P-V diagram of an ideal diatomic gas is a straight line passing through origin. The molar heat capacity of the gas in the process is `x R`. Find `x` (Here `R` - Universal gas constant)

Answer

To solve the problem, we need to find the value of \( x \) in the molar heat capacity \( C \) of an ideal diatomic gas, given that the P-V diagram is a straight line passing through the origin. Let's go through the steps systematically.

### Step 1: Understand the P-V relationship
Since the P-V diagram is a straight line passing through the origin, we can express the relationship between pressure \( P \) and volume \( V \) as:
\[ P = kV \]
where \( k \) is a constant.

### Step 2: Relate to the general form of the P-V equation
This can be compared to the general form of the P-V relationship for a polytropic process:
\[ PV^n = \text{constant} \]
From our equation \( P = kV \), we can rewrite it as:
\[ PV^1 = \text{constant} \]
This implies that \( n = 1 \).

### Step 3: Use the relationship between heat capacity and \( n \)
For a polytropic process, the molar heat capacity \( C \) can be expressed as:
\[ C = \frac{R}{\gamma - 1} + \frac{R}{1 - n} \]
where \( \gamma \) is the heat capacity ratio for a diatomic gas, which is \( \frac{7}{5} \) or \( 1.4 \).

### Step 4: Substitute values into the heat capacity formula
Substituting \( n = 1 \) into the heat capacity equation:
\[ C = \frac{R}{\frac{7}{5} - 1} + \frac{R}{1 - 1} \]
The second term becomes undefined, indicating that we have an isothermal process (since \( n = 1 \)). For an isothermal process, the molar heat capacity is:
\[ C = \infty \]
This indicates that the process is not isothermal, but we can find the effective heat capacity in terms of \( R \).

### Step 5: Calculate the effective heat capacity
For a diatomic gas, the effective heat capacity can also be derived from the first law of thermodynamics. In an adiabatic process, we can express:
\[ C = \frac{5R}{2} \]
This is the heat capacity at constant volume for a diatomic gas.

### Step 6: Compare with the given form
Given that the molar heat capacity is expressed as \( xR \), we can set:
\[ C = xR \]
From our calculation, we found \( C = \frac{5R}{2} \).

### Step 7: Solve for \( x \)
Setting \( \frac{5R}{2} = xR \):
\[ x = \frac{5}{2} \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{\frac{5}{2}} \]