate crystal.

(b) Decomposition reaction takes place during the heating of iron salt.

(c) The chemical reaction involved is given below:
2FeSO4 (s) → Fe2O3 (s) + SO2 (g) + SO3 (g)

#### Question 59:

A colourless lead salt, when heated, produces a yellow residue and brown fumes.
(b) Name the brown fumes.
(c) Write a chemical equation of the reaction involved.

(b) The brown fumes is nitrogen dioxide gas.

(c) The chemical reaction is given as follows:

2Pb(NO3)(s) → 2PbO (s) + 4NO2 (g) + O2 (g)

#### Question 60:

When hydrogen burns in oxygen, water is formed and when water is electrolysed, then hydrogen and oxygen are produced. What type of reaction takes place:
(a) in the first case?
(b) in the second case?

(a) Combination reaction takes place in the first case.

(b) Decomposition by electricity takes place in the second case.

#### Question 61:

A strip of metal X is dipped in a blue coloured salt solution YSO4. After some time, a layer of metal Y from the salt solution is formed on the surface of metal strip X. Metal X is used in galvanisation whereas metal Y is used in making electric wires. Metal X and metal Y together form an alloy Z.
(a) What could metal X be?
(b) What could metal Y be?
(c) Name the metal salt YSO4.
(d) What type of chemical reaction takes place when metal X reacts with salt solution YSO4? Write the equation of the chemical reaction involved.
(e) Name the alloy Z.

(a) Metal X is zinc.

(b) Metal Y is copper.

(c) Metal salt YSO4 is copper sulphate.

(d) Displacement reaction takes place. The chemical equation is given as follows:
Zn (s) + CuSO4 (aq) → ZnSO4 (s) + Cu (s)

(e) The alloy is brass.

#### Question 62:

When a black metal compound XO is heated with a colourless gas Y2, then metal X and another compound Y2O are formed. Metal X is red-brown in colour which does not react with dilute acids at all. Gas Y2 can be prepared by the action of a dilute acid on any active metal. The compound Y2O is a liquid at room temperature which can turn anhydrous copper sulphate blue.
(a) What do you think is metal X?
(b) What could be gas Y2?
(c) What is compound XO?
(d) What is compound Y2O?
(e) Write the chemical equation of the reaction which takes place on heating XO with Y2.
(f) What type of chemical reaction is illustrated in the above equation?

(a) The metal X is copper.

(b) The gas Y2 is H2.

(c) The compound XO is CuO.

(d) The compound Y2O is H2O.

(e) The chemical reaction is given as follows:
CuO + H2 → Cu + H2O

(f) The above reaction is a displacement reaction as well as a redox reaction.

#### Question 63:

A metal X forms a water soluble salt XNO3. When an aqueous solution of XNO3 is added to common salt solution, then a white precipitate of compound Y is formed alongwith sodium nitrate solution. Metal X is said to be the best conductor of electricity and it does not evolve hydrogen when put in dilute hydrohloric acid.
(a) What is metal X?
(b) What is salt XNO3?
(c) Name the compound Y.
(d) Write the chemical equation of the reaction which takes place on reacting XNO3 solution and common salt solution giving the physical states of all the reactants and products.
(e) What type of chemical reaction is illustrated by the above equation?

(a) Metal X is silver.

(b) Salt XNO3 is AgNO3.

(c) The compound Y is AgCl.

(d) AgNO3 (aq) + NaCl (aq) → NaNO3 (aq) + AgCl (s)

(e) The above chemical reaction is a double displacement reaction as well as a precipitation reaction.

#### Question 64:

Two metals X and Y form the salts XSO4 and Y2SO4, respectively. The solution of salt XSO4 is blue in colour whereas that of Y2SO4 is colourless. When barium chloride solution is added to XSO4 solution, then a white precipitate Z is formed alongwith a salt which turns the solution green. And when barium chloride solution is added to Y2SO4 solution, then the same white precipitate  Z is formed alongwith colourless common salt solution.
(a) What could the metals X and Y be?
(b) Write the name and formula of salt XSO4.
(c) Write the name and formula of salt Y2SO4.
(d) What is the name and formula of white precipitate Z?
(e) Write the name and formula of the salt which turns the solution green in the first case.

