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Article Text

A systematic review of cost-effectiveness, comparing traction to intramedullary nailing of femoral shaft fractures, in the less economically developed context
1. Rebekah J Parkes1,
2. Gary Parkes2,
3. Kyle James3
1. 1 School of Medicine and Dentistry, Queen Mary University of London, London, UK
2. 2 Department of Surgery, United Mission Hospital Tansen, Tansen, Nepal
3. 3 Department of Orthopaedics, Beit CURE International Malawi, Blantyre, Malawi
1. Correspondence to Rebekah J Parkes; r.j.parkes{at}smd14.qmul.ac.uk

## Abstract

Introduction Femoral shaft fractures carry considerable morbidity and are increasingly common in less economically developed countries (LEDCs). Treatment options include traction and intramedullary (IM) nailing but in a limited-resource environment; cost-effectiveness is fundamental to policy development.

The objective herein was to evaluate the cost-effectiveness of moving from traction to IM nailing for femoral shaft fractures, in adults, in LEDCs. Incorporating a systematic review of complications and functional outcomes and a cost-minimization analysis.

Methods PubMed, EMBASE, Africa Journals Online and the Cochrane Library were searched from inception using the terms: femur* AND fracture AND traction AND (sign OR nail* OR intramedullary) AND (cost-effectiveness OR cost* OR outcome OR function) NOT paed* NOT child* NOT elastic NOT neck NOT intertrochanteric NOT periprosthetic (where asterisks indicate an unlimited truncation strategy). Abstracts were reviewed for all titles returned and full texts obtained as indicated. References of all relevant papers were also examined for further studies.

Results IM nailing has been successfully used in several institutions and reported infection, union and reoperation rates are encouraging, although no randomised control trials were identified. Three studies assessed the cost aspect and all found IM nailing to be the cheaper strategy.

Conclusion To date, the improved complication profile and reduced cost of treatment suggest that IM nailing is more cost-effective than traction. Evidence, however, is limited and the necessity for appropriate training and audit with the introduction of new techniques must be emphasised.

• Health economics < Health policies and all other topics
• Orthopedic surgery

This is an Open Access article distributed in accordance with the terms of the Creative Commons Attribution (CC BY 4.0) license, which permits others to distribute, remix, adapt and build upon this work, for commercial use, provided the original work is properly cited. See: http://creativecommons.org/licenses/by/4.0/

## Statistics from Altmetric.com

### Key questions

• Femoral shaft fractures are increasingly common in less economically developed countries (LEDCs).

• Although many affluent counties use intramedullary (IM) nailing as a first-line treatment traction remains the primary management strategy in LEDCs.

#### What are the new findings?

• The mean infection rate across the cohort studies of IM nailing discussed herein was 3% compared with 27% in those managed with traction.

• The mean length of inpatient stay is longer in those managed with traction (54 days) versus (28 days) for IM fixation.

• Evidence is limited but suggests cost-effectiveness of moving from traction to IM nailing for both improved outcome and reduced cost.

#### How might this influence practice?

• Service planners considering transition from traction to operative management of femoral fractures should be encouraged but caution in the form of adequate training and monitoring must be emphasised.

## Introduction

In affluent countries, intramedullary (IM) nailing has been well established as the primary management strategy for femoral shaft fractures in mainstream orthopaedics for almost half a century. However, this is not the case worldwide and many hospitals in Less Economically Developed Countries (LEDCs) continue to implement traction as the mainstay of treatment. In their recent review, Kramer et al highlight the high incidence of complications and extended treatment period traction entails.1

As the rate of road traffic accidents in LEDCs rises, the management of femoral fractures which, represent 7.5% of resultant non-fatal injuries, is becoming a significant public health issue.2 3 In the context of limited resources, economic evaluation of the merits of competing therapy options is imperative to inform policy makers’ decisions regarding provision of care.

To be considered cost-effective, an intervention must be cheaper and more effective than the alternative or the increase in cost be outweighed by the benefit or the cost reduced substantially more than the quality of the outcome.4 Therefore, studies assessing various outcome measures and those evaluating costs will be considered.

In addition, logistics must also be considered, such as the lack of a C-arm or a reliable electrical source from which to run one in many institutions.3 This aspect has been recognised by the surgical implant generation network (SIGN) who have designed a solid locking IM nail that can be inserted without the need for a traction table or intraoperative imaging.5 The use of this implant is therefore central to many of the studies discussed herein.

For the purpose of this study, the term LEDCs equates to the countries referred to by the United Nations (UN) as ‘developing economies’.6 These tend to be developing countries but do not all feature among the least developed countries subgroup.6 7

## Aims

1. To compare outcomes in terms of complications (eg, infection and non-union) and function (eg, return to work).

2. To evaluate which method of femoral shaft fracture treatment (traction or IM nailing) is more expensive in LEDCs on a purely hospital stay resource-use basis (cost-minimisation analysis).

3. To combine these and consider the overall cost-effectiveness of moving from traditional traction to IM nailing.

## Research question

In the resource poor environment typical of LEDCs, is the initial outlay of investing in IM nailing for femoral shaft fractures worthwhile from a health economics perspective, in terms of overall treatment cost and health outcome?

It is hypothesised that although the cost of bed days is considerably lower in LEDCs, the early mobilisation associated with IM nailing is worthwhile enough to pursue.

Population:

• patients sustaining femoral shaft fractures.

Intervention:

• femoral nailing.

Comparison:

• traction.

Outcomes:

• absolute cost of treatment (however calculated);

• length of stay;

• complications including infection, malunion and non-union;

• function including timing and ability to mobilise and return to work.

Context:

• general hospitals in LEDCs.

## Method

This systematic review was conducted in accordance with the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) guidelines.8

PubMed, EMBASE, Africa Journals Online and the Cochrane Library (from inception to present) were searched using the terms: femur* AND fracture AND traction AND (sign OR nail* OR intramedullary) AND (cost-effectiveness OR cost* OR outcome OR function) NOT paed* NOT child* NOT elastic NOT neck NOT intertrochanteric NOT periprosthetic (where asterisks indicate an unlimited truncation strategy). Two reviewers independently screened the result, applying the inclusion and exclusion criteria first to the title and abstract and then as necessary to the full text. Any discrepancy was resolved by discussion. The reference list of relevant papers was also examined for further studies in an attempt to minimise omissions.

Inclusion criteria:

• adult patients (≥16 years, skeletally mature or as defined by study);

• with a femoral shaft fracture;

• treating with IM nailing and/or traction;

• in a country listed on the UN developing economies or least developed countries list6;

• discussing an aspect of cost, outcome or function.

Exclusion criteria:

• paediatric patients;

• fractures of the femoral head or neck;

• patients with an ipsilateral acetabular fracture or lower limb amputation;

• periprosthetic and pathological fractures;

• cases using elastic nailing or external fixation;

• mixed long bone fracture cohorts, where figures for femoral fractures are not separately reported;

• descriptions of techniques or case reports;

• hospitals not functioning under restrictions in keeping with an LEDC setting.

The quality of economic studies was assessed using the Critical Appraisal Skills Programme (CASP) Economic Evaluation Checklist and data extracted and collated across all cohort studies for comparison including absolute cost of treatment, length of stay, complication rates, measures of function and time horizon.9 Where multiple traction methods are used, results for Perkins’ method were extracted.

## Results

The search returned three cost-effectiveness studies and 13 cohort studies relating outcomes of traction and or IM fixation (figure 1). Only one of the cohort studies compared the two modalities. Fourteen other papers reported database analysis, qualitative aspects and background information, but no randomised control trials were evident. See online Supplementary file 1 for details of studies excluded during full text review.

### Supplementary Material

Supplementary file 1
Figure 1

Flow diagram of study selection.

Table 1 summarises the details of the included studies.

