For an ideal monoatomic gas during any process T=kV, find out the molar heat capacity of the gas during the process. (Assume vibrational degree of freedom to be active)

Options

A. `(5)/(2)R`
B. 3R
C. `(7)/(2)`R
D. 4R

Answer

To solve the problem of finding the molar heat capacity of an ideal monoatomic gas during a process where \( T = kV \) (with \( k \) being a constant), we will follow these steps:

### Step 1: Understand the relationship between temperature and volume
Given that \( T = kV \), we can infer that temperature \( T \) is directly proportional to volume \( V \). This implies that as the volume changes, the temperature also changes proportionally.

**Hint:** Identify how changes in one variable affect another in thermodynamic processes.

### Step 2: Analyze the implications for pressure
From the ideal gas law, we know that for an ideal gas:
\[ PV = nRT \]
Substituting \( T = kV \) into the equation gives:
\[ P = \frac{nR(kV)}{V} = nkR \]
This indicates that the pressure \( P \) is constant because \( k \) is a constant and does not depend on \( V \).

**Hint:** Use the ideal gas law to relate pressure, volume, and temperature.

### Step 3: Determine the molar heat capacity at constant pressure
For an ideal monoatomic gas, the molar heat capacity at constant pressure \( C_P \) is given by:
\[ C_P = \frac{5}{2}R \]
This value comes from the degrees of freedom of a monoatomic gas, which includes 3 translational and 2 rotational degrees of freedom (and we assume vibrational degrees of freedom are active).

**Hint:** Recall the formula for molar heat capacity based on the degrees of freedom of the gas.

### Step 4: Conclude the molar heat capacity during the process
Since we established that the pressure is constant during the process (as derived from the relationship \( T = kV \)), the molar heat capacity of the gas during this process will be equal to the molar heat capacity at constant pressure:
\[ C = C_P = \frac{5}{2}R \]

**Hint:** Relate the conditions of the process to known heat capacity values.

### Final Answer
Thus, the molar heat capacity of the ideal monoatomic gas during the process where \( T = kV \) is:
\[ C = \frac{5}{2}R \]