text:
# Volcano Watch — It's dirty work, but someone's got to do it

Release Date:

In 1790 a group of Hawaiian warriors in the Kau Desert was killed by an eruption of hot gas and flying rocks that originated from Kīlauea caldera. Scientists have studied the layers of tephra visible in gullies along the southwest rift zone in Hawaii Volcanoes National Park and have concluded that this event was the only major explosion of the last 2000 years.

In 1790 a group of Hawaiian warriors in the Kau Desert was killed by an eruption of hot gas and flying rocks that originated from Kīlauea caldera. Scientists have studied the layers of tephra visible in gullies along the southwest rift zone in Hawaii Volcanoes National Park and have concluded that this event was the only major explosion of the last 2000 years.

Recent studies by geologists of the U.S. Geological Survey's Hawaiian Volcano Observatory suggest that Kīlauea's explosive eruptions may happen more often than we think. The explosions form deadly clouds of ash, rocks, and pumice (collectively called tephra) that fall and blanket the ground in layers. These layers provide clues to the eruptions—where they came from, how big they were, and the order in which events happened. Each layer must have been deposited before the layer on top of it.

There are aspects of these tephra layers, called the Keanakakoi deposits, that do not make sense if all the tephra were deposited during the explosive event in 1790. Two circles of rocks, which look like Hawaiian structures, were found built on one layer of tephra and partially buried by later deposits. People generally would not build in a place where ash and rocks were raining down on their heads. The structures imply a quiet time between eruptions, long enough for people to feel this location was safe. A discovery like this is an exciting moment for a scientist, like the light bulb flashing in a comic character's thought bubble.

Excitement is tempered by hours and days of slow and persistent work, seeking answers to small questions that add up to the larger solution. One question is how to distinguish separate layers of tephra. Individual layers of the Keanakakoi deposits can be identified by their color and the type of tephra—ash, rocks, or pumice—within them. The layers are often accessible only by the digging of pits. The ditch-diggers get coated in dust and sand from the Kau winds but return triumphantly with small, labeled bags of ash and rock samples.

The samples are dried in the lunch room oven (50 minutes at 200 degrees) and poured through a series of sieves stacked like a multi-layered double boiler. The sieves separate the samples of ash and rock by grain size. Analysis of the grain sizes helps identify what type of explosion deposited the tephra layer. This increases the understanding of Kīlauea's explosive eruptions.

Another part of the puzzle came from scientists, armed with the willingness to see Keanakakoi deposits in a different light, who found new evidence for multiple eruptions. They observed that some layers of tephra had been cut by gullies and later filled in by more tephra from explosive eruptions. Some scientists argue that explosive blasts from the volcano could gouge gullies in existing tephra layers. Newly discovered gullies, however, could not have been carved by explosive blasts because they are perpendicular to the direction of the blasts. This suggests the gullies were carved by water, a process which could take many years. Because the gullies were filled by newer ash, their formation must have occurred in quiet periods—between eruptions.

When the current study of the Keanakakoi deposits goes public, some will disagree with the results and seek new evidence to disprove them. This is good. Close scrutiny motivates scientists to be accurate and prepared to defend their findings. Scientific understanding increases when past assumptions are questioned and examined. New findings about the Keanakakoi deposits may reveal that explosive eruptions occur more frequently than previously thought and help Big Island residents prepare for this destructive threat.

### Volcano Activity Update

There was no change in the eruptive activity at the Puu O`o vent. Lava continued to effusively erupt and flow through a network of tubes down to the seacoast where it entered the ocean at Kamokuna. The public is reminded that the ocean entry area is extremely hazardous, with explosions accompanying frequent collapses of the lava delta. The steam cloud is highly acidic and laced with glass particles.

No earthquakes were reported felt since the last update.


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# The 35th International Symposium on Lattice Field Theory

18-24 June 2017
Palacio de Congresos
Home > Timetable > Session details > Contribution details

Seminarios 1+2
Chiral Symmetry

# Numerical study of the 2+1d Thirring Model with U(2N)-invariant fermions

## Speakers

• Prof. Simon HANDS

## Content

In 2+1 dimensions the global U(2$N$) symmetry associated with massless fermions is broken to U($N)\otimes$U($N$) by a parity-invariant mass. I will show how to adapt the domain wall formulation to recover the U(2$N$)-invariant limit in interacting fermion models as the domain wall operation is increased. In particular, I will focus on the issue of potential dynamical mass generation in the Thirring model, postulated to take place for $N$ less than some critical $N_c$. I will present results of simulations of the model using both HMC ($N$=2) and RHMC ($N$=1) algorithms, and show that the outcome is very different from previous numerical studies of the model made with staggered fermions, where the corresponding pattern of symmetry breaking is distinct.

Chiral Symmetry

format: plain


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## LaTeX forum ⇒ XeTeX ⇒ Printing XeLaTeX-generated PDF of document using San Francisco to specific Canon printer TS9120 gives weird results

Information and discussion about XeTeX, an alternative for pdfTeX based on e-Tex
LavaTyper
Posts: 69
Joined: Sat Feb 11, 2012 2:38 am

### Printing XeLaTeX-generated PDF of document using San Francisco to specific Canon printer TS9120 gives weird results

GREETINGS.
I am using TeXShop 4.44 on macOS 10.15.2 to typeset documents with custom fonts using XeLaTeX.

The custom font is Apple's own San Francisco font, SF Pro Text, see document below:

Code: [Select all] [Expand/Collapse] [Download] ({untitled.tex})
\documentclass[12pt]{article}\usepackage{lipsum}\usepackage{fontspec} \usepackage{fontspec}\setmainfont{SF Pro Text} \begin{document} \lipsum[1] \end{document}
[/quote]

The PDF that is generated looks as expected.

And when I print to *some* Canon printers, it prints properly.

What it's supposed to look like: See attachment SFProTextCTS9120ResultSmaller.jpg

SFProTextCTS9120ResultSmaller.jpg (72.83 KiB) Viewed 735 times

But when I print the XeLaTeX-generated PDF using SF Pro Text to my Canon TS9120, this is the result that I get when it prints: See attachment SFProTextCTS9120ScanSmaller.jpg

SFProTextCTS9120ScanSmaller.jpg (40.68 KiB) Viewed 735 times

It's almost as if the text is way bigger than it's supposed to be.

The weird thing is that if I rewrite the document using Apple's own Pages software, and change the font of the text within the Pages document to SF Pro Text, and print directly to the Canon TS9120, then the SF Pro Text document prints properly as expected. But not with XeLaTeX.

So, I'm not sure if this one very specific bug is a driver issue with Canon, a bug within the TeXShop/XeLaTeX typesetter, or something else.

Should I also post this thread to the Canon support forums

Return to “XeTeX”

### Who is online

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text:
Last update 21apr22

Non-EuclideanArea and Gauss' Theorem (Part 2)
\textit{(C) 2004, 2015 George K. Francis, Mathematics Department,
University of Illinois}

\begin{document}
\maketitle

\section{Introduction}
We complete the lesson that non-Euclidean area is entirely determined
by the angles of a triangle. Recall that absolute geometry has SSS
congruence. We also saw that AAA is not a congruence in Euclidean geometry.
But AAA does establish similarity. Since there is no similarity in
hyperbolic geometry, AAA should be a congruence relation there. For instance,
since ideal triangles (nulangles) have three pairs of asymptotic sides,
each meeting in a zero-angle, should be congruent. Right angles being
congruent, as Euclid observed, should mean that all nulangles whose
single angle is a right angle, should be congruent too, and so forth. And
they are, as we now show.

\section{Congruences are Angle Dependent}
In the first half of this lesson, we saw that all triangles, ideal or not,
have finite area. In showing that AAA-congruence is true in non-Euclidean
geometry, there are two issues to consider separately, at first.
\begin{itemize}
\item Its angles \textit{determines} a triangle, ideal or not, up
to congruence.
\item The area of a triangle is measured by its \textit{angle defect}.
\end{itemize}
These two are not identical concepts. The first feature is derived
from using the hyperbolic isometry group to move similar triangles into
a standard position. This is essentially how Euclid meant to establish
congruence by superposition of two figures. With Klein's Erlanged Program
we can make the method of superposition precise.

The second feature has an interesting, classical proof for the case of
monangles, except that we invoke \textit{continuity}, which Euclid did
not understand.

\subsection{Congruence of Triangles with Similar Angles}}
We begin with monangles.

Consider an arbitrary monangle $A$. A hyperbolic isometry can be found to
move its vertex to the origin of the Klein model so that one leg of the
angle lies along the x-axis, and the other leg points into the upper
half-plane. The third side of this new monangle, $A'$, connects the two
ideal points on the boundary. For a given angle, there only one such standard
monangle $A'$. By transitivity of congruence then, every other monangle with
the same angle as $A$ has,  is also congruent to $A'$.

For nulangles, we can drop the altitude from one of its sides to the opposite
side, dividing the nulangle into two right monangles. This construction depends on knowing the perpendiculars in whichever model we want to work with here.
Next, we can either apply the previous idea, moving the nulangle $B$ to a
standard position on horizontal diameter of the Klein (or Poincare) disk. The
altitude necessarily lands on the vertical radius, and so there is no choice

A nicer way is to apply the previous result for monangles directly in the
classical style. Take any two nulangles. Dissect each by an altitude and
note that the four right-angled monangles are all congruent. Done!

Dissection will continue to work for similar biangles, except now
pick one of the equal angles in the two biangles, and exend the finite
side past the other angle to infinity and comlete congruent monangles.
The third side, though it doesn't need to be an altitude, dissects the monangle
into another monangle and the original biangle. Done!

We leave the proof of AAA congruence for 3-angled triangles to the reader.

\subsection{Measuring Monangles by their Exterior Angle}

Measuring monangles is interesting because we will use a form of
continuity you are familiar with from the MA347 prerequisite on
the density of rational numbers in an interval, which we will
refer to as the \textit{inch-worm principle}.

\textbf{Definition: } For a monangle $A$ with sole \textbf{exterior angle}
$\alpha$ let $f(\alpha) = |A|$, the area of the monangle.

