1 mole of rigid diatomic gas performs a work of Q/5 when heat Q is supplied to it. The molar heat capacity of the gas during this transformation is `(xR)/(8)`, The value of x is ........... [K = universal gas constant]

Answer

To solve the problem, we will use the first law of thermodynamics and the definitions of heat capacity and internal energy for a diatomic gas. The steps are as follows:

### Step 1: Understand the first law of thermodynamics
The first law of thermodynamics states:
\[ Q = W + \Delta U \]
where:
- \( Q \) is the heat supplied,
- \( W \) is the work done,
- \( \Delta U \) is the change in internal energy.

### Step 2: Identify the work done and change in internal energy
From the problem, we know:
- Work done \( W = \frac{Q}{5} \)
- Therefore, the change in internal energy can be calculated as:
\[ \Delta U = Q - W = Q - \frac{Q}{5} = \frac{5Q}{5} - \frac{Q}{5} = \frac{4Q}{5} \]

### Step 3: Relate heat supplied to molar heat capacity
The heat supplied can also be expressed in terms of molar heat capacity \( C \) and change in temperature \( \Delta T \):
\[ Q = N C \Delta T \]
For 1 mole of gas (\( N = 1 \)):
\[ Q = C \Delta T \quad \text{(Equation 1)} \]

### Step 4: Calculate change in internal energy for a diatomic gas
For a diatomic gas, the change in internal energy is given by:
\[ \Delta U = \frac{F}{2} N R \Delta T \]
where \( F \) (degrees of freedom) for a diatomic gas is 5:
\[ \Delta U = \frac{5}{2} N R \Delta T = \frac{5}{2} R \Delta T \quad \text{(Equation 2)} \]

### Step 5: Set up the equation using the first law
From the first law, we have:
\[ \Delta U = \frac{4Q}{5} \]
Substituting Equation 1 into this gives:
\[ \frac{5}{2} R \Delta T = \frac{4}{5} C \Delta T \]

### Step 6: Cancel \( \Delta T \) and solve for \( C \)
Assuming \( \Delta T \neq 0 \), we can cancel \( \Delta T \) from both sides:
\[ \frac{5}{2} R = \frac{4}{5} C \]
Now, rearranging for \( C \):
\[ C = \frac{5}{2} R \cdot \frac{5}{4} = \frac{25}{8} R \]

### Step 7: Compare with given form of heat capacity
The problem states that the molar heat capacity is given by:
\[ C = \frac{xR}{8} \]
Setting the two expressions for \( C \) equal:
\[ \frac{25}{8} R = \frac{xR}{8} \]

### Step 8: Solve for \( x \)
By comparing coefficients:
\[ 25 = x \]
Thus, the value of \( x \) is:
\[ x = 25 \]

### Final Answer
The value of \( x \) is \( 25 \).

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