Question

An ideal gas undergoes a quasi static, reversible process in which its molar heat capacity Cremains constant. If during this process the relation of pressure $P$ and volume $V$ is given by $PV^n =$ constant, then $n$ is given by (Here $C_P$ and $C_V$ are molar specific heat at constant pressure and constant volume, respectively) :

• A. $n = \frac{C_P}{C_V} $
• B. $n = \frac{C - C_P}{C - C_V} $
• C. $n = \frac{C_P - C }{C - C_V} $
• D. $n = \frac{C - C_V}{C - C_P}$

Answer 1

The Correct Option is B

The problem involves determining the value of $n$ in the equation $PV^n = \text{constant}$ for an ideal gas undergoing a reversible process where the molar heat capacity $C$ remains constant.

Let's explore the step-by-step solution:

1. An ideal gas follows the relation $PV = nRT$. In a quasi-static process, if $PV^n = \text{constant}$, such a process is called a polytropic process. The parameter $n$ is used to define the process's specific nature.

2. The relation between the molar heat capacities $C_P$ and $C_V$ is given by:

   C_P - C_V = R

3. In a polytropic process where heat capacity is constant, the effective molar heat capacity $C$ is introduced.

4. The relation for polytropic index $n$ in terms of the heat capacities can be derived using the first law of thermodynamics and the equation of state:

   The heat exchanged in a polytropic process per mole is:

   dQ = C_V dT + \left( \frac{P \cdot dV + V \cdot dP}{1 - n} \right)

   Since dQ = C \cdot dT, equating the expressions yields a relation leading to:

   n = \frac{C - C_P}{C - C_V}

5. Therefore, the correct answer is:

   n = \frac{C - C_P}{C - C_V}

Thus, the correct option is indeed n = \frac{C - C_P}{C - C_V}. This expression shows that the polytropic index $n$ depends on how the specific heat capacity varies from conventional heat capacities at constant pressure and volume.