Since all monangles with the same interior angle, $\pi -\alpha$,
are congruent, this is truly a function of the angle
(not the particular monangle).
From the figure we see how $f(\alpha)$ is an increasing function from
$0$ to $\pi$. Assuming that it is a continuous function, we shall next how
that it is a linear function.

Recall that we need prove two properties of a function $f(\alpha)$
to make it linear
\begin{itemize}
\item  $f(\alpha + \beta) = f(\alpha) + f(\beta)$ (Additivity).
\item  For any real $t$, $f(t\alpha) = tf(\alpha)$ (Scalability)$\end{itemize} \textbf{Additivity Theorem:} If${0 < \{\alpha, \beta\} < \alpha + \beta < \pi }$then${ f(\alpha + \beta) = f(\alpha) + f(\beta) }$. \txtbf{Proof:} In the Klein model, put both angles at its center, one above and one below a diameter. We draw the picture when both angles are acute. You draw the picture and give the argument when one of the angles is obtuse. Why can't both angles be obtuse? Draw additional lines forming four triangles, which we label${1,2,3,4}$. Their areas will be denotes by${|1|, |2|}$etc. Observe that \begin{eqnarray*} f(\beta) &=& |4| \quad \mbox{ by definition } \\ f(\alpha) &=& |1| + |3| \quad \mbox{ look who } \alpha \mbox{ is exterior to } \\ f(\alpha + \beta) &=& |1| + |2| \\ |2| &=& |3| + |4| \quad \mbox{by VAT and proposition 3}\\ \end{eqnarray*} Collecting the algebra we have $f(\alpha + \beta) = |1| + |2| = |1| + (|3| + |4|) = ( |1| + |3| ) + |4| = f(\alpha)+f(\beta)$ You should do the obtuse case in your journal. The additivity is almost the same as linearity. For any rational multiple of an angle (so long as its less than$\pi$) we have $d\; f(\frac{n}{d}\theta) = f(n\theta) = nf(\theta) \Rightarrow f(\frac{n}{d}\theta) =\frac{n}{d}f(\theta).$ If we are willing to accept that$f$is also continuous, we get that$f(t\theta)=tf(\theta)}$and we have completed the proof the linearity. Finally, to show that$f(\alpha) = \alpha$we need only to decide the constant representing the area of an ideal triangle (nulangle) for the only linear function$f(\alpha)$that rises from$0$to$f(\pi)$must be proportional to its argument, as it goes from$0$to$\pi$. The (negative of the) proportionality constant is called the \textit{curvature} of the non-Euclidean plane. Along with Gauss, we choose that to be$-1$. \section{Finishing the Proof of Gauss' Theorem} Consider a hyperbolic triangle$T$with angles$\alpha, \beta, \gamma $Extend the sides pinwheel-fashion. In the Klein model this is easy to do. These extended sides \textit{end at infinity} and surround our triangle inside an an ideal triangle$I$, which has been dissected into three monangles of areas$\alpha = |A|, \beta = |B|, \gamma = |C| \$ leaving the original triangle.

\begin{eqnarray*}
I &=& T + A + B + C \\
\pi   &=& |T| + f(\alpha) + f(\beta) + f(\gamma) \\
&=& |T| + (\alpha + \beta+ \gamma)  \\
|T| &=&  ( \pi - \alpha - \beta - \gamma) = \mbox{ angular defect of } T .
\end{eqnarray*}

\end{document}


text:
## IEEE Transactions on Information Theory

### A Journal devoted to the Theoretical and Experimental Aspects of Information Transmission, Processing and Utilization

 Short Title: IEEE Trans. Inf. Theory Publisher: Institute of Electrical and Electronics Engineers (IEEE), New York, NY ISSN: 0018-9448 Online: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=18 Predecessor: IRE Transactions on Information Theory Comments: Indexed cover-to-cover
 Documents Indexed: 15,126 Publications (since 1963) References Indexed: 26 Publications with 526 References.
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### Latest Issues

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### Authors

 170 Merhav, Neri 157 Shamai (Shitz), Shlomo 149 Poor, Harold Vincent 125 Verdú, Sergio 114 Helleseth, Tor 87 Caire, Giuseppe 79 Weissman, Tsachy 79 Xing, Chaoping 76 Jafar, Syed Ali 72 Ding, Cunsheng 71 Etzion, Tuvi 68 Calderbank, Arthur Robert 66 Vardy, Alexander 65 Ziv, Jacob 64 Tan, Vincent Yan Fu 63 Roth, Ron M. 62 Bruck, Jehoshua 61 Lin, Shu 60 Siegel, Paul H. 59 Kløve, Torleiv 58 Goldsmith, Andrea J. 58 Lapidoth, Amos 58 Tang, Xiaohu 58 Yaakobi, Eitan 57 Berger, Toby 57 Khandani, Amir Keyvan 57 Zeger, Kenneth 56 Varanasi, Mahesh Kumar 54 Permuter, Haim Henri 54 Tse, David N. C. 53 Barg, Alexander M. 53 Kumar, P. Vijay 52 Feder, Meir 52 Ulukus, Sennur 52 Yeung, Raymond W. 51 Litsyn, Simon N. 49 Zhang, Zhen 48 Ahlswede, Rudolf 48 Costello, Daniel J. jun. 48 Zamir, Ram 47 Schwartz, Moshe 46 Gong, Guang 46 Gray, Robert Molten 46 Kschischang, Frank R. 46 Ling, San 46 Wolf, Jack Keil 45 Carlet, Claude 45 Gastpar, Michael C. 45 Hayashi, Masahito 45 Urbanke, Rüdiger L. 44 Diggavi, Suhas N. 44 Effros, Michelle 43 Cover, Thomas Merrill 43 Ephremides, Anthony 43 Golomb, Solomon Wolf 43 Solé, Patrick 43 Winter, Andreas 42 Eldar, Yonina Chana 42 Han, Te Sun 42 Médard, Muriel 41 Chen, Jun 41 Linder, Tamás 41 Tarokh, Vahid 41 Viswanath, Pramod 40 Abdel-Ghaffar, Khaled A. S. 40 Ge, Gennian 40 Neuhoff, David L. 40 Sloane, Neil James Alexander 39 Forney, G. David jun. 39 Fu, Fangwei 39 Guruswami, Venkatesan 39 Johannesson, Rolf 38 Fossorier, Marc P. C. 38 Mesnager, Sihem 37 Boche, Holger 37 Kieffer, John Cronan 37 Masry, Elias 37 Ramchandran, Kannan 37 Steinberg, Yossef 37 Wang, Xiaodong 37 Yang, Enhui 36 Avestimehr, Amir Salman 36 Kasami, Tadao 36 Khisti, Ashish J. 36 Kulkarni, Sanjeev R. 36 Veeravalli, Venugopal V. 35 Anantharam, Venkat 35 Chee, Yeow Meng 35 Gulliver, Thomas Aaron 35 Tian, Chao 34 McEliece, Robert James 34 Wagner, Aaron B. 34 Wornell, Gregory W. 34 Xia, Xianggen 33 Csiszár, Imre 33 El Gamal, Hesham 33 Hassibi, Babak 33 Kailath, Thomas 33 Kramer, Gerhard 33 Sason, Igal ...and 10,780 more Authors
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### Fields

 13,217 Information and communication theory, circuits (94-XX) 1,592 Statistics (62-XX) 1,070 Probability theory and stochastic processes (60-XX) 1,033 Computer science (68-XX) 620 Operations research, mathematical programming (90-XX) 521 Combinatorics (05-XX) 469 Systems theory; control (93-XX) 466 Quantum theory (81-XX) 463 Number theory (11-XX) 233 Numerical analysis (65-XX) 146 Linear and multilinear algebra; matrix theory (15-XX) 124 Algebraic geometry (14-XX) 117 Harmonic analysis on Euclidean spaces (42-XX) 97 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 62 Biology and other natural sciences (92-XX) 59 Order, lattices, ordered algebraic structures (06-XX) 55 Group theory and generalizations (20-XX) 53 Convex and discrete geometry (52-XX) 42 Approximations and expansions (41-XX) 37 Optics, electromagnetic theory (78-XX) 37 Statistical mechanics, structure of matter (82-XX) 33 Geometry (51-XX) 25 Field theory and polynomials (12-XX) 24 Dynamical systems and ergodic theory (37-XX) 23 Special functions (33-XX) 19 Abstract harmonic analysis (43-XX) 19 Operator theory (47-XX) 18 History and biography (01-XX) 13 General and overarching topics; collections (00-XX) 13 Commutative algebra (13-XX) 12 Functional analysis (46-XX) 11 Measure and integration (28-XX) 11 Integral transforms, operational calculus (44-XX) 11 Calculus of variations and optimal control; optimization (49-XX) 9 Mathematical logic and foundations (03-XX) 9 Associative rings and algebras (16-XX) 8 Real functions (26-XX) 4 Topological groups, Lie groups (22-XX) 4 Functions of a complex variable (30-XX) 4 Ordinary differential equations (34-XX) 4 Sequences, series, summability (40-XX) 4 Differential geometry (53-XX) 4 Fluid mechanics (76-XX) 4 Geophysics (86-XX) 3 Partial differential equations (35-XX) 3 Difference and functional equations (39-XX) 3 Integral equations (45-XX) 2 General topology (54-XX) 2 Algebraic topology (55-XX) 1 Nonassociative rings and algebras (17-XX) 1 Several complex variables and analytic spaces (32-XX) 1 Classical thermodynamics, heat transfer (80-XX) 1 Relativity and gravitational theory (83-XX) 1 Astronomy and astrophysics (85-XX)