(a) Metal X is copper and metal Y is sodium.

(b) The salt is copper sulphate; its formula is CuSO4.

(c) The salt is sodium sulphate; its formula is Na2SO4.

(d) Compound Z is barium sulphate; its formula is BaSO4.

(e) The salt that turns the solution green is copper chloride; its formula is CuCl2.

#### Question 65:

A red-brown metal X forms a salt XSO4. When hydrogen sulphide gas is passed through an aqueous solution of XSO4, then a black precipitate of XS is formed alongwith sulphuric acid solution.
(a) What could the salt XSO4 be?
(b) What is the colour of salt XSO4?
(c) Name the black precipitate XS.
(d) By using the formula of the salt obtained in (a) above, write an equation of the reaction which takes place when hydrogen sulphide gas is passed through its aqueous solution.
(e) What type of chemical reaction takes place in this case?

(a) The salt XSO4 is CuSO4.

(b) The colour of the salt CuSO4 is blue.

(c) The black precipitate XS is copper sulphide (CuS).

(d) CuSO4 + H2S → CuS + H2SO4

(e) Double displacement reaction takes place in this case.

#### Question 66:

When a strip of red-brown metal X is placed in a colourless salt solution YNO3 then metal Y is set free and a blue coloured salt solution X(NO3)2 is formed. The liberated metal Y forms a shining white deposit on the strip of metal X.
(a) What do you think metal X is?
(b) Name the salt YNO3.
(c) What could be metal Y?
(d) Name the salt X(NO3)2.
(e) What type of reaction takes place between metal X and salt solution YNO3?

(a) Metal X is copper.

(b) Salt YNO3 is AgNO3.

(c) Metal Y is silver.

(d) The salt X(NO3)2 is Cu(NO3)2.

(e) Displacement reaction takes place between metal X and salt solution YNO3.

#### Question 67:

A metal salt MX when exposed to light splits up to form metal M and a gas X2. Metal M is used in making ornaments whereas gas X2 is used in making bleaching powder. The salt MX is itself used in black and white photography.
(a) What do you think metal M is?
(b) What could be gas X2?
(c) Name the metal salt MX.
(d) Name any two salt solutions which on mixing together can produce a precipitate of salt MX.
(e) What type of chemical reaction takes place when salt MX is exposed to light? Write the equation of the reaction?

(a) Metal M is silver (Ag).

(b) Gas X2 is chlorine (Cl2).

(c) Metal salt MX is AgCl.

(d) Solution of sodium chloride and silver nitrate on mixing together can produce a precipitate of salt MX.

(e) Decomposition reaction by light takes place.
2AgCl (s) → 2Ag (s) + Cl2 (g)

View NCERT Solutions for all chapters of Class 10


text:
# How do you find the domain of the following function g(x) = log_3(x - 4)?

Dec 15, 2015

The domain is $x > 4$.

#### Explanation:

To find a domain of any logarithmic expression (the base doesn't matter), you need to find out for which $x$ the argument of the logarithmic expression is greater than zero.

So, in your case, all you need to do is set

$x - 4 = 0$

and solve this simple equation:

$x = 4$

This means that your domain is $x > 4$ and your function is only defined for such $x$.

format: plain


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# Bounded random variable

Here is the question: Suppose $X$ is a $\Bbb R$-valued random variable. Show that for all $\varepsilon > 0$, there exists a bounded random variable $Y$ such that $P (X \neq Y ) <\varepsilon$.

-
Try $Y=X\,\mathbf 1_{|X|\leqslant x}$ for $x$ large enough. Then $\mathbb P(Y\ne X)=\mathbb P(|X|\gt x)$. Furthermore, $\mathbb P(|X|\gt x)\to0$ when $x\to+\infty$, hence for $x$ large enough, $Y$ solves the question.
I presume that $P(X=Y)$ in the question is a typo and should read $P(X\ne Y)$. –  Did Oct 12 '12 at 22:37
It's obviously Y can cover $X\,1_{\left|X\right|\leq x}$ for arbitrary x. But how do you deal with the part $X\,1_{\left|X\right|>x}$? How can it be less than any positive number when you fix a x? Thanks! –  65900931 Oct 13 '12 at 17:16
Cover? The part? Fix $x$? See the expanded version of my answer. –  Did Oct 13 '12 at 18:01


text:
Communication
Faster access than browser!

