Let dw be the work done in a quasistatic reversible thermodynamic process.
Which of the following statements about dw is correct?
a)
dw is a perfect differential if the process is isothermal.
b)
dw is a perfect differential if the process is adiabatic
c)
dw is always a perfect differential.
d)
dw cannot be a perfect differential.
Correct answer is option 'B'. Can you explain this answer?

Question

Let dw be the work done in a quasistatic reversible thermodynamic process.
Which of the following statements about dw is correct?

a)
dw is a perfect differential if the process is isothermal.
b)
dw is a perfect differential if the process is adiabatic
c)
dw is always a perfect differential.
d)
dw cannot be a perfect differential.

Correct answer is option 'B'. Can you explain this answer?

Answer

The first law of thermodynamics can be given dQ = dU + dW
here dQ and dW both are path function and dU is state function.
For making dW a perfect differential, we take dQ = 0 So dU + dW = 0 dW = –dU and dQ = 0 occur in adiabatic process, so, dW0 is a perfect differential if the process is adiabatic.

Answer

Explanation:
In thermodynamics, a perfect differential is a property that is exact and independent of the path taken during a process. If a differential is not perfect, it is referred to as an inexact differential.

To determine whether dw is a perfect differential, we need to consider the conditions under which it occurs in various thermodynamic processes.

a) dw is a perfect differential if the process is isothermal:
In an isothermal process, the temperature of the system remains constant. During such a process, the work done on or by the system can be expressed as:
dw = -PdV
where P is the pressure and dV is the change in volume.

Since the pressure and volume are both state functions, dw is a perfect differential in an isothermal process.

b) dw is a perfect differential if the process is adiabatic:
In an adiabatic process, there is no heat transfer between the system and its surroundings. The work done on or by the system can be expressed as:
dw = -PdV + TdS
where S is the entropy and T is the temperature.

In an adiabatic process, the change in entropy (dS) is zero. Therefore, the second term in the expression for dw becomes zero, and we are left with:
dw = -PdV

Similar to the isothermal case, since the pressure and volume are state functions, dw is a perfect differential in an adiabatic process.

c) dw is always a perfect differential:
This statement is incorrect. As seen from the explanations above, dw is a perfect differential only in isothermal and adiabatic processes. In other thermodynamic processes, such as isobaric or isochoric processes, dw is not a perfect differential.

d) dw cannot be a perfect differential:
This statement is also incorrect. As explained earlier, dw is a perfect differential in both isothermal and adiabatic processes.

Therefore, the correct answer is option 'b': dw is a perfect differential if the process is adiabatic.

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