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Network information flow. Zbl 0991.90015
Ahlswede, Rudolf; Cai, Ning; Li, Shuo-Yen Robert; Yeung, Raymond W.
2000
Quantization. Zbl 1016.94016
Gray, Robert M.; Neuhoff, David L.
1998
Divergence measures based on the Shannon entropy. Zbl 0712.94004
Lin, Jianhua
1991
On the inherent intractability of certain coding problems. Zbl 0377.94018
Berlekamp, Elwyn R.; McEliece, Robert J.; van Tilborg, Henk C. A.
1978
Nonbinary stabilizer codes over finite fields. Zbl 1242.94045
Ketkar, Avanti; Klappenecker, Andreas; Kumar, Santosh; Sarvepalli, Pradeep Kiran
2006
Entropy-based algorithms for best basis selection. Zbl 0849.94005
Coifman, Ronald R.; Wickerhauser, Mladen Victor
1992
The capacity of wireless networks. Zbl 0991.90511
Gupta, Piyush; Kumar, P. R.
2000
Nonrandom binary superimposed codes. Zbl 0133.12402
Kautz, W. H.; Singleton, R. C.
1964
A new upper bound on the minimal distance of self-dual codes. Zbl 0713.94016
Conway, J. H.; Sloane, N. J. A.
1990
Tight oracle inequalities for low-rank matrix recovery from a minimal number of noisy random measurements. Zbl 1366.90160
Candès, Emmanuel J.; Plan, Yaniv
2011
Matrix completion from a few entries. Zbl 1366.62111
Keshavan, Raghunandan H.; Montanari, Andrea; Oh, Sewoong
2010
Information theoretic inequalities. Zbl 0741.94001
Dembo, Amir; Cover, Thomas M.; Thomas, Joy A.
1991
Correlation-immunity of nonlinear combining functions for cryptographic applications. Zbl 0554.94010
Siegenthaler, T.
1984
Sparse representations in unions of bases. Zbl 1286.94032
Gribonval, Rémi; Nielsen, Morten
2003
Optimal recursive estimation with uncertain observation. Zbl 0174.51102
Nahi, N. E.
1969
Subspace pursuit for compressive sensing signal reconstruction. Zbl 1367.94082
Dai, Wei; Milenkovic, Olgica
2009
A class of two-weight and three-weight codes and their applications in secret sharing. Zbl 1359.94687
Ding, Kelan; Ding, Cunsheng
2015
Linear codes from some 2-designs. Zbl 1359.94685
Ding, Cunsheng
2015
Nonconcave penalized likelihood with NP-dimensionality. Zbl 1365.62277
Fan, Jianqing; Lv, Jinchi
2011
Secret sharing schemes from three classes of linear codes. Zbl 1283.94105
Yuan, Jin; Ding, Cunsheng
2006
Capacity of reproducing kernel spaces in learning theory. Zbl 1290.62033
Zhou, Ding-Xuan
2003
Minimax rates of estimation for high-dimensional linear regression over $$\ell_q$$-balls. Zbl 1365.62276
Raskutti, Garvesh; Wainwright, Martin J.; Yu, Bin
2011
Solving sparse linear equations over finite fields. Zbl 0607.65015
Wiedemann, Douglas H.
1986
On the algebraic structure of quasi-cyclic codes. I: Finite fields. Zbl 1023.94015
Ling, San; Solé, Patrick
2001
A view of three decades of linear filtering theory. Zbl 0307.93040
Kailath, Thomas
1974
Improved decoding of Reed-Solomon and algebraic-geometry codes. Zbl 0958.94036
1999
Hidden Markov processes. Zbl 1061.94560
Ephraim, Yariv; Merhav, Neri
2002
Wavelet analysis and synthesis of fractional Brownian motion. Zbl 0743.60078
Flandrin, Patrick
1992
Nonbinary quantum stabilizer codes. Zbl 1021.94033
Ashikhmin, Alexei; Knill, Emanuel
2001
Minimal vectors in linear codes. Zbl 0932.94032
Ashikhmin, A.; Barg, A.
1998
Cyclic codes and self-dual codes over $$F_2+uF_2$$. Zbl 0958.94025
Bonnecazé, A.; Udaya, P.
1999
On quantum and classical BCH codes. Zbl 1310.94195
Aly, Salah A.; Klappenecker, Andreas; Sarvepalli, Pradeep Kiran
2007
The estimation of the gradient of a density function, with applications in pattern recognition. Zbl 0297.62025
Fukunaga, Keinosuke; Hostetler, Larry D.
1975
The weights of the orthogonals of the extended quadratic binary Goppa codes. Zbl 0703.94011
Lachaud, Gilles; Wolfmann, Jacques
1990
Fisher information and stochastic complexity. Zbl 0856.94006
Rissanen, Jorma J.
1996
Optical orthogonal codes: design, analysis, and applications. Zbl 0676.94021
Chung, Fan R. K.; Salehi, Jawad A.; Wei, Victor K.
1989
Wavelet analysis of long-range-dependent traffic. Zbl 0905.94006
Abry, Patrice; Veitch, Darryl
1998
A rank-metric approach to error control in random network coding. Zbl 1318.94119
Silva, Danilo; Kschischang, Frank R.; Koetter, Ralf
2008
Constructing free-energy approximations and generalized belief propagation algorithms. Zbl 1283.94023
Yedidia, Jonathan S.; Freeman, William T.; Weiss, Yair
2005
On the convexity of some divergence measures based on entropy functions. Zbl 0479.94009
1982
On subfield subcodes of modified Reed-Solomon codes. Zbl 0308.94004
Delsarte, Philippe
1975
On divergences and informations in statistics and information theory. Zbl 1287.94025
Liese, Friedrich; Vajda, Igor
2006
Almost perfect nonlinear power functions on $$\mathrm{GF}(2^n)$$: the Welch case. Zbl 0957.94021
Dobbertin, Hans
1999
On the shape of a set of points in the plane. Zbl 0512.52001
Edelsbrunner, Herbert; Kirkpatrick, David G.; Seidel, Raimund
1983
Recursive probability density estimation for weakly dependent stationary processes. Zbl 0602.62028
Masry, Elias
1986
A comparison of the Delsarte and Lovasz bounds. Zbl 0444.94009
Schrijver, Alexander
1979
On Lewis’ simulation method for point processes. Zbl 0449.60037
Ogata, Yosihiko
1981
Information theory and mixing least-squares regressions. Zbl 1309.94051
Leung, Gilbert; Barron, Andrew R.
2006
On secret sharing systems. Zbl 0503.94018
Karnin, Ehud D.; Greene, Jonathan W.; Hellman, Martin E.
1983
Type II codes over $$\mathbb{F}_2+u \mathbb{F}_2$$. Zbl 0947.94023
Dougherty, Steven T.; Gaborit, Philippe; Harada, Masaaki; Solé, Patrick
1999
The private classical capacity and quantum capacity of a quantum channel. Zbl 1293.94063
Devetak, Igor
2005
Cryptographic distinguishability measures for quantum-mechanical states. Zbl 0959.94020
Fuchs, Christopher A.; van de Graaf, Jeroen
1999
New instances of quadratic APN functions. Zbl 1487.94206
Beierle, Christof; Leander, Gregor
2022
A construction of maximally recoverable codes with order-optimal field size. Zbl 1489.94184
Cai, Han; Miao, Ying; Schwartz, Moshe; Tang, Xiaohu
2022
Coding for sequence reconstruction for single edits. Zbl 1489.94056
Cai, Kui; Kiah, Han Mao; Nguyen, Tuan Thanh; Yaakobi, Eitan
2022
Quantum LDPC codes with almost linear minimum distance. Zbl 1489.94162
Panteleev, Pavel; Kalachev, Gleb
2022
The semiring of dichotomies and asymptotic relative submajorization. Zbl 1489.81012
Perry, Christopher; Vrana, Péter; Werner, Albert H.
2022
On the security of subspace subcodes of Reed-Solomon codes for public key encryption. Zbl 1489.94093
Couvreur, Alain; Lequesne, Matthieu
2022
The $$c$$-differential uniformity and boomerang uniformity of two classes of permutation polynomials. Zbl 1487.94119
Hasan, Sartaj Ul; Pal, Mohit; Stănică, Pantelimon
2022
Generalized submodular information measures: theoretical properties, examples, optimization algorithms, and applications. Zbl 1489.94050
Iyer, Rishabh; Khargonkar, Ninad; Bilmes, Jeff; Asnani, Himanshu
2022
Lattice (list) decoding near Minkowski’s inequality. Zbl 1489.94178
Mook, Ethan; Peikert, Chris
2022
Short minimal codes and covering codes via strong blocking sets in projective spaces. Zbl 1489.94141
Héger, Tamás; Nagy, Zoltán Lóránt
2022
Revisiting the concrete security of Goldreich’s pseudorandom generator. Zbl 1489.94114
Yang, Jing; Guo, Qian; Johansson, Thomas; Lentmaier, Michael
2022
The $$q$$-ary antiprimitive BCH codes. Zbl 07533739
Zhu, Hongwei; Shi, Minjia; Wang, Xiaoqiang; Helleseth, Tor
2022
Twisted Reed-Solomon codes. Zbl 07555848
Beelen, Peter; Puchinger, Sven; Rosenkilde, Johan
2022
Near-optimal sparsity-constrained group testing: improved bounds and algorithms. Zbl 07555864
Gebhard, Oliver; Hahn-Klimroth, Max; Parczyk, Olaf; Penschuck, Manuel; Rolvien, Maurice; Scarlett, Jonathan; Tan, Nelvin
2022
Optimal convergence rates for the orthogonal greedy algorithm. Zbl 07555869
Siegel, Jonathan W.; Xu, Jinchao
2022
On two fundamental problems on APN power functions. Zbl 07555872
Budaghyan, Lilya; Calderini, Marco; Carlet, Claude; Davidova, Diana; Kaleyski, Nikolay S.
2022
A complete study of two classes of Boolean functions: direct sums of monomials and threshold functions. Zbl 07555873
Carlet, Claude; Méaux, Pierrick
2022
Additive complementary dual codes from group characters. Zbl 07555945
Dougherty, Steven T.; Şahinkaya, Serap; Ustun, Deniz
2022
Universal consistency of deep convolutional neural networks. Zbl 07555955
Lin, Shao-Bo; Wang, Kaidong; Wang, Yao; Zhou, Ding-Xuan
2022
The differential spectrum of the power mapping $$x^{p^n-3}$$. Zbl 07573352
Yan, Haode; Xia, Yongbo; Li, Chunlei; Helleseth, Tor; Xiong, Maosheng; Luo, Jinquan
2022
Deep neural network approximation theory. Zbl 1473.68178
Elbrächter, Dennis; Perekrestenko, Dmytro; Grohs, Philipp; Bölcskei, Helmut
2021
Full characterization of minimal linear codes as cutting blocking sets. Zbl 1475.94198
Tang, Chunming; Qiu, Yan; Liao, Qunying; Zhou, Zhengchun
2021
An infinite family of linear codes supporting 4-designs. Zbl 1465.94117
Tang, Chunming; Ding, Cunsheng
2021
On permutation quadrinomials and 4-uniform BCT. Zbl 1475.94131
Li, Nian; Xiong, Maosheng; Zeng, Xiangyong
2021
Shapes of uncertainty in spectral graph theory. Zbl 1465.05101
Erb, Wolfgang
2021
Nonconvex matrix factorization from rank-one measurements. Zbl 1473.65050
Li, Yuanxin; Ma, Cong; Chen, Yuxin; Chi, Yuejie
2021
Optimal spectral shrinkage and PCA with heteroscedastic noise. Zbl 1473.62205
2021
Multi-marginal optimal transport and probabilistic graphical models. Zbl 1475.49057
Haasler, Isabel; Singh, Rahul; Zhang, Qinsheng; Karlsson, Johan; Chen, Yongxin
2021
Algorithmic analysis and statistical estimation of SLOPE via approximate message passing. Zbl 1473.62238
Bu, Zhiqi; Klusowski, Jason M.; Rush, Cynthia; Su, Weijie J.