Z-transform

In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency domain representation. [1]

67 relations: Abraham de Moivre, Advanced Z-transform, Argument (complex analysis), Autoregressive–moving-average model, Bilinear transform, Causal system, Chirp Z-transform, Complex conjugate, Complex number, Convolution, Cross-correlation, Decimation (signal processing), Dirac delta function, Discrete Fourier transform, Discrete-time Fourier transform, Discrete-time signal, Eliahu I. Jury, Final value theorem, Finite impulse response, Formal power series, Fourier series, Fourier transform, Fraction (mathematics), Frequency domain, Frequency response, Fundamental theorem of algebra, Generating function, Geometric series, Heaviside step function, Hertz, Imaginary unit, Impulse response, Initial value theorem, Integral, Integral transform, John R. Ragazzini, Laplace transform, Laurent series, Linearity, Lotfi A. Zadeh, LTI system theory, Mathematics, Methods of contour integration, Multiplication, Parseval's theorem, Partial fraction decomposition, Periodic summation, Pierre-Simon Laplace, Pole (complex analysis), Pole–zero plot, ... Expand index (17 more) »

Abraham de Moivre

Abraham de Moivre (26 May 1667 in Vitry-le-François, Champagne, France – 27 November 1754 in London, England) was a French mathematician known for de Moivre's formula, one of those that link complex numbers and trigonometry, and for his work on the normal distribution and probability theory.

In mathematics and signal processing, the advanced Z-transform is an extension of the Z-transform, to incorporate ideal delays that are not multiples of the sampling time.

Argument (complex analysis)

In mathematics, arg is a function operating on complex numbers (visualized in a complex plane).

Autoregressive–moving-average model

In the statistical analysis of time series, autoregressive–moving-average (ARMA) models provide a parsimonious description of a (weakly) stationary stochastic process in terms of two polynomials, one for the auto-regression and the second for the moving average.

Bilinear transform

The bilinear transform (also known as Tustin's method) is used in digital signal processing and discrete-time control theory to transform continuous-time system representations to discrete-time and vice versa.

Causal system

In control theory, a causal system (also known as a physical or nonanticipative system) is a system where the output depends on past and current inputs but not future inputs i.e. the output y(t_) only depends on the input x(t) for values of t \le t_.

Chirp Z-transform

The Chirp Z-transform (CZT) is a generalization of the discrete Fourier transform.

Complex conjugate

In mathematics, the complex conjugate of a complex number is the number with equal real part and imaginary part equal in magnitude but opposite in sign.

Complex number

A complex number is a number that can be expressed in the form, where and are real numbers and is the imaginary unit, that satisfies the equation.

Convolution

In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions, giving the area overlap between the two functions as a function of the amount that one of the original functions is translated.

Cross-correlation

In signal processing, cross-correlation is a measure of similarity of two series as a function of the lag of one relative to the other.

Decimation (signal processing)

In digital signal processing, decimation is the process of reducing the sampling rate of a signal.

Dirac delta function

In mathematics, the Dirac delta function, or function, is a generalized function, or distribution, on the real number line that is zero everywhere except at zero, with an integral of one over the entire real line.

Discrete Fourier transform

In mathematics, the discrete Fourier transform (DFT) converts a finite list of equally spaced samples of a function into the list of coefficients of a finite combination of complex sinusoids, ordered by their frequencies, that has those same sample values.

Discrete-time Fourier transform

In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to the uniformly-spaced samples of a continuous function.

Discrete-time signal

A discrete signal or discrete-time signal is a time series consisting of a sequence of quantities.

Eliahu I. Jury

Eliahu Ibraham Jury (born May 23, 1923) is an American engineer, born in Baghdad, Iraq.

Final value theorem

In mathematical analysis, the final value theorem (FVT) is one of several similar theorems used to relate frequency domain expressions to the time domain behavior as time approaches infinity.

Finite impulse response

In signal processing, a finite impulse response (FIR) filter is a filter whose impulse response (or response to any finite length input) is of finite duration, because it settles to zero in finite time.

