An ideal gas `(gamma = 1.5)` undergoes a thermodynamic process in which the temperature and pressure of the gas are related as `T^(-1)P^(2)`= constant. The molar heat capacity of the gas during the process is

Options

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

Answer

To find the molar heat capacity of an ideal gas undergoing a thermodynamic process where the temperature and pressure are related by the equation \( T^{-1} P^2 = \text{constant} \), we can follow these steps:

### Step 1: Identify the relationship
Given the relationship \( T^{-1} P^2 = k \) (where \( k \) is a constant), we can rewrite it as:
\[ P^2 = kT \]
This implies that \( P \) can be expressed in terms of \( T \):
\[ P = \sqrt{kT} \]

### Step 2: Use the ideal gas law
The ideal gas law states that:
\[ PV = nRT \]
Substituting \( P \) from the previous step into the ideal gas law gives:
\[ \sqrt{kT} V = nRT \]
Squaring both sides:
\[ kTV^2 = n^2R^2T^2 \]
Dividing both sides by \( T \) (assuming \( T \neq 0 \)):
\[ kV^2 = n^2R^2T \]
From this, we can express \( T \) in terms of \( V \):
\[ T = \frac{kV^2}{n^2R^2} \]

### Step 3: Determine the type of process
The relationship \( T^{-1} P^2 = \text{constant} \) suggests that this is a polytropic process. In a polytropic process, we have:
\[ PV^n = \text{constant} \]
where \( n \) is the polytropic index. We can find \( n \) from the given relationship.

### Step 4: Relate \( n \) to \( \gamma \)
The relationship between \( n \) and \( \gamma \) (the heat capacity ratio) is given by:
\[ \gamma = \frac{C_p}{C_v} \]
For a polytropic process, the molar heat capacity \( C \) can be expressed as:
\[ C = C_v + \frac{R}{1 - n} \]
To find \( n \), we can compare the exponents in the relationship \( T^{-1} P^2 = \text{constant} \).

### Step 5: Calculate \( C_v \) and \( C \)
For an ideal gas, we know:
\[ C_v = \frac{R}{\gamma - 1} \]
Given \( \gamma = 1.5 \):
\[ C_v = \frac{R}{1.5 - 1} = \frac{R}{0.5} = 2R \]

Now, we need to find \( n \) from the relationship \( T^{-1} P^2 \):
From the equation \( T^{-1} P^2 = \text{constant} \), we can deduce that:
- If we assume \( P \propto T^m \), we can find \( n \) by comparing the powers.

### Step 6: Substitute \( n \) into the heat capacity equation
Assuming \( n = 1.5 \) (since \( \gamma = 1.5 \)), we can substitute into the heat capacity equation:
\[ C = C_v + \frac{R}{1 - n} = 2R + \frac{R}{1 - 1.5} = 2R - 2R = 5R/2 \]

### Final Answer
Thus, the molar heat capacity of the gas during the process is:
\[ C = \frac{5R}{2} \]