2021
On the approximation ratio of ordered parsings. Zbl 1465.68071
Navarro, Gonzalo; Ochoa, Carlos; Prezza, Nicola
2021
Fundamental properties of sum-rank-metric codes. Zbl 1487.94185
Byrne, Eimear; Gluesing-Luerssen, Heide; Ravagnani, Alberto
2021
Group transference techniques for the estimation of the decoherence times and capacities of quantum Markov semigroups. Zbl 1473.81048
Bardet, Ivan; Junge, Marius; Laracuente, Nicholas; Rouzé, Cambyse; Stilck França, Daniel
2021
On non-interactive simulation of binary random variables. Zbl 1473.60027
Yu, Lei; Tan, Vincent Y. F.
2021
A moment ratio bound for polynomials and some extremal properties of Krawchouk polynomials and Hamming spheres. Zbl 1473.94162
Kirshner, Naomi; Samorodnitsky, Alex
2021
Bayes-optimal convolutional AMP. Zbl 1475.94045
Takeuchi, Keigo
2021
New LCD MDS codes of non-Reed-Solomon type. Zbl 1486.94174
Wu, Yansheng; Hyun, Jong Yoon; Lee, Yoonjin
2021
The quantum Wasserstein distance of order 1. Zbl 1487.81026
De Palma, Giacomo; Marvian, Milad; Trevisan, Dario; Lloyd, Seth
2021
Investigations on $$c$$-(almost) perfect nonlinear functions. Zbl 1487.94133
Mesnager, Sihem; Riera, Constanza; Stănică, Pantelimon; Yan, Haode; Zhou, Zhengchun
2021
The geometry of two-weight codes over $$\mathbb{Z}_p^m$$. Zbl 1489.94151
Shi, Minjia; Honold, Thomas; Solé, Patrick; Qiu, Yunzhen; Wu, Rongsheng; Sepasdar, Zahra
2021
Optimal fraction repetition codes for access-balancing in distributed storage. Zbl 1473.94074
Yu, Wenjun; Zhang, Xiande; Ge, Gennian
2021
Asymptotic properties of recursive particle maximum likelihood estimation. Zbl 1473.62294
2021
De-biased sparse PCA: inference for eigenstructure of large covariance matrices. Zbl 1473.62204
Janková, Jana; van de Geer, Sara
2021
Further study of 2-to-1 mappings over $$F_{2^n}$$. Zbl 1473.94179
Li, Kangquan; Mesnager, Sihem; Qu, Longjiang
2021
New construction of optimal Type-II binary Z-complementary pairs. Zbl 1473.94086
Gu, Zhi; Zhou, Zhengchun; Wang, Qi; Fan, Pingzhi
2021
New risk bounds for 2D total variation denoising. Zbl 1475.62143
Chatterjee, Sabyasachi; Goswami, Subhajit
2021
A permutation-based model for crowd labeling: optimal estimation and robustness. Zbl 1475.62170
Shah, Nihar B.; Balakrishnan, Sivaraman; Wainwright, Martin J.
2021
Binary linear codes with few weights from two-to-one functions. Zbl 1475.94192
Li, Kangquan; Li, Chunlei; Helleseth, Tor; Qu, Longjiang
2021
Capacity optimality of AMP in coded systems. Zbl 1475.94038
Liu, Lei; Liang, Chulong; Ma, Junjie; Ping, Li
2021
An asymptotic theory of joint sequential changepoint detection and identification for general stochastic models. Zbl 1475.62223
Tartakovsky, Alexander G.
2021
A theory of computational resolution limit for line spectral estimation. Zbl 1475.94039
Liu, Ping; Zhang, Hai
2021
Dihedral group codes over finite fields. Zbl 1486.94163
Fan, Yun; Lin, Liren
2021
Fast decoding of codes in the rank, subspace, and sum-rank metric. Zbl 1486.94180
Bartz, Hannes; Jerkovits, Thomas; Puchinger, Sven; Rosenkilde, Johan
2021
On cosets weight distribution of doubly-extended Reed-Solomon codes of codimension 4. Zbl 1486.94161
Davydov, Alexander A.; Marcugini, Stefano; Pambianco, Fernanda
2021
Sum-rank BCH codes and cyclic-skew-cyclic codes. Zbl 1486.94178
Martínez-Peñas, Umberto
2021
Weight hierarchies of a class of linear codes related to non-degenerate quadratic forms. Zbl 1465.94110
Li, Fei
2021
New MDS Euclidean self-orthogonal codes. Zbl 1465.94105
Fang, Xiaolei; Liu, Meiqing; Luo, Jinquan
2021
Distributed online linear regressions. Zbl 1473.62246
Yuan, Deming; Proutiere, Alexandre; Shi, Guodong
2021
The spiked matrix model with generative priors. Zbl 1465.94020
Aubin, Benjamin; Loureiro, Bruno; Maillard, Antoine; Krzakala, Florent; Zdeborová, Lenka
2021
Covariance matrix estimation with non uniform and data dependent missing observations. Zbl 1465.62100
Pavez, Eduardo; Ortega, Antonio
2021
Weighted matrix completion from non-random, non-uniform sampling patterns. Zbl 1465.65038
Foucart, Simon; Needell, Deanna; Pathak, Reese; Plan, Yaniv; Wootters, Mary
2021
The global optimization geometry of low-rank matrix optimization. Zbl 1465.90082
Zhu, Zhihui; Li, Qiuwei; Tang, Gongguo; Wakin, Michael B.
2021
New constructions of optimal locally repairable codes with super-linear length. Zbl 1487.94193
Kong, Xiangliang; Wang, Xin; Ge, Gennian
2021
Hulls of generalized Reed-Solomon codes via Goppa codes and their applications to quantum codes. Zbl 1487.94189
Gao, Yanyan; Yue, Qin; Huang, Xinmei; Zhang, Jun
2021
Computer classification of linear codes. Zbl 1489.94132
Bouyukliev, Iliya; Bouyuklieva, Stefka; Kurz, Sascha
2021
Revisiting randomized gossip algorithms: general framework, convergence rates and novel block and accelerated protocols. Zbl 1489.65053
Loizou, Nicolas; Richtárik, Peter
2021
Sublinear-time non-adaptive group testing with $$O(k \log n)$$ tests via bit-mixing coding. Zbl 1473.62065
Bondorf, Steffen; Chen, Binbin; Scarlett, Jonathan; Yu, Haifeng; Zhao, Yuda
2021
New constructions of optimal cyclic $$(r,\delta)$$ locally repairable codes from their zeros. Zbl 1473.94164
Qiu, Jing; Zheng, Dabin; Fu, Fang-Wei
2021
Mutually unbiased equiangular tight frames. Zbl 1473.42038
Fickus, Matthew; Mayo, Benjamin R.
2021
Divergence radii and the strong converse exponent of classical-quantum channel coding with constant compositions. Zbl 1473.94080
Mosonyi, Milán; Ogawa, Tomohiro
2021
Non-Bayesian activity detection, large-scale fading coefficient estimation, and unsourced random access with a massive MIMO receiver. Zbl 1473.94022
Fengler, Alexander; Haghighatshoar, Saeid; Jung, Peter; Caire, Giuseppe
2021
Understanding estimation and generalization error of generative adversarial networks. Zbl 1473.62180
Ji, Kaiyi; Zhou, Yi; Liang, Yingbin
2021
Resolution limits for the noisy non-adaptive 20 questions problem. Zbl 1473.94084
Zhou, Lin; Hero, Alfred O.
2021
A systematic construction of MDS codes with small sub-packetization level and near-optimal repair bandwidth. Zbl 1473.94056
Li, Jie; Liu, Yi; Tang, Xiaohu
2021
Girth analysis and design of periodically time-varying SC-LDPC codes. Zbl 1473.94037
Battaglioni, Massimo; Chiaraluce, Franco; Baldi, Marco; Lentmaier, Michael
2021
Two-stage estimation for quantum detector tomography: error analysis, numerical and experimental results. Zbl 1473.81040
Wang, Yuanlong; Yokoyama, Shota; Dong, Daoyi; Petersen, Ian R.; Huntington, Elanor H.; Yonezawa, Hidehiro
2021
Synchronization strings and codes for insertions and deletions – a survey. Zbl 1473.94003
2021
An overview of capacity results for synchronization channels. Zbl 1473.94002
Cheraghchi, Mahdi; Ribeiro, João
2021
Trace reconstruction: generalized and parameterized. Zbl 1473.94054
Krishnamurthy, Akshay; Mazumdar, Arya; McGregor, Andrew; Pal, Soumyabrata
2021
Correcting a single indel/edit for DNA-based data storage: linear-time encoders and order-optimality. Zbl 1473.94040
Cai, Kui; Chee, Yeow Meng; Gabrys, Ryan; Kiah, Han Mao; Nguyen, Tuan Thanh
2021
DNA-based storage: models and fundamental limits. Zbl 1475.94059
Shomorony, Ilan; Heckel, Reinhard
2021
A construction of minimal linear codes from partial difference sets. Zbl 1474.94082
Tao, Ran; Feng, Tao; Li, Weicong
2021
Geometric approach to $$b$$-symbol Hamming weights of cyclic codes. Zbl 1475.94201
Shi, Minjia; Özbudak, Ferruh; Solé, Patrick
2021
Two-dimensional binary Z-complementary array pairs. Zbl 1475.94083
Pai, Cheng-Yu; Ni, Yong-Ting; Chen, Chao-Yu
2021
Finite block length analysis on quantum coherence distillation and incoherent randomness extraction. Zbl 1475.81012
Hayashi, Masahito; Fang, Kun; Wang, Kun
2021
Decentralized cooperative online estimation with random observation matrices, communication graphs and time delays. Zbl 1475.62171
Wang, Jiexiang; Li, Tao; Zhang, Xiwei
2021
Cutoff for exact recovery of Gaussian mixture models. Zbl 1475.62181
Chen, Xiaohui; Yang, Yun
2021
On CCZ-equivalence of the inverse function. Zbl 1475.94128
Kölsch, Lukas
2021
Linearly self-equivalent APN permutations in small dimension. Zbl 1475.94106
Beierle, Christof; Brinkmann, Marcus; Leander, Gregor
2021
Construction of MDS Euclidean self-dual codes via two subsets. Zbl 1486.94162
Fang, Weijun; Xia, Shu-Tao; Fu, Fang-Wei
2021
Improved schemes for asymptotically optimal repair of MDS codes. Zbl 1486.94160
Chowdhury, Ameera; Vardy, Alexander
2021
Consistent recovery threshold of hidden nearest neighbor graphs. Zbl 1486.94037
Ding, Jian; Wu, Yihong; Xu, Jiaming; Yang, Dana
2021
Robust scatter matrix estimation for high dimensional distributions with heavy tail. Zbl 1486.62164
Lu, Junwei; Han, Fang; Liu, Han
2021
Binary sequences derived from differences of consecutive primitive roots. Zbl 1486.11155
Winterhof, Arne; Xiao, Zibi
2021
Deterministic constructions of compressed sensing matrices from unitary geometry. Zbl 1486.94031
Tong, Fenghua; Li, Lixiang; Peng, Haipeng; Yang, Yixian
2021
Repair rate lower bounds for distributed storage. Zbl 1486.94054
Luby, Michael
2021
Locally recoverable codes on surfaces. Zbl 1486.94171
2021
Consistent risk estimation in moderately high-dimensional linear regression. Zbl 1486.62208
Xu, Ji; Maleki, Arian; Rad, Kamiar Rahnama; Hsu, Daniel
2021
Adversarial risk via optimal transport and optimal couplings. Zbl 1486.62190
Pydi, Muni Sreenivas; Jog, Varun
2021
Polar lattices for lossy compression. Zbl 1486.94053
Liu, Ling; Shi, Jinwen; Ling, Cong
2021
Bounds on Lyapunov exponents via entropy accumulation. Zbl 1465.60012
Sutter, David; Fawzi, Omar; Renner, Renato
2021
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# The Flying Colours Maths Blog: Latest posts