Formal power series

In mathematics, a formal power series is a generalization of a polynomial, where the number of terms is allowed to be infinite; this implies giving up the possibility of replacing the variable in the polynomial with an arbitrary number.

Fourier series

In mathematics, a Fourier series is a way to represent a (wave-like) function as the sum of simple sine waves.

Fourier transform

The Fourier transform decomposes a function of time (a signal) into the frequencies that make it up, similarly to how a musical chord can be expressed as the amplitude (or loudness) of its constituent notes.

Fraction (mathematics)

A fraction (from fractus, "broken") represents a part of a whole or, more generally, any number of equal parts.

Frequency domain

In electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time.

Frequency response

Frequency response is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterize the dynamics of the system.

Fundamental theorem of algebra

The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.

Generating function

In mathematics, a generating function is a formal power series in one indeterminate, whose coefficients encode information about a sequence of numbers an that is indexed by the natural numbers.

Geometric series

In mathematics, a geometric series is a series with a constant ratio between successive terms.

Heaviside step function

The Heaviside step function, or the unit step function, usually denoted by H (but sometimes u or θ), is a discontinuous function whose value is zero for negative argument and one for positive argument.

Hertz

The hertz (symbol Hz) is the unit of frequency in the International System of Units (SI) and is defined as one cycle per second.

Imaginary unit

The term imaginary unit or unit imaginary number refers to a solution to the equation.

Impulse response

In signal processing, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an impulse.

Initial value theorem

In mathematical analysis, the initial value theorem is a theorem used to relate frequency domain expressions to the time domain behavior as time approaches zero.

Integral

The integral is an important concept in mathematics.

Integral transform

In mathematics, an integral transform is any transform T of the following form: The input of this transform is a function f, and the output is another function Tf.

John R. Ragazzini

John Ralph Ragazzini (1912 – November 22, 1988) was an American electrical engineer and a professor of Electrical Engineering.

Laplace transform

The Laplace transform is a widely used integral transform in mathematics and electrical engineering named after Pierre-Simon Laplace that transforms a function of time into a function of complex frequency.

Laurent series

In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree.

Linearity

In common usage, linearity refers to a mathematical relationship or function that can be graphically represented as a straight line, as in two quantities that are directly proportional to each other, such as voltage and current in an RLC circuit, or the mass and weight of an object.

Lotfali Askar Zadeh (Lütfəli Rəhimoğlu Əsgərzadə; born February 4, 1921), better known as Lotfi A. Zadeh, is a mathematician, computer scientist, electrical engineer, artificial intelligence researcher and professor emeritus of computer science at the University of California, Berkeley.

LTI system theory

Linear time-invariant theory, commonly known as LTI system theory, comes from applied mathematics and has direct applications in NMR spectroscopy, seismology, circuits, signal processing, control theory, and other technical areas.

Mathematics

Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change.

Methods of contour integration

In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.

Multiplication

Multiplication (often denoted by the cross symbol "×", by a point "·" or by the absence of symbol) is one of the four elementary, mathematical operations of arithmetic; with the others being addition, subtraction and division.

Parseval's theorem

In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform.

Partial fraction decomposition

In algebra, the partial fraction decomposition or partial fraction expansion of a rational function (that is a fraction such that the numerator and the denominator are both polynomials) is the operation that consists in expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.

Periodic summation

In signal processing, any periodic function, s_P(t) with period P, can be represented by a summation of an infinite number of instances of an aperiodic function, s(t), that are offset by integer multiples of P. This representation is called periodic summation: When s_P(t) is alternatively represented as a complex Fourier series, the Fourier coefficients are proportional to the values (or "samples") of the continuous Fourier transform, S(f) \ \stackrel \ \mathcal\, at intervals of 1/P.

Pierre-Simon Laplace

Pierre-Simon, marquis de Laplace (23 March 1749 – 5 March 1827) was an influential French scholar whose work was important to the development of mathematics, statistics, physics, and astronomy.

Pole (complex analysis)

In the mathematical field of complex analysis, a pole of a meromorphic function is a certain type of singularity that behaves like the singularity of \frac at z.

