Chapter 20, Problem 1Q

Point i in Fig. 20-19 represents the initial state of an ideal gas at temperature T. Taking algebraic signs into account, rank the entropy changes that the gas undergoes as it moves, successively and reversibly, from point i to points a, b, c, and d, greatest first.

Figure 20-19 Question 1.

To determine

To rank:

The entropy changes that the gas undergoes as it moves successively and reversibly from point $i$ to points $a$, $b$, $c$, and $d$ greatest first.

Answer to Problem 1Q

Solution:

The ranking of the change of entropy of the gas is $b>a>c>d$.

Explanation of Solution

1) Concept:

We can compare the entropy changes of the gas at different points from specific heat and temperature at that points using the relation between change in entropy, specific heat, and temperature at the given point.

2) Formulae:

i) ${∆S}_{gas}=n{C}_{P}ln\frac{T_f}{T_i}$

ii) ${∆S}_{gas}=n{C}_{V}ln\frac{T_f}{T_i}$

3) Given:

The figure showing point $i$ is the initial state of ideal gas at temperature $T$ which undergoes a reversible process.

4) Calculations:

In $P-V$ plot, there is initial point i with temperature $T$. There are two isothermal states with temperatures

$T+∆T and T-∆T .$

There are four processes in which two of them are at a higher temperature and two of them are at a lower temperature. The points $a$ and $c$ are at constant volume. So ${ V}_f=V_i$.

The process in which heat is absorbed leads to an increase in the temperature and entropy of the gas. S, o the change of entropy of the gas is positive. $∆S>0$.

The process that releases energy in the form of heat leads to decrease in entropy. i.e. $∆S<0$

The molar specific heat at constant pressure is greater than constant volume, i.e. $C_P>C_V$.

The points $a$ and $b$ are at higher temperature.

For an isobaric process,

${∆S}_{gas}=n{C}_{P}ln\frac{T_f}{T_i}$

For an isochoric process,

${∆S}_{gas}=n{C}_{V}ln\frac{T_f}{T_i}$

So the change of entropy is larger for the isobaric process. ${∆S}_P>{∆S}_V$.

Hence, entropy change is greater at point b and d than at point a and c.

Since b is at a higher temperature than that of d and a is at a higher temperature than that of c.

 ${∆S}_b>{∆S}_d$ and  ${∆S}_a>{∆S}_c$

Therefore, the ranking of the entropy changes of the gas is $b>a>c>d$

Conclusion:

Entropy change depends on temperature and specific heat of an ideal gas.