Table 1

Summary of effectiveness study characteristics

The earliest cohort study, 1954–1971, involved treatment with traction in Nigeria. The study includes paediatric patients; however, the format of data presentation allows extraction of findings for 33 patients, 20 years of age and over. Of these, only 11 had skeletal traction and although they do not provide figures for pin site infection, they indicate that it was common, even stating that one patient, excluded from analysis, died from pin site related septicaemia. Their focus is on duration of inpatient treatment, and aside from the absence of any cases of non-union or readmission, they do not comment on complications and follow-up patterns are unclear. Surprisingly, their mean of 65 inpatient bed days is only marginally higher than in later studies despite physiotherapy only being available during 2 years of the study, perhaps because multiply injured patients were excluded.10

Another study over 3 years during this period, in South Africa, compared cohorts of 54 patients treated with Perkins’ method, 67 with Küntscher nailing and a further group treated in Thomas’ splints. No detail is provided on how the method of treatment was selected. They highlight that pin site related ‘sepsis’ was seen only with Steinmann’s and not Denham’s pins, although there is no detail of diagnostic criteria. The mean hospital stay in the IM nailing group was as protracted as 5.5 weeks (seven for traction); however, there is a suggestion of crossover into this group when clinicians became frustrated with slow progress. The prevalence of severe injuries in the cohort, suggested by the mortality rate of 3.9%–11.8% across the groups, may also have delayed rehabilitation.11

Bewes in his 1974 series of 15 patients treated with Perkin’s traction in Tanzania focused largely on treatment methodology and subjective comparisons to other forms of traction. He reported clinical union in all cases but indicates that two patients were managed with IM nailing due to tissue interposition; cases that might have presented greater challenge to union.12

The fourth cohort, 1991–1999, was of 109 patients across Médecins Sans Frontières and Red Cross hospitals in East Africa receiving Perkins’ traction. In this study, there were a high proportion of open injuries (40%), the majority caused by ballistics. Amazingly, after appropriate debridement and wound management, only two patients developed deep infections. Again they allude to the presence of pin site infection but do not attempt to quantify this. The other complications reported are five refractures from mobilising too quickly and one delayed union. However, there was poor (not quantified) follow-up and complications were based on clinical assessment alone.13

The remaining studies cover shorter time intervals.

The next study, 2003–2005, of 41 patients in Sierra Leone had a much lower proportion of open fractures (7.6%); however, 47.2% had other significant injuries. Here, a minimum of 3 months follow-up was required for inclusion so some complications may represent overestimates, the assumption being that someone without problems would be less likely to return. Pin site infection was a major issue in this study with 42.6% of patients affected despite the relatively strict diagnostic criteria of loosening, purulent discharge or X-ray evidence. Two patients required operative intervention for sequestrum.14 Despite wider use of X-ray, a similar rate of refracture was seen as in the older East Africa study; however, the mean number of inpatient days was lower (52 compared with 65).13 14

A study focusing on neglected femoral fractures, 2003–2007 in India reviewed 17 cases of IM nailing. Unlike many of the other studies, an image intensifier was available. Nevertheless, the study was included as the delay in treatment is in keeping with an LEDC healthcare environment. Given the extent of the initial deformities, a single case of shortening and the requirement for blood transfusion (29%) are relatively acceptable. Further operations were for thrombectomy and locking screw adjustment/removal. The authors report that all patients were able to return to work.15

Another group in India studied a much larger cohort of 200 patients with fresh fractures (<14 days old), recording blood loss of only 50–100 mL. However, although they did not have the instantaneous feedback of a C-arm, they did have some access to intraoperative X-ray meaning a large percentage of cases could be completed with a closed technique (all except eight). Along with an absence of open fractures, this access to imaging goes some way to explaining the total absence of infective complications in these two studies.16

A cohort of 60 femoral fractures, in a provincial hospital in Kenya, was managed with SIGN nailing. Forty-one of these were fresh fractures but the study also includes cases where complications were being managed as well as an additional 20 patients with tibial nailings; not all outcomes are reported separately clouding the picture. However, the infective complication rate appears to be low (3% in femoral cases) and postoperative length of stay was a mean of 10 days across the cohort, despite a high proportion of open fractures among the tibial group. Four cases of screw loosening are noted but this is not expanded on and may represent further cases of infection.17

Fifty patients in Uganda were treated with SIGN nailing of closed femoral fractures and followed for a minimum of 6 months or until their fracture was healed. Another 20 were excluded for inadequate follow-up, which may bias results as mentioned previously. The results suggest technical difficulties with two intraoperative patella fractures (attributed to knee stiffness in cases of delayed retrograde nailing), four malunions (>10 degrees coronal plane) and two delayed unions.18 These issues could be related to individuals’ learning curves, fracture configuration or timing of treatment.

A small mixed cohort of 16 tibia and six femur fractures in Nepal evaluated the introduction of SIGN nailing. One case of over distraction in a femur required dynamisation.19

One of the more recent cohorts of traction (2010–2011) reviewed 20 patients in Malawi. The treatment in this group was heterogeneous with both skin traction and Braun frame methods employed. Seven of the 17 patients with skeletal traction required oral antibiotics for pin site infections, giving a rate similar to other series.20

Another study from Malawi focuses on IM nailing in 137 patients (77% fresh fractures). This group went to great lengths to maximise follow-up rates and found no infections in the patients that had not attended for follow-up. However, they still saw one of the highest infection rates 6% (5% deep) and on statistical testing similar rates independent of HIV status. They also highlight areas of inefficiency in terms of mean 17 (SD 10) bed days preoperatively and delay in mobilisation postoperatively ‘for the carpenter to make their crutches’.21

The final cohort of patients treated with Perkins’ traction in Ethiopia consists of 68 cases. They saw pin site infection in only 12% of cases despite 36% being open fractures. Sixteen per cent of patients had leg-length discrepancy; however, this may be largely explained by their using the strictest definition across the studies (>2 cm).22

Young et al reviewed the SIGN database looking at infection rates in 2010 and revisited it the following year. They found that follow-up rates above 5% did not increase the infection rate seen.23 If rates were based only on patients attending follow-up, 3.2% (95% CI 2.9 to 3.5) had infection, dropping to 0.8% (95% CI 0.7 to 0.9) against the total number of those treated (slightly higher percentages are seen in the least economically developed countries).24 The first figure is similar to the mean infection rate across the cohort studies of IM nailing discussed herein (3%); however, prophylactic antibiotics, which can reduce the relative risk of infection by as much as 29%, are rarely mentioned.24

The overall infection rate is higher in those cohorts treated with traction (27%) as is the need for further surgery (7% vs 5%) and the mean length of stay is longer (54 vs 28 days), although some bias is suggested by higher open fracture rates.

### Economic Studies

The first economic study was of 97 patients in 2007 in Cambodia on the introduction of the SIGN nail to their hospital. The study sought to evaluate the change in cost and outcome associated with the introduction of IM nailing in a hospital run by a non-government organisation (NGO) and collected data retrospectively. The time horizon although not specifically defined was approximately 6 months.

They saw the mean price of treatment drop from US$941 to 820 with fewer bed days and higher rates of union.25 However, the subgroup delineation was not pure, with a number of patients receiving a period of traction before undergoing fixation (mean 20.7 days). Complications were reported in natural units to allow comparison of effectiveness but quality of life measures were not available. Costs were clearly displayed from a payer/provider perspective and there was evidence of incremental cost analysis, for instance, in estimates around the cost of later intervention for non-unions. The paper describes a change in the profile of complications encountered and concludes that better outcome was achieved at lower expenditure with IM nailing. Variation in cost, particularly with contamination of the operative group with periods of traction likely lengthening stay, was considered and if deducted would strengthen the argument for change.25 The other two economic studies were performed in larger hospitals in Kenya including 148 and 50 patients, respectively. Opondo et al prospectively studied care in a Provincial Hospital in 2010. They aimed to evaluate the difference in costs (hospital bills) and consequences based on patient choice of whether to be treated with traction or IM nailing. Outcomes were reported in terms of bed days, ability to mobilise unaided and ‘complications’. However, there is a disconcerting lack of detail about ‘complications’, for example, what they were, what defines them as local or systemic or indeed whether they were self-limiting or required costly intervention. Overall, there was a reduction in price from US$167 to 120. However, the components included in the economic calculations are listed but only the total cost is presented and no breakdown given. Therefore, neither incremental nor sensitivity analysis are evident. They found that IM nailing was the cheaper option with no significant change in ‘complications’ and improved ability to mobilise at 12 weeks (the time horizon).26

