For polytropic process `PV^(n)` = constant, molar heat capacity `(C_(m))` of an ideal gas is given by:

Options

A. `C_(v,m)+(R)/((n-1))`
B. `C_(v,m)+(R)/((1-n))`
C. `C_(v,m)+R`
D. `C_(p,m)+(R)/((n-1))`

Explanation

To find the molar heat capacity \( C_m \) of an ideal gas during a polytropic process defined by \( PV^n = \text{constant} \), we can follow these steps:

### Step 1: Understand the First Law of Thermodynamics
The first law of thermodynamics states:
\[ \Delta U = Q + W \]
where \( \Delta U \) is the change in internal energy, \( Q \) is the heat added to the system, and \( W \) is the work done by the system.

### Step 2: Express Internal Energy and Heat
For an ideal gas, the change in internal energy can be expressed as:
\[ \Delta U = n C_v \Delta T \]
where \( n \) is the number of moles, \( C_v \) is the molar heat capacity at constant volume, and \( \Delta T \) is the change in temperature.

The heat added to the system can be expressed in terms of the molar heat capacity \( C_m \):
\[ Q = n C_m \Delta T \]

### Step 3: Work Done in a Polytropic Process
The work done during a polytropic process can be expressed as:
\[ W = -P \Delta V \]
Using the relation \( PV^n = \text{constant} \), we can derive the relationship for work done.

### Step 4: Substitute into the First Law
Substituting the expressions for \( \Delta U \) and \( Q \) into the first law gives:
\[ n C_v \Delta T = n C_m \Delta T - P \Delta V \]

### Step 5: Rearranging the Equation
Rearranging the equation to isolate \( C_m \):
\[ n C_m \Delta T = n C_v \Delta T + P \Delta V \]
Dividing through by \( n \Delta T \):
\[ C_m = C_v + \frac{P \Delta V}{n \Delta T} \]

### Step 6: Finding \( \frac{\Delta V}{\Delta T} \)
To find \( \frac{\Delta V}{\Delta T} \), we can differentiate the polytropic equation \( PV^n = k \) with respect to \( T \):
1. From \( PV^n = k \), we can express \( P \) as \( P = \frac{k}{V^n} \).
2. Differentiate both sides with respect to \( T \) to find \( \frac{\Delta V}{\Delta T} \).

### Step 7: Substitute \( \frac{\Delta V}{\Delta T} \) into \( C_m \)
After differentiating, we can substitute \( \frac{\Delta V}{\Delta T} \) back into the equation for \( C_m \).

### Step 8: Final Expression for \( C_m \)
After simplification, we arrive at the final expression for the molar heat capacity during a polytropic process:
\[ C_m = C_v + \frac{R}{1 - n} \]
where \( R \) is the universal gas constant.

### Conclusion
Thus, the molar heat capacity \( C_m \) of an ideal gas during a polytropic process is given by:
\[ C_m = C_v + \frac{R}{1 - n} \]