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 derive 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 relationship between internal energy, heat, and work
According to the first law of thermodynamics, the change in internal energy \( dU \) can be expressed as:
\[ dU = \delta Q - \delta W \]
For an ideal gas, the change in internal energy can also be expressed as:
\[ dU = nC_{V_m} dT \]
where \( C_{V_m} \) is the molar heat capacity at constant volume.

### Step 2: Express heat transfer in terms of molar heat capacity
The heat added to the system can be expressed as:
\[ \delta Q = nC_m dT \]
where \( C_m \) is the molar heat capacity during the process.

### Step 3: Work done in a polytropic process
The work done \( \delta W \) in a polytropic process can be expressed as:
\[ \delta W = PdV \]
Using the ideal gas law, we can relate pressure and volume:
\[ P = \frac{nRT}{V} \]
Thus, the work done can be rewritten as:
\[ \delta W = \frac{nRT}{V} dV \]

### Step 4: Combine the equations
From the first law, we have:
\[ nC_{V_m} dT = nC_m dT - \frac{nRT}{V} dV \]
Rearranging gives:
\[ nC_{V_m} dT + \frac{nRT}{V} dV = nC_m dT \]

### Step 5: Differentiate the polytropic equation
From the polytropic process, we have:
\[ PV^n = K \quad \text{(constant)} \]
Differentiating this with respect to \( V \) gives:
\[ P \cdot nV^{n-1} dV + V^n dP = 0 \]
Rearranging gives:
\[ dP = -\frac{nP}{V} dV \]

### Step 6: Substitute \( dP \) into the heat capacity equation
Substituting \( dP \) back into the equation we derived from the first law, we get:
\[ nC_{V_m} dT + \frac{nRT}{V} dV = nC_m dT \]
Now, we can express \( dT \) in terms of \( dV \) using the relationships from the polytropic process.

### Step 7: Solve for \( C_m \)
After manipulating the equations, we arrive at:
\[ C_m = C_{V_m} + \frac{R}{1 - n} \]

### Final Expression
Thus, the molar heat capacity \( C_m \) for an ideal gas undergoing a polytropic process is given by:
\[ C_m = C_{V_m} + \frac{R}{1 - n} \]