The final study performed by Kamau et al was undertaken in 2012/2013 in the National Hospital, Kenya. They reviewed cost as compared with fracture union from the perspective of hospital cost with a 3-month (or hospital discharge if later) time horizon. To qualify in the operative group, patients had to have their procedure within 1 week, clearly defining the study groups. Effectiveness was viewed from the attainment of union.27 The record of effects was somewhat limited, with a time horizon not long enough to evaluate for non-union and no comment was made on complications. Contacting the author revealed that only uncomplicated cases were evaluated, with all cases of infection excluded (DM Kamau, personal communication, 2016). This is understandable in a small study where such cases could dramatically skew results. However, this means that opportunity costs related to the complications of nailing do not form part of the analysis. The costs detailed do incorporate many elements and the relative weightings of these contributors are discussed. They found that routine management with IM nailing in uncomplicated cases is cheaper than traction (US$798 compared with US$640) and is associated with higher union rates.27

Overall, reduction in price was seen in all studies but in monetary terms the absolute values were very variable.

Table 2 summarises the comparative prices across the economic studies.

Table 2

Collated costing and estimated comparative cost breakdown for economic studies

## Discussion

The standard management of femoral shaft fractures in LEDCs is that of traction and the studies collated herein demonstrate the success of this treatment with a mean non-union rate of only 4%, delayed union of 5% and refracture of 4%. However, malunion is seen in 13% with significant shortening in 6% which may have significant implication for the functional part these individuals are able to play in society. Further, some of these patients may require delayed operative intervention with resource implications. There is little in the literature to indicate how many of these individuals seek such further treatment.

The main difficulty with traction, however, is the longevity of treatment (mean 54 inpatient days) which will inevitably have negative economic impact on many patients and in many cases will result in another family member also being removed from the work force as they provide personal care at the hospital. The studies here see this reduced to a mean of 28 days with IM nailing. There is likely to be scope for reducing this period further as some of the studies from which this mean is derived has considerable lead time before surgery was undertaken (however, various barriers may exist to expedient surgery).

The critical area that has traditionally resulted in IM nailing being avoided is fear over infection in the LEDC context but the results here are very reassuring with only 3% having infection compared with 27% receiving traction. However, the reporting of infection for the two modalities is from different studies so direct comparison should be cautious as the necessary surgical management of deep infection will vary.

In 1973, Carr and Wingo published a study of the cost-effectiveness of IM nailing as compared with traction in the USA. Despite a few inconsistencies seen among the figures for complications presented in their paper, they showed relatively convincingly that IM nailing as compared with traction resulted in reduced hospital stay and earlier return to work. They modelled cost of treatment, projecting forward 5 years and noted that IM nailing cost 20%–30% less than traction with the difference expected to increase. They noted that the cost of treatments and their adjuncts were growing in cost more quickly than that of accommodation. Breakdowns of this calculation are not provided, so it is not possible to define which aspects contributed most to this projection (for instance, it is possible that the cost of spica casting in the traction population added disproportionately to the overall expense). However, the USA’s health system sits in stark contrast to that of many LEDCs so the benefit seen there may not be transferable.28

Some decades on, the first economic analysis performed on this topic in an LEDC was by Gosselin et al in 2007, in a hospital run by an NGO, in Cambodia. A further two economic studies later followed in larger hospitals in Kenya. In all cases, similarly to the earlier USA-based paper traction rather than no treatment is used as the baseline with it widely viewed that outcomes are not acceptable in the majority of cases where no intervention is made. All three studies noted a reduction in price associated with a reduction in bed days. In the studies that provided cost breakdown, this is demonstrated in a reduction in accommodation cost which forms a large percentage of the overall price of treatment.

All the economic studies were limited in justifying effectiveness from the literature and largely relied on their own outcomes to define this. None of the researchers employed discounting (techniques in economics for adjustment based on the concept that society values early reward over delayed gratification), but this is not unreasonable given their short time horizons. The context of these hospitals is appropriate and they do fulfil the inclusion criteria set out here. However, by nature of being centres of research and higher level institutions or in one case an NGO hospital supported by an affluent country, these hospitals will likely be better resourced, with better trained staff than many other centres in LEDCs. Indeed, the very fact that SIGN nails are used in these institutions means that they have satisfied the charity of their experience and follow-up abilities, which other groups may struggle to demonstrate.

Regarding resources, all three groups included the cost of implants in the calculations for the IM group although this aspect is only presented in the Cambodian study. Kamau et al used three kinds of IM nail, SIGN (supplied free of charge), Küntscher nails and occasionally TREU nails, which complicates the cost profile (DM Kamau, personal communication, 2016). With theatre costs, including that of the implant, making up 75.3% of the overall treatment cost in this study, which implant was used is likely to have a significant impact on the cost-minimisation calculations. In the study by Opondo et al, the costs are very different from those in Kamau et al’s study despite being conducted in the same country, only a year apart. There are some anomalies in the data in Opondo’s study with one bill as low as 400Ksh which equates to about US$5 in the conservative group and the mean cost in the operative group is approximately US$120 (which the author informs me includes the price of the nail) (E Opondo, personal communication, 2016), while the cost of providing a nail is US\$150 according to SIGN (J St John, personal communication, 2016). It is unclear whether this is a miscalculation or rather represents a degree of subsidisation, as if you charge the patient in your hospital SIGN charge for the nail, while if care is provided for free they do not charge for implants.

Considering equipment expense, one group compared the cost of sourcing traction pins for Malawi, Kenya and Tanzania, finding local manufacture to be significantly cheaper (p<0.001).29 A laudable attempt at simple innovation. However, such saving would have only minimal impact on overall cost of traction treatment, with traction pins making up 0.33% of the cost in the Cambodian study and the entirety of the traction assembly contributing only 2.9% of costs in Kamau et al’s paper.25 26 Also the authors acknowledge, using local metal fabrication shops, appropriate quality for clinical use cannot be guaranteed making the ethical position of pursuing this idea complex.29 This is an issue that does not affect the current manufacture process for SIGN nails as they are produced under an internationally approved standard.5

None of the cost-effectiveness studies have attempted to evaluate costs outside of direct treatment, for example, loss of earnings for the patient and carers. This is a difficult area to study with patients coming from different backgrounds, with high levels of subsistence and informal employment in many regions. One qualitative study in Uganda of a mixed cohort of 35 patients with tibial and femoral fractures has attempted to explore the issues. They found the effects wide ranging, with patients supporting a mean of 5.7 dependants (and anxious about ongoing ability to do so), delaying treatment due to prohibitive travel costs and children being taken out of school due to lack of finance.30 This highlights access issues regarding how broadly a policy of IM nailing can be rolled out, especially as not all hospitals have ready access to orthopaedic surgeons.2 However, SIGN have been strategic in establishing their technique through maintaining device simplicity, allowing hospitals to become regional training centres and providing a supply line integrated with cases reporting, affording a degree of quality assurance.3 5

### Limitations

None of the studies described herein are randomised control trials, rather they are highly heterogeneous and many only consider one of the therapeutic methods under discussion. Furthermore, the majority summarise data as means in the absence of a full data set or SD making statistical comparison and combination unreliable.

Knee stiffness has not been considered among the outcomes as it is variably reported and success is largely the product of the quality of physiotherapy received (which is difficult to evaluate). Also physiotherapy is the area that has subjectively seen the most improvement across the time spanned by the studies further complicating comparison (Gosselin RA. 2016, personal communication, January 23).

Aside from the one qualitative paper discussed above, there is a noticeable silence in the literature regarding the wider impact of treatment in terms of the functional (eg, return to work) and societal (eg, loss of children’s education) outcomes.

The economic studies are small observational cohorts, and the focus is on a relatively simplistic price in the form of hospital fees (payee interface) rather than true cost of the treatments, with short time horizons and no information regarding wider social impact or patients’ later contribution to society (eg, though return to work).