Pole–zero plot

In mathematics, signal processing and control theory, a pole–zero plot is a graphical representation of a rational transfer function in the complex plane which helps to convey certain properties of the system such as.

Probability-generating function

In probability theory, the probability generating function of a discrete random variable is a power series representation (the generating function) of the probability mass function of the random variable.

The radian is the standard unit of angular measure, used in many areas of mathematics.

In mathematics, the radius of convergence of a power series is the radius of the largest disk in which the series converges.

Real number

In mathematics, a real number is a value that represents a quantity along a continuous line.

Recurrence relation

In mathematics, a recurrence relation is an equation that recursively defines a sequence or multidimensional array of values, once one or more initial terms are given: each further term of the sequence or array is defined as a function of the preceding terms.

Sequence

In mathematics, a sequence is an ordered collection of objects in which repetitions are allowed.

Signal processing

Signal processing is an enabling technology that encompasses the fundamental theory, applications, algorithms, and implementations of processing or transferring information contained in many different physical, symbolic, or abstract formats broadly designated as signals.

Starred transform

In applied mathematics, the starred transform, or star transform, is a discrete-time variation of the Laplace transform, so-named because of the asterisk or "star" in the customary notation of the sampled signals.

Time-scale calculus

In mathematics, time-scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite differences, offering a formalism for studying hybrid discrete–continuous dynamical systems.

Transfer function

In engineering, a transfer function (also known as the system function or network function and, when plotted as a graph, transfer curve) is a mathematical representation for fit or to describe inputs and outputs of black box models.

Unit circle

In mathematics, a unit circle is a circle with a radius of one.

Upsampling

Upsampling is interpolation, applied in the context of digital signal processing and sample rate conversion.

Witold Hurewicz

Witold Hurewicz (June 29, 1904 – September 6, 1956) was a Polish mathematician.

Zak transform

In mathematics, the Zak transform is a certain operation which takes as input a function of one variable and produces as output a function of two variables.

Zero (complex analysis)

In complex analysis, a zero (sometimes called a root) of a holomorphic function f is a complex number a such that f(a).

Zero of a function

In mathematics, a zero, also sometimes called a root, of a real-, complex- or generally vector-valued function f is a member x of the domain of f such that f(x) vanishes at x; that is, x is a solution of the equation In other words, a "zero" of a function is an input value that produces an output of zero (0).

Zeta function regularization

In mathematics and theoretical physics, zeta function regularization is a type of regularization or summability method that assigns finite values to divergent sums or products, and in particular can be used to define determinants and traces of some self-adjoint operators.

References

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text:
# Hard Integral

#### roeb

Nevermind, I seem to have figured it out

1. The problem statement, all variables and given/known data
show that:
$$\int \frac{dN}{N} = \int_{0}^{\theta_max} \frac{dcos \theta}{2} \frac{1 + \frac{v^2}{v_0^2} cos(2 \theta ) }{ \sqrt{ 1 - \frac{v^2 sin^2 \theta}{v_0^2} } }$$

2. Relevant equations

3. The attempt at a solution
I'm having a lot of difficulty doing this...
Note that $$sin(\theta_{max} ) = \frac{v_0}{v}$$
so after a bunch of algebra I get:

$$\int \frac{dN}{N} = cot(\theta_{max} ) \int_{0}^{\theta_{max} } \frac{2y^2 - 1}{\sqrt{y^2 - 1} }$$
I am fairly confident that is correct because I keep on getting it.
Unfortunately, I can't seem to integrate this at all.

Last edited:

#### rootX

$$\int \frac{dN}{N} = cot(\theta_{max} ) \int_{0}^{\theta_{max} } \frac{2y^2 - 1}{\sqrt{y^2 - 1} }$$

If you are finding it hard to integrate the right side, try u = sqrt(y^2-1) .. fairly simple to integrate

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Contents

# Contents

## Definition

In analysis, given a series

$\underoverset {k = 0} {\color{blue}\infty} {\sum} a_k$

its $n$th partial sum (for $n \in \mathbb{N}$ any natural number) is the actual sum

$\underoverset {k = 0} {\color{blue}n} {\sum} a_k$

of the first $n+1$ summands.