Letters - November 2009

Confusing copperas

In his recent Chemlingo,¹ Peter Childs refers to the confusion that might relate to the names connected with iron compounds. I can add a related source of confusion. 'Copperas' is not a copper-containing substance but is, in fact, iron(II) sulfate, FeSO₄. 

The compilers of the Oxford English dictionary write: '...etymologically "Copperas" belonged to the copper salt, but in English use... it has always most commonly, and is now exclusively, applied to green copperas, the proto-sulfate of iron, also called green vitriol, used in dyeing, tanning and making ink'.² 

Alan Dronsfield, University of Derby

Entropy - a subtle concept

I offer several points to clarify remarks in the recent article What is entropy? by Cockcroft and Wheeler³  which teachers and their students should be aware of. 

1. The authors imply that their equation (i), ΔS  = Q _reversible/T, has been replaced by a 'more modern' expression, (ii) ΔS  = ΔH/T. If that is true, then it is a grave error. Equation (i) remains the fundamental definition of thermodynamic entropy; equation (ii) is a consequence of that definition for isothermal processes restricted to constant pressure. 

2.'The units of entropy are Joules per Kelvin'. Modern (SI) practice is for the names of units derived from the names of people to be lowercase, their symbols to be uppercase - eg joules per kelvin (JK^-1). 

3.'But the degree of disorder has no units'. The Boltzmann formula for entropy, S = k lnW, is an expression that relates the 'degree of disorder' W (specifically, the weight of the most probable configuration) to the entropy, and unambiguously has the same units as thermodynamic entropy. 

4.'Systems tends (sic) to adopt the broadest possible distribution'. They do not: the broadest possible distribution would be one in which all states are equally occupied (corresponding to infinite temperature). In fact, systems are typically found in the most probable distribution, not the broadest distribution. 

5.'The three laws of thermodynamics'. There are four laws. 

6. The authors come close to suggesting that standard entropies are for 298K and later in the article are explicit. However, temperature is not a part of the definition of standard states; 298K is simply the conventional temperature at which data are commonly tabulated. The erroneous inclusion of 298K into the definition of the standard state is widespread and should be avoided. (Examiners need to be made aware of the point too.) 

7.'Ludwig Boltzmann was able to show that as the energy is spread more widely in a system, the number of possible distributions increases to a peak... and then declines (Fig 1)'. The Boltzmann distribution is in fact an exponentially decaying function of the energy, and does not go through a peak of the kind indicated in the illustration. A peak in the distribution arises when molecular states exhibit degeneracies. Thus Fig 1 is appropriate to translation and rotation, but is not in general the form of the Boltzmann distribution. 

8.'Energy is constantly being exchanged between different energy levels'. This remark would make sense if the first 'energy' were replaced by 'molecules'. 

9.'The standard state of... a liquid is the pure liquid in equilibrium with its vapour... at 1 standard atmosphere'. Equilibrium between a liquid and its vapour is not relevant, and indeed not always accessible. The modern definition of standard state (which has been extant for several decades) is for a pressure of 1 bar not '1 standard atmosphere' (that is, 1 atm). 

10.'[The surroundings] are generally assumed to be large enough not to be significantly affected by changes occurring in the system'. Whereas the temperature, pressure, and volume of the surroundings are indeed unchanged, that is certainly not true of all their properties. The whole point of chemical thermodynamics, and the implicit role of the Gibbs energy, is to take into account the change in entropy of the surroundings.  

11.'It is important that misconceptions should be avoided, particularly the idea that energy represents the degree of disorder in the system'. The authors seem at this point to reject the Boltzmann expression for entropy and the whole of statistical thermodynamics, and also apparently undermine their earlier discussion. I have already pointed out that the fact that the degree of disorder (that is, W) is dimensionless is irrelevant. 

Entropy is a subtle concept (but no more subtle than energy), and although these comments might seem pedantic, students are likely to consider a subject difficult and confusing if its presentation is itself imprecise.  

Peter Atkins, University of Oxford

Entropy and temperature

Lawson Cockcroft and Graham Wheeler's article on entropy³ and Mike Shipton's letter on temperature⁴ prompt me to make several points. 

1. Thermodynamics is concerned with the behaviour of a system at a macroscopic level. Entropy (S) is therefore strictly a macroscopic property. It can be interpreted at a microscopic level, but when it is, temperature (T) needs to be interpreted too. 

2. The second law states that the entropy of an isolated system increases in any change. Whether this applies to the universe is an open question.⁵ When considering a non-isolated system, therefore, it is better to specify the surroundings and not refer to them as 'the rest of the universe'. 

3. At a microscopic level, entropy does not correspond to the number of microstates (Ω) a system ranges over, but to the logarithm of this. This is because change in entropy (δS) corresponds to the relative change in the number of states, δΩ/Ω.⁶

4. The quantity corresponding to temperature is the reciprocal of the rate of change of lnΩ with energy (ie 1/T  corresponds to this rate). 

5. The Boltzmann constant (k) links S  and 1/T to their statistical-mechanical analogues. The former have the units JK^-1 and K^-1 respectively, while the latter have the units 1 and J^-1. The units of k  are therefore (JK^-1)/1 or K^-1/J^-1, ie J K^-1. 

6. The simplest way of teaching chemical equilibrium is not to use thermodynamics, but Boltzmann's distribution law.⁷

Peter Nelson, University of Hull

Reading the runes of the summer exams

The summer exam season has come and gone. Call me a nerd if you like, but I always analyse the numbers with great interest to see what they tell us, not so much about grades as about the numbers of students who are coming forward to study sciences, particularly chemistry. 