In line with the concept of discounting, it is also unknown how willing patients are to pay for a one-off surgical intervention in comparison to a longer hospital stay and whether this will vary between cultures.

## Conclusion

The studies discussed herein suggest that a transition from traction methods to IM nailing for femoral shaft fractures is beneficial in terms of union, hospital stay and cost-minimisation. Therefore, this would appear to be a cost-effective strategy. However, the evidence is limited and the necessity for appropriate training and audit with the introduction of new techniques must be emphasised. In regions where the logistical barriers to IM nailing are ongoing (eg, absence of a surgeon), traction should remain the modus operandi.

## Acknowledgments

Librarians Lisa Toyne and Barbara Pierce for their help in accessing some of the more difficult to locate papers.

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## Footnotes

• Contributors RJP designed the study with the assistance of KJ (with some refining from GP). RJP generated the literature search. RJP and GP undertook the review process. RJP largely compiled the final article with editorial contributions from GP and KJ. All authors reviewed the final manuscript and revisions. Librarians Lisa Toyne and Barbara Pierce helped in accessing some of the more difficult to locate papers.

• Competing interests GP works in hospitals that uses the SIGN nails but none of the authors directly benefit from their use.

• Provenance and peer review Not commissioned; externally peer reviewed.

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## Algebra: A Combined Approach (4th Edition)

Begin by moving the decimal two places to the right. $331.2\div92$ 92 goes into 331 3 times. Write the 3 in the ones place. Multiply by 92 and subtract from 331. 92 does not go into the remaining amount of 55 so bring down the 2 to make it 552. 92 goes into 552 exactly 6 times. Write 6 in the tenths place for an answer of 3.6.

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# Find $$y$$?

Calculus Level 3

$\lim_{x\to0 } \left [\left \lfloor \dfrac {\sin x}x \right \rfloor + \left \lfloor \dfrac x{\sin x} \right \rfloor + \left \lfloor \dfrac {\tan x}x \right \rfloor + \left \lfloor \dfrac x {\tan x} \right \rfloor + \lfloor \cos x \rfloor \right] = y + \dfrac1y$

Find the value of $$y$$ satisfying the equation above.

Notation: $$\lfloor \cdot \rfloor$$ denotes the floor function.

×


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# Number of sequences with $k$ subsequences of $11$

Let $S_n = \{0,1\}^n$.

Let $S_{i,n} \subseteq S_n$ be a set such that $s \in S_{i,n}$ if and only if $s$ contains exactly $i$ occurences of $11$.

Example for $n=3$. \begin{align*} S_3 &= \{000, 001, 010, 011, 100, 101, 110, 111\}\\ S_{0,3} &= \{000,001,010,100,101\}\\ S_{1,3} &= \{011,110\} \\ S_{2,3} &=\{111\} \end{align*}

Example for $n=4$. \begin{align*} S_4 &= \{0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111 \}\\ S_{0,4} &= \{0000,0001,0010,0100,0101,1000,1001,1010\}\\ S_{1,4} &= \{0110,0110,1100, 1011,1101\} \\ S_{2,4} &=\{0111,1110\}\\ S_{3,4} &=\{1111\} \end{align*} Therefore $|S_{0,4}| = 8 , |S_{1,4}| = 5, |S_{2,4}| = 2, |S_{3,4}| = 1$.

Is it possible to find a formula for $|S_{i,n}|$ ? Upper and lower bounds would also be nice to have

This sequence can also be found in OEIS but no formula is available

• Maybe get a recurrence by looking at the number of length n that end in 0, 01, 11. – marty cohen Jan 25 '18 at 2:39

We can derive a two-variable two generating functions. Define $a_{i,n}:=|S_{i,n}(0)|,\,b_{i,n}:=|S_{i,n}(1)|.$ and $f(x,y)=\sum_{n=0}^\infty\sum_{i=0}^na_{i,n}x^iy^n,\,g(x,y)=\sum_{n=0}^\infty\sum_{i=0}^{n-1}b_{i,n}x^iy^n$. From the recursion relation of the previous answer, we obtain $$\begin{cases} f&=y\,f+y\,g, \\ g &= y\,f+xy\,g+\frac1{1-x}\Big(\frac1{1-y}-\frac1{1-xy}\Big) \end{cases}$$ Then solve for $(f,g)$. \begin{align} f &= \frac{\frac1{1-x}\Big(\frac1{1-y}-\frac1{1-xy}\Big)}{\frac1y-1-x-y+xy}, \\ g &= \Big(\frac1y-1\Big)\,f \end{align}
• @vkonton: You mean an asymptotics for, say, $i=\lfloor \frac n2\rfloor$ as $n\to\infty$? It is possible. I can look into that. – Hans Feb 5 '18 at 0:51
Let $S_{i,n}(0)$ be the subset of $S_{i,n}$ with its elements each having $0$ at its left most position, and $S_{i,n}(1)$ be that having $1$ at its left most position. We have the recursion relation
$$\begin{cases} |S_{i,n}|&=|S_{i,n}(0)|+|S_{i,n}(1)|, \\ |S_{i,n}(0)|& =|S_{i,n-1}|, \\ |S_{i,n}(1)| &= |S_{i,n-1}(0)|+|S_{i-1,n-1}(1)|+1 \end{cases}$$ Perhaps we can form a system of two two-variable generating functions.


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wikiHow to Find the Reciprocal

Reciprocals are helpful in all sorts of algebraic equations. For example when you are dividing one fraction by another you multiply the first by the Reciprocal of the 2nd. You might also need reciprocals when finding equations of lines.

Method 1 Finding the Reciprocal of a Fraction or Whole Number

1. 1
Find the reciprocal of a fraction by flipping it. The definition of "reciprocal" is simple. To find the reciprocal of any number, just calculate "1 ÷ (that number)." For a fraction, the reciprocal is just a different fraction, with the numbers "flipped" upside down (inverted).[1]
• For instance, the reciprocal of 3/4 is 4/3.
• Any number times its reciprocal will give you 1.
2. 2
Write the reciprocal of a whole number as a fraction. Again, the reciprocal of a number is always 1 ÷ (that number). For a whole number, write that as a fraction; there's no point in calculating it out to a decimal.
• For instance, the reciprocal of 2 is 1 ÷ 2 = 1/2.

Method 2 Finding the Reciprocal of a Mixed Number

1. 1
Identify a mixed number. Mixed numbers are part whole number and part fraction, such as 24/5. There are two steps to finding the reciprocal of a mixed number, explained below.
2. 2
Change it to an improper fraction. Remember, the number 1 can always be written as (number)/(the same number), and fractions with the same denominator (lower number) can be added together. Here's an example with 24/5:
• 24/5
• = 1 + 1 + 4/5
• = 5/5 + 5/5 + 4/5
• = (5+5+4)/5
• = 14/5.
3. 3
Flip the fraction. Once the number is written entirely as a fraction, you can find the reciprocal just like you would with any fraction: by flipping it.
• In the example above, the reciprocal of 14/5 is 5/14.

Method 3 Finding the Reciprocal of a Decimal

1. 1
Change it to a fraction if possible. You might recognize some common decimal numbers that can easily be turned into fractions. For instance, 0.5 = 1/2, and 0.25 = 1/4. Once in fraction form, just flip the fraction to find the reciprocal.
• For instance, the reciprocal of 0.5 is 2/1 = 2.
2. 2
Write out a division problem. If you can't change it to a fraction, calculate the reciprocal of that number as a division problem: 1 ÷ (the decimal). You can use a calculator to solve this, or continue on to the next step to solve it by hand.
• For example, you can find the reciprocal of 0.4 by calculating 1 ÷ 0.4.
3. 3
Change the division problem to use whole numbers. The first step to dividing decimals is to move the decimal point until all the numbers involved are whole numbers. As long as you move the decimal point the same number of spaces for both numbers, you'll get the correct answer.
• For example, you can take 1 ÷ 0.4 and rewrite it as 10 ÷ 4. In this case, you've moved each decimal place one space to the right, which is the same as multiplying each number by ten.
4. 4
Solve the problem using long division. Use long division techniques to calculate the reciprocal. If you calculate it for 10 ÷ 4, you'll get the answer 2.5, the reciprocal of 0.4.