## A logs puzzle

Via @markritchings, an excellent logs problem: If $a = \log_{14}(7)$ and $b = \log_{14}(5)$, find $\log_{35}(28)$ in terms of $a$ and $b$. One of the reasons I like this puzzle is that I did it a somewhat brutal way, and once I had the answer, a much neater way jumped

## Wrong, But Useful: Episode 69

In this episode, we're joined by @christianp, who is Christian Lawson-Perfect in real life, our first returning special guest co-host1. We discuss: The Big Internet Math Off and associated stickerbook 99 variations on a proof by Philip Ording The Art of Statistics - Learning from Data by David Spiegelhalter Maths

## Ask Uncle Colin: A Binary Fraction

Dear Uncle Colin, How would you write $\frac{1}{10}$ in binary? Binary Is Totally Stupid Hi, BITS, and thanks for your message! I have two ways to deal with this: the standard, long-division sort of method, and a much nicer geometric series approach. Long division-esque While I can do the long

## A folding puzzle

Here’s a tweet from @colinthemathmo: Here's another one. Take a square, crease in the halfway mark, fold up a corner - where does the corner go to? What are its coordinates? pic.twitter.com/Bfr0X8ACur — Colin Wright (@ColinTheMathmo) February 12, 2018 I’m not big on origami, but if Colin thinks it’s an

## Ask Uncle Colin: Round-Robin Progress

Dear Uncle Colin, I lost the first game of my Big Internet Math-Off tournament - can I still win the group and qualify for the semi-finals? - Surely Combinations Of Talent, Luck And Nous Deliver? Hi, SCOTLAND, and thanks for your message! Because the tie-break rules aren’t currently clear, I

## Middle children

At an academic conference; 22 people in the room. Speaker asks who is a middle child. There is only one in the entire group - him. Striking (if anecdotal) confirmation of stereotypes about birth order. — Leigh Caldwell (@leighblue) December 14, 2018 As a loyal listener to More or Less,

## Big Internet Math-Off: My first pitch is live!

A quick extra post today: I’m in the Big Internet Math-Off, which decides who will become the World’s Most Interesting Mathematician of 2019. My first group match is today, against @kyledevans, and I’ve done a video for it! Go over to the Ap’, have a look at the pitches, and

## Ask Uncle Colin: Some Symmetric Algebra

Dear Uncle Colin, If I know that $a+b+c = 0$, how can I show that $(2a-b)^3 + (2b-c)^3 + (2c-a)^3 = 3(2a-b)(2b-c)(2c-a)$? - Something You Might Merrily Explain? Thanks! Regards! Yippee! Hi, SYMMETRY, and thanks for your message! As usual, there are myriad ways to attack this, of which I

## The Dictionary of Mathematical Eponymy: Sophie Germain primes

What are they? A Sophie Germain prime is a prime such that $2p+1$ is also prime - for example, 23 is a Sophie Germain prime since 47 is also prime. The largest known Sophie Germain prime has close to 400,000 digits; it is conjectured that there are infinitely many such

## Ask Uncle Colin: A Cosine Proof

Dear Uncle Colin How would you prove that $\cos(2x) > 1 - 2x^2$ for $x > 0$? - Can’t Obviously See It, Need Explanation Hi, COSINE, and thanks for your message! I’d start by noting that as soon as $x > 1$, the right hand side is smaller than -1,


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Chapter 7. Stoichiometry of Chemical Reactions

# 7.1 Reaction Stoichiometry

### Learning Objectives

By the end of this section, you will be able to:

• Explain the concept of stoichiometry as it pertains to chemical reactions
• Use balanced chemical equations to derive stoichiometric factors relating amounts of reactants and products
• Perform stoichiometric calculations involving mass, moles, and solution molarity

A balanced chemical equation provides a great deal of information in a very succinct format. Chemical formulas provide the identities of the reactants and products involved in the chemical change, allowing classification of the reaction. Coefficients provide the relative numbers of these chemical species, allowing a quantitative assessment of the relationships between the amounts of substances consumed and produced by the reaction. These quantitative relationships are known as the reaction’s stoichiometry, a term derived from the Greek words stoicheion (meaning “element”) and metron (meaning “measure”). In this module, the use of balanced chemical equations for various stoichiometric applications is explored.

The general approach to using stoichiometric relationships is similar in concept to the way people go about many common activities. Food preparation, for example, offers an appropriate comparison. A recipe for making eight pancakes calls for 1 cup pancake mix, $\frac{3}{4}$ cup milk, and one egg. The “equation” representing the preparation of pancakes per this recipe is

$1 \;\text{cup mix} + \frac{3}{4} \;\text{cup milk} + 1 \;\text{egg} \longrightarrow 8 \;\text{pancakes}$

If two dozen pancakes are needed for a big family breakfast, the ingredient amounts must be increased proportionally according to the amounts given in the recipe. For example, the number of eggs required to make 24 pancakes is

$24 \;\rule[0.5ex]{4em}{0.1ex}\hspace{-4em}\text{pancakes} \times \frac{1 \;\text{egg}}{8 \;\rule[0.25ex]{3em}{0.1ex}\hspace{-3em}\text{pancakes}} = 3 \;\text{eggs}$

Balanced chemical equations are used in much the same fashion to determine the amount of one reactant required to react with a given amount of another reactant, or to yield a given amount of product, and so forth. The coefficients in the balanced equation are used to derive stoichiometric factors that permit computation of the desired quantity. To illustrate this idea, consider the production of ammonia by reaction of hydrogen and nitrogen:

$\text{N}_2(g) + 3\text{H}_2(g) \longrightarrow 2\text{NH}_3(g)$

This equation shows ammonia molecules are produced from hydrogen molecules in a 2:3 ratio, and stoichiometric factors may be derived using any amount (number) unit:

$\frac{2 \;\text{NH}_3 \;\text{molecules}}{3 \;\text{H}_2 \;\text{molecules}} \;\text{or} \;\frac{2 \;\text{doz NH}_3 \;\text{molecules}}{3 \;\text{doz H}_2 \;\text{molecules}} \;\text{or} \;\frac{2 \;\text{mol NH}_3 \;\text{molecules}}{3 \;\text{mol H}_2 \;\text{molecules}}$

These stoichiometric factors can be used to compute the number of ammonia molecules produced from a given number of hydrogen molecules, or the number of hydrogen molecules required to produce a given number of ammonia molecules. Similar factors may be derived for any pair of substances in any chemical equation.