At A-level, last year's growth in numbers taking chemistry has continued, albeit modestly by 2 per cent. The picture at AS-level is stronger, with a 7 per cent increase in chemistry AS numbers on the previous year. At GCSE the encouraging story was the continued growth of numbers taking Triple Science where, following last year's 30 per cent increase in numbers, this year we saw a further 20 per cent increase. This means there were 92,000 UK candidates for separate GCSE chemistry in 2009, up from 60,000 in 2007. 

There is a strong drive to have Triple Science available in all maintained schools. The Government has committed to 90 per cent of all state schools offering Triple Science by 2014 and, to achieve this, has funded the Triple Science Support Programme. For a long time, Triple Science was largely confined to selective and independent schools, but that is changing. In 2006, just 32 per cent of maintained schools offered Triple Science: the proportion is now over 50 per cent, and rising.

When schools are asked what benefits come from introducing Triple Science their answers include:

• young people enjoy the course;
• it is an excellent preparation for A-level;
• it satisfies an increasing parental demand;
• it creates a 'halo' effect that improves the ethos towards science in the school;
• science staff enjoy teaching the course, and it helps to recruit specialist teachers.

That Triple Science is a good preparation for A-level is hardly surprising, given the additional content. For example, separate GCSE chemistry gives more time to deal with the quantitative chemistry that is so important in A-level than is available with Double Science. Teachers say that the final modules of the separate GCSE sciences often provide an opportunity to capture students' interest in areas of science beyond GCSE. Schools that offer Triple Science consistently report that students who follow these courses are well prepared for A-level and get off to a flying start.

However, if choice is to be preserved, it is important that Double Science is still seen as a viable preparation for A-level. At present the majority of schools still use this combination as the entry point to A-level and this must remain if students are to have a serious prospect of keeping their A-level options open until 16. One of the risks of introducing Triple Science is that it might by default become the assumed preparation for A-level at the expense of the Double Science option: this must not be allowed to happen.

IGCSE (International GCSE) sciences are proving popular with some schools, especially in the independent and selective sectors and, while I cannot claim to be an expert on these courses, my impression is that while they are a good preparation for A-level, the curriculum tends to be narrow with limited emphasis on applications and the methods of scientific enquiry. My personal view is that modern chemistry GCSEs offer greater breadth and more modern content than IGCSEs - though it is essential that Ofqual puts pressure on awarding bodies to prevent standards in GCSE chemistry exams slipping.

When I was head of a large secondary school over 10 years ago, the school offered (and continues to offer) every pupil the option of studying Double or Triple Science. If students took the Triple Science option, they used one of their optional GCSE choices to do so, thus expanding their time allocation for science. However, not every school is in a position to do it this way, and schools have proved endlessly inventive in the models they have used. But, if Triple Science is introduced without allocating additional time to it, the risk is that teaching will be rushed, practical work will be reduced and at worst this could have the opposite effect to increasing the popularity of science.

I know that three years of growth do not make a renaissance, but there are encouraging signs, particularly with Triple Science and AS chemistry numbers, that suggest we may have begun to turn the corner in the popularity of our subject. Here's looking forward to the exam results in Summer 2010.

John Holman, director, National Science Learning Centre 

What value 'the total change in entropy'?

When, in the course of their article, Cockcroft and Wheeler³ address the question of the relation of entropy (S) to disorder, they make cautionary remarks which should be welcomed. Likewise, it should be welcomed that they express caution about invoking 'the total change in entropy' and 'the change in entropy of the surroundings'. Their reservations deserve some reinforcement. 

In arguments in terms of a 'total ΔS', it is taken that any value of ΔS for a composite entity, system + surroundings, will necessarily be either positive or zero. Such an assumption is legitimate only in certain circumstances, in particular when the composite entity constitutes an isolated system. 

Instead of starting from a relation which pertains in particular to isolated systems, it is more straightforward to begin from a generalisation of the relation for changes in entropy in reversible processes for closed systems which are not isolated. It is then possible to obtain inequalities for changes in Helmholtz energy (A) and Gibbs energy (G) by an argument which involves only minimal reference to external influences on the system. 

For processes which might not be reversible, occurring in a closed but non-isolated system, the generalisation required is that in the case of small changes (to the first order of small quantities, and for positive T): 

q  < TδS   (i) 

or the equivalent: 

δU - w<TδS   (ii) 

where δU  is the change in energy and w  is work done on the system. Then, the normal inequalities - ie  those for situations in which no work is done other than by virtue of any changes in volume - are: 

• for situations in which w may be equated to zero, equation (ii) becomes: 

δU <TδS  (iii) 

or  

δU - TδS < 0 (iv) 

giving the proposition that for closed systems at constant T  and V, with no 'other work', A can decrease but cannot increase; 

• for situations in which w may be equated to 

  -pδV, the corresponding expression is: 

δU  + pδV<TδS   (v) 

or  

δU + pδV - TδS< 0 (vi) 

giving the proposition that for closed systems at constant T and p, with no 'other work', G can decrease but cannot increase. 

An obvious reason for preferring to proceed in this way, rather than by reference to the 'entropy of the surroundings', is that in chemical thermodynamics the normal subject of concern is the situation in the reaction vessel, and not what might be happening in the thermostat in which it is sitting - or in any other surrounding objects. Cockcroft and Wheeler are fully justified in drawing attention to arguments which do not invoke a 'total ΔS'. 

P. G. Wright, Dundee