Community Q&A

Search
• What is the reciprocal of a squared number?
It is 1 divided by the squared number.
• What is the reciprocal of a+b?
1 / (a+b).
• How do I use a mathematical table to find reciprocal of a decimal number?
Some tables do have a column devoted to reciprocals, but not typically reciprocals of decimal numbers. Your best bet is simply to divide the decimal number into 1.
• What is the reciprocal of 8?
wikiHow Contributor
1/8. Since every whole number is invisibly divided by 1 (in this case, 8/1), you just interchange the numerator and denominator, putting the denominator over the numerator and there is your reciprocal, 8/1. In other words, a reciprocal takes a number a/b and turns it into b/a.
• How do I get the reciprocal of a mixed number?
wikiHow Contributor
Convert the mixed number into an improper fraction, then find the reciprocal as you normally would.
• How do I find the reciprocal of 4?
wikiHow Contributor
The number 4 as a fraction is 4/1. To find the reciprocal of a number, the numerator and denominator switch places, so the reciprocal of 4 is 1/4.
• What is the reciprocal of 2.72?
wikiHow Contributor
The reciprocal of 2.72 is (1 divided by 2.72)=0.367647 to six significant figures.
• What is the reciprocal of 2?
wikiHow Contributor
1/2. For reciprocating any number, just put it as a fraction and reverse the numerator and denominator. For example, 3 becomes 1/3 ; 97 becomes 1/97 ; 1/23 becomes 23.
• R /a-b, what is the reciprocal please?
(a-b) / R.
• Reciprocal of negative number?
The reciprocal of a negative number is the same as the reciprocal of the positive number, except it's a negative value.
200 characters left

Tips

• A numbers negative reciprocal is the same as the regular reciprocal, multiplied by negative one.[2] For instance, the negative reciprocal of 3/4 is -4/3.
• The reciprocal is sometimes called the "multiplicative inverse."[3]
• The number 1 is its own reciprocal, since 1 ÷ 1 = 1.
• The number 0 does not have a reciprocal, since 1 ÷ 0 is undefined.[4]

Article Info

Categories: Multiplication and Division | Fractions

In other languages:

Português: Achar o Recíproco de um Número, Deutsch: Den Kehrwert bestimmen, Español: encontrar el recíproco, Русский: найти обратное число, Français: trouver l'inverse, Italiano: Calcolare il Reciproco, Bahasa Indonesia: Mencari Resiprok atau Kebalikan

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text:
# nth power

Not Reviewed
y =
Tags:
Rating
ID
DavidC.nth power
UUID
e6cf66a8-da27-11e2-8e97-bc764e04d25f

The nth power calculator computes the real number (y) equal to another real number (x) to a specified power (n).

INSTRUCTIONS: Enter the following:

• (x) This is the value of the base number
• (n) This is the value of the exponent

nth power (xn):  The calculator returns the real number

#### The Math / Science

The equation for the nth power is:

y = xn

where:

Note: The value of x must be a positive real number, otherwise the answer may be an imaginary number


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# Chemistry Exam Paper Worked Solutions

Keen to see how you went in the 2020 HSC Chemistry exam? Well, our Chemistry team have been hard at work cooking up the solutions for you!

Below are the answers and worked solutions to this year’s Chemistry HSC Exam.

## Section 1: Multiple Choice Solutions

 Question Solution Explanation 1 B Charged particles travel along a curved path in a magnetic field 2 C Phenol red will be yellow at pH 5 (for face cleanser) and red at pH 9 (for soap). All other indicators will show the same colour for both the face cleanser and the soap. 3 C Alkanes, alcohols and esters are neutral. Amines are basic. 4 A Infrared spectroscopy identifies different types of bonds. Butane and propane both only have C-C and C-H bonds. 5 A Two signals in the spectrum indicate the molecule only has two carbon environments. Chloroethane has two carbon environments. 1-chloropropane has 3 carbon environments, 1,2-dichloroethane has 1 carbon environment and 1,2-dichloropropane has 3 carbon environments. 6 D Hydroxyl group has highest priority, therefore carbon bearing the OH group will have the lowest number, and the chain is numbered right to left. The substituents are named in alphabetical order, hence bromo is named before fluoro. 7 C Functional group isomers result when the atoms in the molecules are arranged to form different functional groups. Compound 2 and compound 3 have different functional groups. Chain isomers involve rearrangement of the carbons in the backbone. Compounds 1 and 2 and compounds 1 and 4 have the same carbon chain (4 carbons in the backbone) therefore are not chain isomers. Position isomers result when molecules have the same carbon chain but the functional group is at a different location. Compounds 2 and 4 do not have the same functional groups. 8 D A solution is most effective as a buffer at the pH corresponding to the half-equivalence point. From the titration curve, this occurs at pH 9. 9 A The compound in A is a carboxylic acid, which reacts readily with sodium hydrogen carbonate (a base). The other compounds are not acidic: compound in B is a ketone and compound in C and D are alcohols. 10 D $$NaCl$$ is a neutral salt, $$NH_4Cl$$ is an acidic salt and $$NaCH_3COO$$ is a basic salt. From least to most acidic (basic → neutral → acidic): $$NaCH_3COO, NaCl, NH_4Cl$$ 11 B Mixing solutions of $$Ba(OH)_2$$ and $$MgSO_4$$ results in the formation of the precipitates $$BaSO_{4(s)}$$ and $$Mg(OH)_{2(s)}$$. Each of the other multiple choice options only produces one precipitate. 12 C The softening of plastics occurs due to a weakening of the intermolecular forces between polymer chains. Based on the portion of the polymer shown, the intermolecular forces that are present are dipole-dipole forces and dispersion forces. (hydrogen bonding cannot occur as there is no N-H, O-H, or F-H bond). 13 B The conversion of methanoic acid to methanol results in the VSEPR shape around the carbon atom changing from trigonal planar to tetrahedral. 14 B $$K_w = [H_3O^+][OH^-] = 5.5 \times 10^{-14}$$ $$[H_3O^+] = [OH^-] = \sqrt{K_w} = 2.34 \times 10^{-7}$$ $$\text{pH} = \text{-log}_{10}[H_3O^+] = 6.63$$ 15 D Option A cannot produce a peak in the mass spectrum as it is uncharged. Option B has an m/z of 16. Option C contains 4 carbon atoms in the fragment, while chloroacetamide only contains 2 carbon atoms. 16 C The yield increases with temperature indicating that the forward reaction is endothermic. The yield decreases with pressure, indicating that the product side of the equation contains more moles of gas than the reactant side. 17 D The following equilibrium occurs in the saturated solution of copper(II) sulfate : $$CuSO_{4(s)} ⇌ Cu^{2+}_{(aq)} + SO_4^{2-}$$$$_{(aq)}$$ Addition of sodium hydroxide results in the precipitation of $$Cu(OH)_{2(s)}$$, which reduces the concentration of $$Cu^{2+}_{(aq)}$$ and disturbs the equilibrium. As a result, more $$CuSO_{4(s)}$$ will dissolve to minimise the disturbance (Le Chatelier’s Principle). The blue colour will fade as the concentration of $$Cu^{2+}$$ ions in the final solution is lower than in the original solution. 18 A $$HCO_3^-$$ is amphiprotic, meaning that it can act as either an acid or base. As the pH of the solution is greater than 7, $$HCO_3^-$$ is acting as a base and accepts a proton from water. This indicates that water is a stronger acid than $$HCO_3^-$$. 19 D Cooling the system causes a reduction in the forward and reverse reaction rates. As exothermic reactions possess a lower activation energy than endothermic reactions, the forward reaction rate is less affected by cooling than the reverse reaction rate. 20 C $$K_{sp} = [Ag^+]^2[SO_4^{2-}]$$ From the graph, when $$[Ag^+] = 1 \times 10^{-4} \ \text{mol L}^{-1}, [CrO_4^{2-}] = 11 \times 10^{-5} \ \text{mol L}^{-1}$$ Therefore, $$K_{sp} = (1 \times 10^{-4})^2(11 \times 10^{-5}) = 1.1 \times10^{-12}$$

## Section 2: Long response Questions

### Question 21

The molecular ion peak is at $$m/z = 44$$.