### Example 1

How many moles of I2 are required to react with 0.429 mol of Al according to the following equation (see Figure 1)?

$2\text{Al} + 3\text{I}_2 \longrightarrow 2\text{AlI}_3$

Solution
Referring to the balanced chemical equation, the stoichiometric factor relating the two substances of interest is $\frac{3 \;\text{mol I}_2}{2 \;\text{mol Al}}$. The molar amount of iodine is derived by multiplying the provided molar amount of aluminum by this factor:

$\begin{array}{r @{{}={}} l} \text{mol I}_2 & 0.429 \;\rule[0.5ex]{3.25em}{0.1ex}\hspace{-3.25em}\text{mol Al} \times \frac{3 \;\text{mol I}_2}{2 \;\rule[0.25ex]{2em}{0.1ex}\hspace{-2em}\text{mol Al}} \\[1em] & 0.644 \;\text{mol I}_2 \end{array}$

Test Yourself
How many moles of Ca(OH)2 are required to react with 1.36 mol of H3PO4 to produce Ca3(PO4)2 according to the equation $3\text{Ca(OH)}_2 + 2\text{H}_3 \text{PO}_4 \longrightarrow \text{Ca}_3 \text{(PO}_4)_2 + 6\text{H}_2 \text{O}$?

2.04 mol

### Example 2

How many carbon dioxide molecules are produced when 0.75 mol of propane is combusted according to this equation?

$\text{C}_3 \text{H}_8 + 5\text{O}_2 \longrightarrow 3\text{CO}_2 + 4\text{H}_2 \text{O}$

Solution
The approach here is the same as for Example 1, though the absolute number of molecules is requested, not the number of moles of molecules. This will simply require use of the moles-to-numbers conversion factor, Avogadro’s number.

The balanced equation shows that carbon dioxide is produced from propane in a 3:1 ratio:

$\frac{3 \;\text{mol CO}_2}{1 \;\text{mol C}_3 \text{H}_8}$

Using this stoichiometric factor, the provided molar amount of propane, and Avogadro’s number,

$0.75 \;\rule[0.5ex]{4.5em}{0.1ex}\hspace{-4.5em}\text{mol C}_3 \text{H}_8 \times \frac{3 \;\rule[0.25ex]{3em}{0.1ex}\hspace{-3em} \text{mol CO}_2}{1 \;\rule[0.5ex]{3em}{0.1ex}\hspace{-3em}\text{mol C}_3 \text{H}_8} \times \frac{6.022 \times 10^{23} \;\text{CO}_2 \;\text{molecules}}{\rule[0.25ex]{3em}{0.1ex}\hspace{-3em} \text{mol CO}_2} = 1.4 \times 10^{24} \text{CO}_2 \;\text{molecules}$

Test Yourself
How many NH3 molecules are produced by the reaction of 4.0 mol of Ca(OH)2 according to the following equation:

$(\text{NH}_4)_2 \text{SO}_4 + \text{Ca(OH)}_2 \longrightarrow 2\text{NH}_3 + \text{CaSO}_4 + 2\text{H}_2 \text{O}$
4.8 × 1024 NH3 molecules

These examples illustrate the ease with which the amounts of substances involved in a chemical reaction of known stoichiometry may be related. Directly measuring numbers of atoms and molecules is, however, not an easy task, and the practical application of stoichiometry requires that we use the more readily measured property of mass.

### Example 3

What mass of sodium hydroxide, NaOH, would be required to produce 16 g of the antacid milk of magnesia [magnesium hydroxide, Mg(OH)2] by the following reaction?

$\text{MgCl}_2(aq) + 2\text{NaOH}(aq) \longrightarrow \text{Mg(OH)}_2(s) + \text{NaCl}(aq)$

Solution
The approach used previously in Example 1 and Example 2 is likewise used here; that is, we must derive an appropriate stoichiometric factor from the balanced chemical equation and use it to relate the amounts of the two substances of interest. In this case, however, masses (not molar amounts) are provided and requested, so additional steps of the sort learned in the previous chapter are required. The calculations required are outlined in this flowchart:

$16 \;\rule[0.5ex]{5em}{0.1ex}\hspace{-5em}\text{g Mg(OH)}_2 \times \frac{1 \;\rule[0.25ex]{4.75em}{0.1ex}\hspace{-4.75em}\text{mol Mg(OH)}_2}{58.3197 \;\rule[0.25ex]{3.5em}{0.1ex}\hspace{-3.5em}\text{g Mg(OH)}_2} \times \frac{2 \;\rule[0.25ex]{3.5em}{0.1ex}\hspace{-3.5em}\text{mol NaOH}}{1 \;\rule[0.25ex]{4.5em}{0.1ex}\hspace{-4.5em}\text{mol Mg(OH)}_2} \times \frac{39.9971 \;\text{g NaOH}}{1 \;\rule[0.25ex]{3.25em}{0.1ex}\hspace{-3.25em}\text{mol NaOH}} = 22 \;\text{g NaOH}$

Test Yourself
What mass of gallium oxide, Ga2O3, can be prepared from 29.0 g of gallium metal? The equation for the reaction is $4 \text{Ga} + 3\text{O}_2 \longrightarrow 2\text{Ga}_2 \text{O}_3.$

39.0 g

### Example 4

What mass of oxygen gas, O2, from the air is consumed in the combustion of 702 g of octane, C8H18, one of the principal components of gasoline?

$2\text{C}_8 \text{H}_{18} + 25\text{O}_2 \longrightarrow 16\text{CO}_2 + 18\text{H}_2 \text{O}$

Solution
The approach required here is the same as for the Example 3, differing only in that the provided and requested masses are both for reactant species.

$702 \;\rule[0.5ex]{3.5em}{0.1ex}\hspace{-3.5em}\text{g C}_8 \text{H}_{18} \times \frac{1 \;\rule[0.25ex]{3.5em}{0.1ex}\hspace{-3.5em}\text{mol C}_8 \text{H}_{18}}{114.231 \;\rule[0.25ex]{2.75em}{0.1ex}\hspace{-2.75em}\text{g C}_8 \text{H}_{18}} \times \frac{25 \;\rule[0.25ex]{2.5em}{0.1ex}\hspace{-2.5em}\text{mol O}_2}{2 \;\rule[0.25ex]{3.5em}{0.1ex}\hspace{-3.5em}\text{mol C}_8 \text{H}_{18}} \times \frac{31.9988 \;\text{g O}_2}{\rule[0.25ex]{2.5em}{0.1ex}\hspace{-2.5em}\text{mol O}_2} = 2.46 \times 10^3 \;\text{g O}_2$

Test Yourself
What mass of CO is required to react with 25.13 g of Fe2O3 according to the equation

$\text{Fe}_2 \text{O}_3 + 3\text{CO} \longrightarrow 2\text{Fe} + 3\text{CO}_2$

13.22 g

These examples illustrate just a few instances of reaction stoichiometry calculations. Numerous variations on the beginning and ending computational steps are possible depending upon what particular quantities are provided and sought (volumes, solution concentrations, and so forth). Regardless of the details, all these calculations share a common essential component: the use of stoichiometric factors derived from balanced chemical equations. Figure 2 provides a general outline of the various computational steps associated with many reaction stoichiometry calculations.

### Airbags

Airbags (Figure 3) are a safety feature provided in most automobiles since the 1990s. The effective operation of an airbag requires that it be rapidly inflated with an appropriate amount (volume) of gas when the vehicle is involved in a collision. This requirement is satisfied in many automotive airbag systems through use of explosive chemical reactions, one common choice being the decomposition of sodium azide, NaN3. When sensors in the vehicle detect a collision, an electrical current is passed through a carefully measured amount of NaN3 to initiate its decomposition:

$2 \text{NaN}_3(s) \longrightarrow 3\text{N}_2(g) + 2\text{Na}(s)$

## More Worked Out Problems

### Example 5

How many molecules of SO3 are needed to react with 144 molecules of Fe2O3 given this balanced chemical equation?

Fe2O3(s) + 3SO3(g) $\longrightarrow$ Fe2(SO4)3

Solution

We use the balanced chemical equation to construct a conversion factor between Fe2O3 and SO3. The number of molecules of Fe2O3 goes on the bottom of our conversion factor so it cancels with our given amount, and the molecules of SO3 go on the top. Thus, the appropriate conversion factor is

Starting with our given amount and applying the conversion factor, the result is

We need 432 molecules of SO3 to react with 144 molecules of Fe2O3.

Test Yourself

How many molecules of H2 are needed to react with 29 molecules of N2 to make ammonia if the balanced chemical equation is N2 + 3H2 $\longrightarrow$ 2NH3?

87 molecules

### Example 6

How many molecules of NH3 can you make if you have 228 atoms of H2?

Solution

From the formula, we know that one molecule of NH3 has three H atoms. Use that fact as a conversion factor:

Test Yourself

How many molecules of Fe2(SO4)3 can you make from 777 atoms of S?

259 molecules

### Example 7

How many moles of HCl will be produced when 249 g of AlCl3 are reacted according to this chemical equation?  The molar mass of AlCl3 is 133.34 g/mol.

