Calculating the Change in Entropy of a Reaction

SAMPLE PROBLEM 20.1 Predicting Relative Entropy Values 

Problem Choose the member with the higher entropy in each of the following pairs, and justify your choice [assume constant temperature, except in part (e)]  

Plan In general, we know that particles with more freedom of motion or more dispersed 

FOLLOW-UP PROBLEM 20.1 For 1 mol of substance at a given temperature, select the member in each pair with the higher entropy, and give the reason for your choice  

In addition to understanding trends in S° values for different substances or for the same substance in different phases, chemists are especially interested in learning how to predict the sign and calculate the value of the change in entropy as a reaction occurs. 

Entropy Changes in the System Standard Entropy of Reaction (AS°xn) 

CHAPTER 20 Thermodynamics Entropy, Free Energy, and the Direction of Chemical Reactions 

For example, when H2(g) and l2(s) form HI(g), the total number of moles of substance stays the same, but we predict that the entropy increases because the number of moles of gas increases  

Hzfg) + h(s) — 2HI(g) ASScn products cl clalu b When ammonia forms frc n its elements, 4 mol of gas produces 2 mol of gas, so we predict that the entropy decreases  

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To calculate the change in entropy that accompanies a reaction, we need to know the molar entropies of all the substances taking part then we calculate the difference between the entropies of the products and those of the reactants. More specifically, the standard reaction entropy, AS°, is the difference between the standard molar entropies of the products and those of the reactants, taking into account their stoichiometric coefficients ... 

Bywater and Roberts have made a detailed comparison of such reactions for a series of compounds from which the changes in entropy of activation can be considered in some detail. Table XII.3 lists the calculated standard (300°K, 1 atm) molar entropies of translation and rotation of complexes made of H or CH3 and RH and the standard entropy changes in the reactions H + HR H2R, CII3 + HR = CH4R (neglecting vibration) calculated by these authors. 

Since the last two terms in Eq. (11.74) can be calculated from heat capacities and heats of reaction, the only unknown quantity is ASS, the change in entropy of the reaction at 0 K. In 1906, Nernst suggested that for all chemical reactions involving pure crystalline solids, ASS is zero at the absolute zero the Nernst heat theorem. In 1913, Planck suggested that the reason that ASS is zero is that the entropy of each individual substance taking part in such a reaction is zero. It is clear that Planck s statement includes the Nernst theorem. 

To calculate the entropy change in the surroundings when a reaction takes place at constant pressure, we use eqn 2.6, interpreting the AH in that expression as the reaction enthalpy. For example, for the formation of the NAD -enzyme complex discussed above, with AjH = -24.2 kj mol", the change in entropy of the surroundings (which are maintained at 25 C, the same temperature as the reaction mixture) is... 

In Chapter 6, we learned how to calculate standard changes in enthalpy (AH°xJ for chemical reactions. We now turn to calculating standard changes in entropy for chemical reactions. Recall from Section 6.9 that the standard enthalpy change for a reaction (AH°x ) is the change in enthalpy for a process in which all reactants and products are in their standard states. Recall also the definition of the standard state ... 

The entropy of any chemical substance increases as temperature increases. These changes in entropy as a function of temperature can be calculated, but the techniques require calculus. Fortunately, temperature affects the entropies of reactants and products similarly. The absolute entropy of every substance increases with temperature, but the entropy of the reactants often changes with temperature by almost the same amount as the entropy of the products. This means that the temperature effect on the entropy change for a reaction is usually small enough that we can consider A Sj-eaction he independent of temperature. 

As we mentioned, it is necessary to have information about the standard enthalpy change for a reaction as well as the standard entropies of the reactants and products to calculate the change in Gibbs function. At some temperature T, A// j can be obtained from Af/Z of each of the substances involved in the transformation. Data on the standard enthalpies of formation are tabulated in either of two ways. One method is to list Af/Z at some convenient temperature, such as 25°C, or at a series of temperatures. Tables 4.2 through 4.5 contain values of AfZ/ at 298.15 K. Values at temperatures not listed are calculated with the aid of heat capacity equations, whose coefficients are given in Table 4.8. 

A quantum mechanical approach to ion-water interactions has the up side that it is the kind of development one might think of as inevitable. On the other hand, there is a fundamental difficulty that attends all quantum mechanical approaches to reactions in chemistry. It is that they concern potential energy and do not account for the entropic aspects of the situation. The importance of the latter (cf. the basic thermodynamic equation AG = AH - TAS) depends on temperature, so that at T = 0, the change in entropy in a reaction, AS, has no effect. However, in calculations of solvation at ordinary temperatures, the inaease in order brought about by the effect of the ion on the water molecules is an essential feature of the situation. Thus, a quantum mechanical approach to solvation can provide information on the energy of individual ion-water interactions (clusters in the gas phase have also been calculated), but one has to ask whether it is relevant to solution chemistry. 

We now pass to the explicit calculation of entropy production. We shall consider here the very important special case in which mechanical and thermal equilibrium are already established. Mechanical equilibrium excludes the production of entropy by viscous flow, while uniformity of temperature, which is necessary for thermal equilibrium, excludes the internal production of entropy arising from the transport of heat between two regions at different temperatures. Similarly we assume that diffusion equilibrium has been attained within each phase of the system. The only production of entropy which can take place in a system of this kind is that associated with chemical reactions, with the transport of matter from one phase to another, or in general with any change which can be expressed in terms of a reaction co-ordinate... 

It should be noted that the barriers presented here only correspond to the enthalpy part (AiG ) of the free energy of activation (AG = AiG — TAS ). The entropic part is usually much harder to calculate accurately. It is therefore assumed that AG AiG, which in many cases is a quite valid assumption because the change in entropy (AG ) in going from the reactant to the transition state often is small. Obvious exceptions are when a gas molecule is bound or released in the reaction. Special care has to be taken in these cases. 

Whether or not a chemical reaction will proceed spontaneously is determined by the change in standard free energy (AG" ). The standard free energy is calculated from the change in enthalpy (H" ) and the change in entropy (S" ) by the Gibbs free energy equation AG° = AH - TAS This calcrdation assumes that the reaction is done under standard conditions in solution with a concentration of 1 M and at 298.15 K. 

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