Given that the general formula for an alkane is $$C_nH_{2n+2}$$, the alkane is propane $$(C_3H_8)$$

### Question 22

Cation

Perform a flame test:

• Brick red flame indicates $$Ca^{2+}$$
• Apple green flame indicates $$Ba^{2+}$$
• No characteristic flame colour indicates $$Mg^{2+}$$

Anion

### Question 23

Temperature: Using temperatures that are beneficial to rate and yield. Optimising temperature can also improve the longevity of catalysts, safety, and cost.

By-products: Creating by-products that are valuable and easy to separate from ethane-1,2-diol.

Availability of reactants: Reactants should be widely available and inexpensive. In this case, oxygen can be obtained from air, and ethene from cracking hydrocarbons.

### Question 24a

The functional group present in both the oil and the biodiesel is an ester.

### Question 24b

 $$C_{14}H_{30}O_{2(l)} + 41/2O_{2(g)} → 14CO_{2(g)} + 15H_2O_{(l)}$$ $$C_{18}H_{38 (l)} + 55/2O_{2(g)} → 18CO_{2(g)} + 19H_{2}O_{(l)}$$

Soot is produced when there is insufficient oxygen for a carbon containing fuel to undergo complete combustion. As biodiesel is more oxygenated than diesel, less $$O_2$$ is required for complete combustion, meaning that soot is less likely to form.

### Question 24c

Energy density of biodiesel in

 $$\text{MJ L}^{-1} = 38\text{ MJ kg}^{-1} \times 0.90\text{ kg L}^{-1} = 34\text{ MJ L}^{-1}$$

Volume of biodiesel required to produce

 $$2141\text{ MJ} = 2141\text{ MJ} / \ 34\text{ MJ L}^{-1} = 63\text{ L (2 s.f.)}$$

### Question 24d

 Advantages Disadvantages Renewable since it can be produced through fermentation of sugar from sugarcane Arable land (which is finite) is required to produce biomass. $$CO_2$$ produced from combustion is recycled through the growth of biomass required for biodiesel Lower energy density meaning more fuel is required to produce the same amount of energy.

### Question 25

 \begin{align*} n(\text{citric acid}) &= 1.0 \ \text{mol L}^{-1} \times 0.0024 \text{ L} = 0.0024 \ \text{mol} \\ n(NaOH) &= 3 \times n(\text{citric acid}) = 3 \times 0.0024 \ \text{mol} = 0.0072 \ \text{mol in }8.0 \text{ mL} \\ c(NaOH) &= 0.0072 / 0.0080 = 0.90 \ \text{mol L}^{-1} \ \text{(2 s.f.)} \\ \end{align*}

### Question 26a

$$N_{2(g)} + O_{2(g)} ⇌ 2NO_{(g)}$$

### Question 26b

The energy profile diagram shows that the enthalpy of the products is greater than the enthalpy of the reactants indicating that the forward reaction is endothermic and the reverse reaction is exothermic. Moreover, the energy profile diagram shows that the activation energy of the forward reaction is larger than the activation energy for the reverse reaction.

Increasing the temperature increases the average kinetic energy of the collisions, thus a greater proportion of collisions will have energy greater than the activation energy. As a result, increasing the temperature increases both the forward and reverse rates of reaction. However, the percentage increase in the rate of the forward, endothermic reaction is greater than that of the reverse, exothermic reaction. As a result, the concentration of the $$NO$$ increases while the concentration of the reactant $$N_2$$ and $$O_2$$ gases decreases.

As the equilibrium constant ($$K_{eq}$$) for this process is:

$$K_{eq}= \frac{[NO]^2}{[N_2][O_2]}$$

The value of $$K_{eq}$$ will increase with temperature.

### Question 27b

$$K_b= \frac{[C_3 H_7 NH_3^+ ][OH^-]}{[C_3 H_7 NH_2]}$$

 Concentration (mol L-1) $$C_3H_7NH_2$$ $$C_3H_7NH_3^+$$ $$OH^-$$ Initial $$1.00$$ $$0$$ $$0$$ Change $$-x$$ $$+x$$ $$+x$$ Equilibrium $$1.00-x$$ $$x$$ $$x$$

$$K_b = \frac{[x][x]}{[1.00-x]}$$

Assuming $$x$$ is small since $$K_b$$ is small: $$[1.00 – x] ≈[1.00]$$

\begin{align*}
K_b &= \frac{[x][x]}{[1.00]} =4.37 \times 10^{-4} \\
x &=0.0209045 \\
\end{align*}

Therefore, the concentration of hydroxide ions is $$0.0209 \ \text{mol L}^{-1} (\text{3 s.f.})$$

Question 28

Inaccurate as:

• $$NaOH_{(s)}$$ is hygroscopic and therefore not suitable as a primary standard.
• Bromocresol green is used as an indicator. As a weak acid is being titrated by a strong base, the equivalence point would be basic (pH > 7). As a result, bromocresol green will not change colour at the equivalence point.

Unreliable as the results are inconsistent. The experiment should be repeated until enough concordant results have been recorded.

### Question 30

Infrared spectrum:

Signal between $$1680-1750 \text{ cm}^{-1}$$ indicates the presence of a carbonyl group (C=O). The absence of a broad trough between $$2850-3550 \ \text{ cm}^{-1}$$ indicates that the compound does not contain a hydroxyl group.

$$^{13}\text{C NMR}$$:

The $$^{13}\text{C NMR}$$ spectrum has 4 peaks, indicating there are 4 unique carbon environments, this is consistent with the proposed structure. The peak at 170 ppm is characteristic of the carbonyl carbon and the peak at 70 ppm corresponds to a carbon nucleus adjacent to an oxygen, hence confirms the presence of an ester group.

$$^1\text{H NMR}$$:

The $$^1\text{H NMR}$$ spectrum shows 3 peaks, indicating there are 3 unique proton environments.

The peak at 1.2 ppm corresponds to the protons labelled A. The chemical shift is consistent for alkyl protons, the peak is a doublet which indicates these protons are adjacent to one other non-equivalent proton. Its integration of 6 indicates 6 protons are present in this environment.

The peak at 2.0 ppm corresponds to the protons labelled B. The chemical shift is consistent for protons that are adjacent to a carbonyl group. The singlet indicates there are zero non-equivalent protons on the adjacent carbon atom. An integration of 3 indicates 3 protons are present in this environment.

The peak at 5.0 ppm corresponds to proton labelled C. The chemical shift is consistent for protons that are adjacent to the oxygen of an ester group. The septet indicates there are 6 non-equivalent protons on the adjacent carbon atoms. An integration of 1 indicates 1 proton is present in this environment.

### Question 31

Reaction 1: $$Ag^+_{(aq)} + Cl^-_{(aq)} → AgCl_{(s)}$$
Reaction 2: $$Ag^+_{(aq)} + SCN^-_{(aq)} → AgSCN_{(s)}$$

 \begin{align*} n(SCN^-) &= 0.0500 \times 0.02865 = 0.0014325 \ \text{mol} \\ n(Ag^+)_{\text{Reaction 2}}  &= n(SCN^-) = 0.0014325 \ \text{mol} \\ n(Ag^+)_\text{total} = 0.100 \times 0.025 &= 0.0025 \ \text{mol} \\ n(Ag^+)_{\text{Reaction 1}}  &= 0.0025 – 0.0014325 = 0.0010675 \ \text{mol} \\ n(Cl^-) \text{in} \ 100 \text{ mL} &= n(Ag^+)_{\text{Reaction 1}}  = 0.0010675 \ \text{mol} \\ n(Cl^-) \text{in}\ 1 \text{ L} &= 0.010675 \ \text{mol} \\ m(Cl^-) \text{in} \ 1 \text{ L} &= 0.37842875 \text{ g} \\ \end{align*}

Therefore, concentration of $$Cl^- = 378 \text{ mg/L} \text{(3 s.f.)}$$

### Question 32

All three compounds are polar and can form dispersion forces and dipole-dipole forces. Methanol and propanoic acid can also form hydrogen bonding.