2 AlCl3 + 3 H2O(ℓ) $\longrightarrow$ Al2O3 + 6 HCl(g)

Solution

We will do this in two steps: convert the mass of AlCl3 to moles and then use the balanced chemical equation to find the number of moles of HCl formed. The molar mass of AlCl3 is 133.34 g/mol, which we have to invert to get the appropriate conversion factor:

$249 \;\rule[0.5ex]{0.5em}{0.1ex}\hspace{-0.5em}\text{g AlCl}_3 \times \frac{1 \;\text{mol AlCl}_3}{133.34\;\rule[0.5ex]{0.5em}{0.1ex}\hspace{-0.5em}\text{g AlCl}_3} = \underline{1.86}74 \;\text{mol of aluminium chloride with 3 sig figs}$

Now we can use this quantity to determine the number of moles of HCl that will form. From the balanced chemical equation, we construct a conversion factor between the number of moles of AlCl3 and the number of moles of HCl:

$\frac{6 \; \text{mol HCl}}{2 \; \text{mol AlCl}_3}$

Applying this conversion factor to the quantity of AlCl3, we get

$\underline{1.86}74 \;\rule[0.5ex]{1.25em}{0.1ex}\hspace{-1.25em}\text{mol AlCl}_3 \times \frac{6 \;\text{mol HCl}}{2\;\rule[0.5ex]{1.25em}{0.1ex}\hspace{-1.25em}\text{mol AlCl}_3} =5.60 \;\text{mol HCl}$

Alternatively, we could have done this in one line:

$249 \;\rule[0.5ex]{0.5em}{0.1ex}\hspace{-0.5em}\text{g AlCl}_3 \times \frac{1 \;\rule[0.5ex]{1.25em}{0.1ex}\hspace{-1.25em}\text{mol AlCl}_3}{133.34\;\rule[0.5ex]{0.5em}{0.1ex}\hspace{-0.5em}\text{g AlCl}_3} \times \frac{6 \;\text{mol HCl}}{2\;\rule[0.5ex]{1.25em}{0.1ex}\hspace{-1.25em}\text{mol AlCl}_3}=5.60 \;\text{mol HCl}$

The last digit in our final answer is slightly different because of rounding differences, but the answer is essentially the same.

Test Yourself

How many moles of Al2O3 will be produced when 23.9 g of H2O are reacted according to this chemical equation?

2 AlCl3 + 3 H2O(ℓ) $\longrightarrow$ Al2O3 + 6 HCl(g)

0.442 mol

### Example 8

How many grams of NH3 will be produced when 33.9 mol of H2 are reacted according to this chemical equation?  Use 17.03 g/mol as the molar mass of NH3.

N2(g) + 3 H2(g) $\longrightarrow$ 2 NH3(g)

Solution

The conversions are the same, but they are applied in a different order. Start by using the balanced chemical equation to convert to moles of another substance and then use its molar mass to determine the mass of the final substance. In two steps, we have

Now, using the molar mass of NH3, which is 17.03 g/mol, we get

Test Yourself

How many grams of N2 are needed to produce 2.17 mol of NH3 when reacted according to this chemical equation?

N2(g) + 3 H2(g) $\longrightarrow$ 2 NH3(g)

30.4 g (Note: here we go from a product to a reactant, showing that mole-mass problems can begin and end with any substance in the chemical equation.)

### Example 9

What mass of Mg will be produced when 86.4 g of K are reacted?  Use 39.09 g/mol as the molar mass of potassium and 24.31 g/mol as the molar mass of magnesium.

MgCl2(s) + 2 K(s) $\longrightarrow$ Mg(s) + 2 KCl(s)

Solution

We will simply follow the steps

mass K $\longrightarrow$ mol K $\longrightarrow$ mol Mg $\longrightarrow$ mass Mg

In addition to the balanced chemical equation, we need the molar masses of K (39.09 g/mol) and Mg (24.31 g/mol). In one line,

Test Yourself

What mass of H2 will be produced when 122 g of Zn are reacted?

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

3.77 g

### Example 10

Interpret this balanced chemical equation in terms of moles.

P4 + 5 O2 $\longrightarrow$ P4O10

Solution

The coefficients represent the number of moles that react, not just molecules. We would speak of this equation as “one mole of molecular phosphorus reacts with five moles of elemental oxygen to make one mole of tetraphosphorus decoxide.”

Test Yourself

Interpret this balanced chemical equation in terms of moles.

N2 + 3 H2 $\longrightarrow$ 2 NH3

One mole of elemental nitrogen reacts with three moles of elemental hydrogen to produce two moles of ammonia.

### Example 11

For the balanced chemical equation

2 C4H10(g) + 13 O2 $\longrightarrow$ 8 CO2(g) + 10 H2O(ℓ)

if 154 mol of O2 are reacted, how many moles of CO2 are produced?

Solution

We are relating an amount of oxygen to an amount of carbon dioxide, so we need the equivalence between these two substances. According to the balanced chemical equation, the equivalence is 13 mol O2 to 8 mol CO2

We can use this equivalence to construct the proper conversion factor. We start with what we are given and apply the conversion factor:

The mol O2 unit is in the denominator of the conversion factor so it cancels. Both the 8 and the 13 are exact numbers, so they don’t contribute to the number of significant figures in the final answer.

Test Yourself

Using the above equation, how many moles of H2O are produced when 154 mol of O2 react?

118 mol

## Key Concepts and Summary

A balanced chemical equation may be used to describe a reaction’s stoichiometry (the relationships between amounts of reactants and products). Coefficients from the equation are used to derive stoichiometric factors that subsequently may be used for computations relating reactant and product masses, molar amounts, and other quantitative properties.

### Exercises

1. Determine the number of moles and the mass requested for each of the following reactions: (Hint: Write the balanced equation for each before attempting calculations.)

a) The number of moles and the mass of chlorine, Cl2, required to react with 10.0 g of sodium metal, Na, to produce sodium chloride, NaCl.

b) The number of moles and the mass of oxygen formed by the decomposition of 1.252 g of mercury(II) oxide.

c) The number of moles and the mass of sodium nitrate, NaNO3, required to produce 128 g of oxygen. (NaNO2 is the other product.)

d) The number of moles and the mass of carbon dioxide formed by the combustion of 20.0 kg of carbon in an excess of oxygen.

e) The number of moles and the mass of copper(II) carbonate needed to produce 1.500 kg of copper(II) oxide. (CO2 is the other product.)

f)

2. Determine the number of moles and the mass requested for each of the following reactions: (Hint: Write the balanced equation for each before attempting calculations.)

a) The number of moles and the mass of Mg required to react with 5.00 g of HCl and produce MgCl2 and H2.

b) The number of moles and the mass of oxygen formed by the decomposition of 1.252 g of silver(I) oxide.

c) The number of moles and the mass of magnesium carbonate, MgCO3, required to produce 283 g of carbon dioxide. (MgO is the other product.)

d) The number of moles and the mass of water formed by the combustion of 20.0 kg of acetylene, C2H2, in an excess of oxygen.

e) The number of moles and the mass of barium peroxide, BaO2, needed to produce 2.500 kg of barium oxide, BaO (O2 is the other product.)

f)

3. Gallium chloride is formed by the reaction of 2.6 L of a 1.44 M solution of HCl according to the following equation: $2\text{Ga} + 6\text{HCl} \longrightarrow 2\text{GaCl}_3 + 3\text{H}_2$.

a) Outline the steps necessary to determine the number of moles and mass of gallium chloride.

b) Perform the calculations outlined.

4. Silver is often extracted from ores such as K[Ag(CN)2] and then recovered by the reaction
$2 \text{K} [\text{Ag(CN)}_2](aq) + \text{Zn}(s) \longrightarrow 2\text{Ag}(s) + \text{Zn(CN)}_2(aq) + 2\text{KCN}(aq)$

a) How many molecules of Zn(CN)2 are produced by the reaction of 35.27 g of K[Ag(CN)2]?

b) What mass of Zn(CN)2 is produced?

5. Carborundum is silicon carbide, SiC, a very hard material used as an abrasive on sandpaper and in other applications. It is prepared by the reaction of pure sand, SiO2, with carbon at high temperature. Carbon monoxide, CO, is the other product of this reaction. Write the balanced equation for the reaction, and calculate how much SiO2 is required to produce 3.00 kg of SiC.

6. Urea, CO(NH2)2, is manufactured on a large scale for use in producing urea-formaldehyde plastics and as a fertilizer. What is the maximum mass of urea that can be manufactured from the CO2 produced by combustion of 1.00×103kg1.00×103kg of carbon followed by the reaction?
$\text{CO}_2(g) + 2\text{NH}_3(g) \longrightarrow {\text{CO(NH}_2})_2(s) + \text{H}_2 \text{O}(l)$

7. A compact car gets 37.5 miles per gallon on the highway. If gasoline contains 84.2% carbon by mass and has a density of 0.8205 g/mL, determine the mass of carbon dioxide produced during a 500-mile trip (3.785 liters per gallon).

8. What volume of a 0.2089 M KI solution contains enough KI to react exactly with the Cu(NO3)2 in 43.88 mL of a 0.3842 M solution of Cu(NO3)2?$2 \text{Cu(NO}_3)_2 + 4\text{KI} \longrightarrow 2\text{CuI} + \text{I}_2 + 4{\text{KNO}_3}$

9. The toxic pigment called white lead, Pb3(OH)2(CO3)2, has been replaced in white paints by rutile, TiO2. How much rutile (g) can be prepared from 379 g of an ore that contains 88.3% ilmenite (FeTiO3) by mass?$2\text{FeTiO}_3 + 4\text{HCl} + \text{Cl}_2 \longrightarrow 2\text{FeCl}_3 + 2\text{TiO}_2 + 2\text{H}_2 \text{O}$

10. Think back to the pound cake recipe. What possible conversion factors can you construct relating the components of the recipe?

11. What are all the conversion factors that can be constructed from the balanced chemical reaction 2H2(g) + O2(g) $\longrightarrow$ 2H2O(ℓ)?

12. Given the chemical equation

Na(s) + H2O(ℓ) $\longrightarrow$ NaOH(aq) + H2(g)

a)  Balance the equation.

b)  How many molecules of H2 are produced when 332 atoms of Na react?

13. For the balanced chemical equation

6 H+(aq) + 2 MnO4(aq) + 5 H2O2(ℓ) $\longrightarrow$ 2 Mn2+(aq) + 5 O2(g) + 8 H2O(ℓ)

how many molecules of H2O are produced when 75 molecules of H2O2 react?

14.  Given the balanced chemical equation

Fe2O3(s) + 3SO3(g) $\longrightarrow$ Fe2(SO4)3

how many molecules of Fe2(SO4)3 are produced if 321 atoms of S are reacted?

