Question

For an ideal gas, consider only P-V work in going from an initial state X to the final state Z . The final state Z can be reached by either of the two paths shown in the figure. Which of the following choice(s) is(are) correct?
[Take $△S$ as change in entropy and $w$ as work done]

• A. $△S_{X\to Z}=△S_{X\to Y}+△S_{Y\to Z}$
• B. $w_{X\to Z}=w_{X\to Y}+w_{Y\to Z}$
• C. $w_{X\to Y\to Z}\ne w_{X\to Y}$
• D. $△S_{X\to Y\to Z}=△S_{X\to Y}$

Answer 1

The correct option is A $△S_{X\to Z}=△S_{X\to Y}+△S_{Y\to Z}$

As $△$S does not depend on path and only depends on initial and final state of the system i.e., it is a state function thus $△S_{X\to Z}=△S_{X\to Y}+△S_{Y\to Z}$

As we know that work is not a state function and depends on path,
Thus, $w_{X\to Z}\ne w_{X\to Y}+w_{Y\to Z}$

$w_{X\to Y}=PdV$ ($P$ is constant)
$w_{Y\to Z}=0$ (as V is constant)

$w_{X\to Y\to Z}=w_{X\to Y}+w_{Y\to Z}$
As $w_{Y\to Z}=0,hence{w}_{X\to Y\to Z}=w_{X\to Y}$