However, even though methyl propanoate cannot form hydrogen bonding, it exhibits stronger dispersion forces compared to methanol and propanoic acid due to its larger size. Hence the combined strength of its dispersion forces and dipole-dipole forces are stronger than methanol’s intermolecular forces. Thus, more thermal energy is required to break the intermolecular forces in methyl propanoate to cause boiling.

On the other hand, although propanoic acid has weaker dispersion forces compared to methyl propanoate, propanoic acid has the ability to form two hydrogen bonds per molecule, hence the overall strength of the hydrogen bonds compensates the weaker dispersion forces, allowing propanoic acid to have stronger intermolecular forces compared to methyl propanoate.

### Question 33 a

 \begin{align*} ca(OH)_{2(s)}+ 2HCl_{(aq)} → CaCl_{2(aq)} + 2H_{2}O_{(l)} \\ n(HCl)  = 0.100 x 2.00 = 0.200 \ mol \\ n(Ca(OH) 2 )  = ½ \times n(HCl) = 0.100 \ mol \\ \end{align*}

### Question 33 b

 \begin{align*} n(Ca(OH)_2) \text{in }0.1 \text{ L} &= 0.100 \ \text{mol} \\ c(Ca(OH)_2) &= 0.100 / 0.1 = 1 \ \text{mol/L} \\ \end{align*}

 Concentration (M) $$Ca(OH)_2$$ ⇌ $$Ca^{2+}$$ $$2OH^-$$ Initial – $$1$$ $$0$$ Change – $$+ x$$ $$+ 2x$$ Equilibrium – $$1+x$$ $$2x$$

Since $$K_{sp}$$ is small, assume $$[Ca^{2+}]_{eqm} ≈ [Ca^{2+}]_{\text{initial}} = 1 \text{ M}$$

 \begin{align*} [Ca^{2+}][OH^-]^2 &= 5.02 \times 10^{-6} \\ 1 \times (2?)^2 &= 5.02 \times 10^{-6} \\ 4?^2 &= 5.02 \times 10^{-6} \\ ? &= 0.001120267825 \\ [OH^-] &= 2? = 0.00224053565 \\ pH &= 14 + \text{log}_{10}(0.00224053565) = 11.350 \ (3 \text{s.f.}) \\ \end{align*}

### Question 34

Hydrochloric acid is a strong acid that completely ionises in water:

$$HCl_{(aq)} → H^+_{(aq)} + Cl^-_{(aq)}$$

Hence the $$[H^+] = [HCl]$$

An increase in the concentration of hydrochloric acid by 10 fold results in an increase in the concentration of $$H^+$$ by 10 fold. Since $$pH = \text{-log}_{10}[H^+]$$, each 10 fold increase in $$[H^+]$$ results in a decrease of the pH by 1. Thus the pH decreases incrementally by 1.

Acrylic acid is a weak acid that only partially ionises in water:

$$C_2H_3COOH_{(aq)} ⇌ H^+_{(aq)} + C_2H_3COO^-_{(aq)}$$

Hence the $$[H^+] < [C_2H_3COOH]$$

Thus, an increase in the concentration of acrylic acid by 10 fold will increase the concentration of $$H^+$$ by less than 10 fold. Since $$\text{pH} = \text{-log}_{10}[H^+]$$, this will result in a decrease in the pH by less than 1.
As the concentration of acrylic acid increases, the concentration of all components increase, hence $$Q > K$$. This results in a shift in the reverse direction, which lowers the degree of ionisation. This effectively results in a smaller and smaller change in the $$[H^+]$$ which results in smaller change to the pH as concentration increases.

### Question 35

 \begin{align*} [I_3^- ]_{eq} &= \frac{0.745}{2.76 \times 10^4} = 2.699275 \times 10^{-5} \text{ M} \\ [I_2 ]_{eq} &= \frac{[I^- ]_{\text{initial}} – 3[I_3^- ]_{eq}}{4} = \frac{7.00 \times 10^{-4} – 3 \times (2.699275 × 10^{-5})}{4} = 1.54755 \times 10^{-4} \text{ M} \\ [I^- ]_{eq} &=2 \times [I_2 ]_{eq} =3.09510 \times 10^{-4} \text{ M} \\ K &= \frac{[I_3^-]}{[I^-][I_2]} = \frac{2.699275 \times 10^{-5}}{(3.09510 × 10^{-4})(1.54755 \times 10^{-4})} = 564 \text{ (3 s.f.)} \\ \end{align*}

### Question 36

 \begin{align*} n(NaOH) &= 10.0 / 39.998 = 0.2500125 \ \text{mol} \\ ∆H_{\text{soln}} &= 0.2500125 \times -44.5 = -11.12555628 \ \text{kJ} \\ n(HCl) &= 2.00 \times 0.100 = 0.200 \ \text{mol} \\ \end{align*}

HCl is limiting reactant

 \begin{align*} ∆H_{\text{reaction}} &= 0.200 \times -56.1 = -11.22 \ \text{kJ} \\ ∆H_{\text{total}} &= -11.12555628 + -11.22 = -22.34555628 \text{ kJ} \\ q &= mc∆T \\ q &= 22345.55628 \text{ J} \\ 22345.55628 &= 113 \times 3.99 \times (T_{\text{final}} – 22.5) \\ T_{\text{final}} &= 72.1 \ °\text{C}  \text{ (3 s.f.)} \\ \end{align*}


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zbMATH — the first resource for mathematics

Calculus. II: Analytic functors. (English) Zbl 0776.55008
The paper is a sequel to “Calculus I” (Zbl 0741.57021), where derivatives of homotopy functors were introduced. Its contents is the development of the theory of $$\rho$$-analytic functors, i.e. homotopy functors $$F$$ from spaces to (based) spaces or spectra satisfying the following property: Given an $$n$$-cube diagram $$\mathcal X$$ of spaces, which is an iterated homotopy pushout, and suppose the $$n$$ maps from the initial vertex are all $$(\rho+1)$$-connected, then there is an integer $$q$$ independent of $$n$$ such that $$F({\mathcal X})$$ behaves like an $$n$$- dimensional homotopy pullback diagram up to the dimension $$(n-1)\rho-q$$. A number of results are proved which imply the equivalence of $$\rho$$- analytic functors under additional connectivity assumptions once the equivalence of their differentials or derivatives is established.
To give a flavour of the main results we quote the “First Derivative Criterion”: Let $$F\to G$$ be a natural transformation of $$\rho$$-analytic functors from spaces to spectra which induces homotopy equivalences of the differentials $$D_ X F(Y)\to D_ X G(Y)$$ for all $$X$$, $$Y$$. Then for any $$(\rho+1)$$-connected map $$Y\to X$$ the following diagram is a homotopy pullback $\begin{matrix} F(Y) & \longrightarrow & G(Y)\\ \downarrow && \downarrow\\ F(X) & \longrightarrow & G(X).\end{matrix}$ As examples may serve Waldhausen’s $$A$$-theory functor, which is 1-analytic, or the functor $$X\mapsto$$ $$\Omega^ \infty \Sigma^ \infty({\mathcal T}\text{ op}(K,X)_ +)$$ with a finite CW-complex $$K$$, which is $$\dim K$$-analytic. A sketch of applications can be found in [the author, Proc. Int. Congr. Math., Kyoto/Japan 1990, Vol. I, 621-630 (1991; Zbl 0759.55011)].
The theory of analytic functors relies heavily on the homotopy theory of cubical diagrams. In the first three sections of the paper the author collects those results about cubes, $$n$$-dimensional homotopy pushouts and pullbacks and their homotopy theory spread over the literature which are needed for his theory.
For the convenience of the reader many of the results are reproved. Moreover, they are extended where extensions are necessary for the development of the theory. In the final two sections the author introduces $$\rho$$-analytic functors and proves the implications of analyticity mentioned above. For applications to $$A$$-theory, topological cyclic homology, and the free loop space functor $$\Lambda$$ he shows in an appendix that the homotopy equivalence from the derivative $$\partial_ X \Omega^ \infty \Sigma^ \infty(\Lambda X_ +)$$ to $$\Lambda\Sigma^ \infty(\Omega X_ +)$$ established in “Calculus I” respects the $$S^ 1$$-action on both functors.