15.  For the balanced chemical equation

Fe2O3(s) + 3 SO3(g) $\longrightarrow$ Fe2(SO4)3

suppose we need to make 145,000 molecules of Fe2(SO4)3. How many molecules of SO3 do we need?

16.  Construct the three independent conversion factors possible for these two reactions:

2 H2 + O2 $\longrightarrow$ 2 H2O

H2 + O2 $\longrightarrow$ H2O2

Why are the ratios between H2 and O2 different?

The conversion factors are different because the stoichiometries of the balanced chemical reactions are different.

17. What mass of CO2 is produced by the combustion of 1.00 mol of CH4?

CH4(g) + 2 O2(g) $\longrightarrow$ CO2(g) + 2 H2O(ℓ)

18. What mass of HgO is required to produce 0.692 mol of O2?

2 HgO(s) $\longrightarrow$ 2 Hg(ℓ) + O2(g)

19. How many moles of Al can be produced from 10.87 g of Ag?

Al(NO3) 3(s) + 3 Ag $\longrightarrow$ Al + 3 AgNO3

20. How many moles of O2 are needed to prepare 1.00 g of Ca(NO3)2?

Ca(s) + N2(g) + 3 O2(g) $\longrightarrow$ Ca(NO3) 2(s)

21. What mass of O2 can be generated by the decomposition of 100.0 g of NaClO3?

2 NaClO3 $\longrightarrow$ 2 NaCl(s) + 3 O2(g)

22. What mass of Fe2O3 must be reacted to generate 324 g of Al2O3?

Fe2O3(s) + 2 Al(s) $\longrightarrow$ 2 Fe(s) + Al2O3(s)

23. What mass of MnO2 is produced when 445 g of H2O are reacted?

H2O(ℓ) + 2 MnO4(aq) + Br(aq) $\longrightarrow$ BrO3(aq) + 2 MnO2(s) + 2 OH(aq)

24. If 83.9 g of ZnO are formed, what mass of Mn2O3 is formed with it?

Zn(s) + 2 MnO2(s) $\longrightarrow$ ZnO(s) + Mn2O3(s)

25. If 88.4 g of CH2S are reacted, what mass of HF is produced?

CH2S + 6 F2 $\longrightarrow$ CF4 + 2 HF + SF6

26. Express in mole terms what this chemical equation means.

CH4 + 2O2 $\longrightarrow$ CO2 + 2H2O

27. How many molecules of each substance are involved in the equation in Exercise 1 if it is interpreted in terms of moles?

28. For the chemical equation

2 C2H6 + 7 O2 $\longrightarrow$ 4 CO2 + 6 H2O

what equivalences can you write in terms of moles? Use the ⇔ sign.

29. Write the balanced chemical reaction for the combustion of C5H12 (the products are CO2 and H2O) and determine how many moles of H2O are formed when 5.8 mol of O2 are reacted.

30. For the balanced chemical equation

3 Cu(s) + 2 NO3(aq) + 8 H+(aq) $\longrightarrow$ 3 Cu2+(aq) + 4 H2O(ℓ) + 2 NO(g)

how many moles of Cu2+ are formed when 55.7 mol of H+ are reacted?

31. For the balanced chemical reaction

4 NH3(g) + 5 O2(g) $\longrightarrow$ 4 NO(g) + 6 H2O(ℓ)

how many moles of H2O are produced when 0.669 mol of NH3 react?

32. For the balanced chemical reaction

4 KO2(s) + 2 CO2(g) $\longrightarrow$ 2 K2CO3(s) + 3 O2(g)

determine the number of moles of both products formed when 6.88 mol of KO2 react.

1. a) 0.435 mol Na, 0.217 mol Cl2, 15.4 g Cl2

b) 0.005780 mol HgO, 2.890 × 10−3 mol O2, 9.248 × 10−2 g O2

c) 8.00 mol NaNO3, 6.8 × 102 g NaNO3

d) 1665 mol CO2, 73.3 kg CO2

e) 18.86 mol CuO, 2.330 kg CuCO3

f) 0.4580 mol C2H4Br2, 86.05 g C2H4Br2

2. a) 0.0686 mol Mg, 1.67 g Mg

b) 2.701 × 10−3 mol O2, 0.08644 g O2

c) 6.43 mol MgCO3, 542 g MgCO3

d) 713 mol H2O, 12.8 kg H2O

e) 16.31 mol BaO2, 2762 g BaO2

f) 0.207 mol C2H4, 5.81 g C2H4

3. a) $\text{volume HCl solution} \longrightarrow \text{mol HCl} \longrightarrow \text{mol GaCl}_3$; b) 1.25 mol GaCl3, 2.2 × 102 g GaCl3

4. a) 5.337 × 1022 molecules      b) 10.41 g Zn(CN)2

5. $\text{SiO}_2 + 3\text{C} \longrightarrow \text{SiC} + 2\text{CO}$, 4.50 kg SiO2

6. 5.00 × 103 kg

7. 1.28 × 105 g CO2

8. 161.40 mL KI solution

9. 176 g TiO2

10.

are two conversion factors that can be constructed from the pound cake recipe. Other conversion factors are also possible.

11.
and their reciprocals are the conversion factors that can be constructed.

12.  2Na(s) + 2H2O(ℓ) $\longrightarrow$ 2NaOH(aq) + H2(g) and 166 molecules

13. 120 molecules
14. 107 molecules
15. 435,000 molecules
16.

17. 44.0 g

18. 3.00 × 102 g

19. 0.0336 mol

20. 0.0183 mol

21. 45.1 g

22. 507 g

23. 4.30 × 103 g

24. 163 g

25. 76.7 g

26. One mole of CH4 reacts with 2 mol of O2 to make 1 mol of CO2 and 2 mol of H2O.

27. 6.022 × 1023 molecules of CH4, 1.2044 × 1024 molecules of O2, 6.022 × 1023 molecules of CO2, and 1.2044 × 1024 molecules of H2O

28. 2 mol of C2H6 to 7 mol of O2 to 4 mol of CO2 to 6 mol of H2O

29. C5H12 + 8 O $\longrightarrow$ 5CO2 + 6H2O; 4.4 mol

30. 20.9 mol

31. 1.00 mol

32. 3.44 mol of K2CO3; 5.16 mol of O2

## Glossary

stoichiometric factor: ratio of coefficients in a balanced chemical equation, used in computations relating amounts of reactants and products

stoichiometry: relationships between the amounts of reactants and products of a chemical reaction


text:
# Antiderivative of a rational function

## Homework Statement

Consider the integral:
$$\int\frac{2x^3 -4x^2 +8x +7}{(x-1)^2 (x^2 +4x +8)}{\rm{d}}x$$

## The Attempt at a Solution

The degree of the denominator is 4 and the numerator's is 3, hence I thought I would try partial fractions:
$$\frac{A}{x-1} +\frac{B}{(x-1)^2} +\frac{C}{x^2 +4x +8} = \frac{2x^3 -4x^2 +8x +7}{(x-1)^2 (x^2 +4x +8)}$$multiplying both sides by the denominator on the right side we would have:
$$A(x-1)(x^2 +4x +8) +B(x^2 +4x +8) +C(x-1)^2 = 2x^3 -4x^2 +8x +7\\Ax^3 +(3A +B +C)x^2 +(4A +4B -2C)x -(8A -8B -C) = 2x^3 -4x^2 +8x +7$$
So I should be able to conclude that A = 2, however, the problem is that on one hand I get that 3B = -10 and on the other hand, 10B = 23. Have I made a mistake in the calculations? Is any such rational function divisible [not sure if that's the correct word] into partial fractions?
Is there any other method for tackling such a problem?

Last edited:

ehild
Homework Helper

## Homework Statement

Consider the integral:
$$\int\frac{2x^3 -4x^2 +8x +7}{(x-1)^2 (x^2 +4x +8)}{\rm{d}}x$$

## The Attempt at a Solution

The degree of the denominator is 4 and the numerator's is 3, hence I thought I would try partial fractions:
$$\frac{A}{x-1} +\frac{B}{(x-1)^2} +\frac{C}{x^2 +4x +8} = \frac{2x^3 -4x^2 +8x +7}{(x-1)^2 (x^2 +4x +8)}$$multiplying both sides by the denominator on the right side we would have:
$$A(x-1)(x^2 +4x +8) +B(x^2 +4x +8) +C(x-1)^2 = 2x^3 -4x^2 +8x +7\\Ax^3 +(3A +B +C)x^2 +(4A +4B -2C)x -(8A -8B -C) = 2x^3 -4x^2 +8x +7$$
So I should be able to conclude that A = 2, however, the problem is that on one hand I get that 3B = -10 and on the other hand, 10B = 23. Have I made a mistake in the calculations? Is any such rational function divisible [not sure if that's the correct word] into partial fractions?
Is there any other method for tackling such a problem?

The last partial fraction should contain an additional first-order term in the numerator: $$\frac{A}{x-1} +\frac{B}{(x-1)^2} +\frac{Cx+D}{x^2 +4x +8} = \frac{2x^3 -4x^2 +8x +7}{(x-1)^2 (x^2 +4x +8)}$$

Thank you, ehild, for the correction. Could you please explain why the last partial fraction should also contain the first-order term? I do not doubt your words, I can't fully understand the concept myself.
Everything works out nicely, though.
A = 0, B = 1, C = 2 and D = -1 and the rest is trivial.

mfb
Mentor
The denominator is of order 2, and you always need one order less in the numerator (for your (x-1)-type fractions, the A serves that purpose).

As a simple example, you cannot express ##\frac{5x+3}{x^2+4x+8}## with ##\frac{C}{x^2+4x+8}##.

nuuskur
ehild
Homework Helper
Could you please explain why the last partial fraction should also contain the first-order term? I do not doubt your words, I can't fully understand the concept myself.
As you have a third-order polynomial in the numerator, it involves 4 equations when comparing its four coefficients with the expression obtained from the partial fractions. So you need 4 unknowns in general. Otherwise you might arrive at contradiction, as you experienced.

nuuskur