MSC:
 55P65 Homotopy functors in algebraic topology 19D10 Algebraic $$K$$-theory of spaces
Full Text:
References:
 [1] Barratt, M. and Whitehead, J. H. C.: The first non-vanishing group of an (n+1)-ad, Proc. London Math. Soc (3) 6 (1956), 417-439. · Zbl 0072.18002 · doi:10.1112/plms/s3-6.3.417 [2] Bökstedt, M., Carlsson, G., Cohen, R., Goodwillie, T., Hsiang, W.-c. and Madsen, I.: On the algebraic K-theory of simply-connected spaces, preprint. · Zbl 0867.19003 [3] Bousfield, D. and Kan, D.: Homotopy Limits, Completions, and Localizations, Lecture Notes in Mathematics 304, Springer-Verlag, New York (1972). · Zbl 0259.55004 [4] Ellis, G. and Steiner, R.: Higher-dimensional crossed modules and the homotopy groups of (n+1)-ads, J. Pure Appl. Algebra 46 (1987), 117-136. · Zbl 0622.55010 · doi:10.1016/0022-4049(87)90089-2 [5] Goodwillie, T.: Calculus I, The first derivative of pseudoisotopy theory, K-Theory 4 (1990), 1-27. · Zbl 0741.57021 · doi:10.1007/BF00534191 [6] Goodwillie, T.: Calculus III, The Taylor series of a homotopy functor, in preparation. · Zbl 1067.55006 [7] Goodwillie, T.: A multiple disjunction lemma for smooth concordance embeddings, Mem. Amer. Math. Soc. 431 (1990), 1-317. · Zbl 0717.57011 [8] Waldhausen, F.: Algebraic K-theory of topological spaces. II, in Lecture Notes in Mathematics 763, Springer-Verlag, New York (1979), pp. 356-394. · Zbl 0431.57004 [9] Waldhausen, F.: Algebraic K-theory of spaces, in Lecture Notes in Mathematics 1126, Springer-Verlag, New York (1985), pp. 318-419. · Zbl 0579.18006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


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# Automatic / variable vertical alignment within a minipage?

I have defined a new environment that I use to highlight warnings in my text. Here's a minimal example of the code I use:

\documentclass[parskip]{scrartcl}
\usepackage{fourier} % for the bomb
\usepackage{lipsum}

\newenvironment{warning}{%
\begin{minipage}[t][\height][c]{0.12\textwidth}%
{\Huge{\bomb}}%
\end{minipage}%
\begin{minipage}[t]{0.88\textwidth}
\begin{sloppypar}%
}{%
\end{sloppypar}%
\end{minipage}%
\hspace{-\parfillskip}%
}

\begin{document}
\lipsum[1]

\begin{warning}
\lipsum[2]
\end{warning}

\lipsum[3]

\begin{warning}
This is just a short warning.
\end{warning}

\lipsum[4]

\end{document}

This mostly works as intended, at least for longer warnings. If the text of the warning consists of only one or two lines, the vertical alignment of the text next to the symbol starts looking weird. If I change the vertical alignment of the minipages from t to c, the overall spacing improves for the short warnings, but this also changes the longer warnings in what I consider a worse layout. I realize that I could solve this with two distinct environments for short and long warnings, but is there a way to automate this? Like if the contents of the right-hand column exceed a height of ..., use this alignment, else use the other one?

(I'll also accept pointers to packages that provide a similar highlighting in a more robust/professional/cool way :-))

A modification of my answer to Aligning images in the marginal notes based on a specific paragraph height rule

Caveat: warnings should be one paragraph only for the measurement to be correct.

\documentclass[a4paper]{article}
\usepackage[top=2.5cm,bottom=2.5cm]{geometry}
\usepackage{environ}
\usepackage{fourier}

\makeatletter
\NewEnviron{warning}{%
%% Count the number of lines
\setbox\z@=\vbox{\hsize=.88\textwidth\BODY\par
\xdef\vw@nlines{\the\prevgraf}}%
%% Typeset the warning; .12+.88=1
\begin{list}{}{%
\leftmargin=.12\textwidth
\clubpenalties=4 10000 10000 10000 0
}%
\ifcase\vw@nlines\or % no empty paragraphs
\vw@placefigure{0}%
\or
\vw@placefigure{.5}%
\else
\vw@placefigure{1}%
\fi
\BODY
\end{list}%
}
\def\vw@placefigure#1{\item[\leavevmode\smash{\makebox[0pt][r]{%
\vbox{\hbox{\Huge\bomb}
\vskip-9pt % half the height of the bomb
\vskip-#1\baselineskip % shift down
}}}]}
\makeatother

\usepackage{lipsum}

\begin{document}
\lipsum[3]

\begin{warning}
\lipsum[3]
\end{warning}

\lipsum[3]

\begin{warning}
This is just a short warning.
\end{warning}

\lipsum[3]

\begin{warning}
A two line\\
warning
\end{warning}

\lipsum[3]

\begin{warning}
A three\\
line\\
warning
\end{warning}

\begin{warning}
\lipsum[3]
\end{warning}

\end{document}

This allows also for page breaks, but only after three lines of the warning text.

• Just out of curiosity - why the \makeatletter and \makeatother instructions in this case? There aren't any @s in this code, or am I missing something? – vwegert Jul 17 '13 at 13:59
• @vwegert Oh, there were @'s in the original code. I can put some, if you prefer. ;-) – egreg Jul 17 '13 at 14:01

To get what you wanted, I had to change the warning environment to \warningmacro, since I needed to measure the height of the text in order to decide where to place the bomb. Basically, I place it appropriately for a one-line warning and then start shifting it down 70% of the text depth, until I reach a maximum limit of shift. The two parameters to play with are the length \reflength and the scalar value of \shiftmult, currently set at 22pt and 0.7. The maximum vertical shift of the bomb will be \shiftmult\reflength.

\documentclass[parskip]{scrartcl}
\usepackage{fourier} % for the bomb
\usepackage{lipsum}
\usepackage{stackengine}
\newlength\warndepth

\newsavebox\warnpar
\newlength\reflength
\setlength{\reflength}{22pt}
\def\shiftmult{0.7}
\newcommand\warningmacro[1]{%
\sbox\warnpar{\parbox[t]{0.88\textwidth}{%
\begin{sloppypar}%
#1%
\end{sloppypar}%
}
}%
\def\bigbomb{\Huge{\bomb}}%
\setlength\warndepth{\depthof{\usebox{\warnpar}}}%
\ifdim\warndepth>\reflength \setlength\warndepth{\reflength}\fi%
\belowbaseline[\shiftmult\warndepth-1.5\baselineskip]{%
\makebox[.12\textwidth][l]{\bigbomb}%
}%
\usebox{\warnpar}%
\hspace{-\parfillskip}%
}

\begin{document}
This is normal text%\lipsum[1]

\warningmacro{%
Let's see what happens when we get a two-line warning
Let's see what happens when we get a two-line warning
}

This is more normal text

\warningmacro{%
Let's see what happens when we get a three-line warning
Let's see what happens when we get a three-line warning
Let's see what happens when we get a three-line warning
Let's see what happens when we get a three-line warning
}

This is more normal text

\warningmacro{%
\lipsum[2]
}

\lipsum[3]

\warningmacro{%
This is just a short warning.
}

\lipsum[4]

\end{document}

• A good solution as well, but I'd have to edit about 19 kLOLC to switch from environments to warnings... – vwegert Jul 17 '13 at 14:41
• @vwegert That's what search and replace is for in editors ;^) – Steven B. Segletes Jul 17 '13 at 14:50