An ideal diatomic gas is expanded so that the amount of heat transferred to the gas is equal to the decrease in its internal energy.
The molar specific heat of the gas in this process is given by `C` whose value is

Options

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

Explanation

To solve the problem, we need to analyze the relationship between heat transfer, internal energy change, and the specific heat capacities of an ideal diatomic gas during an expansion process.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We are given that the heat transferred to the gas (dq) is equal to the negative of the change in internal energy (du). This can be expressed mathematically as:
\[ dq = -du \]

2. **Expressing Heat Transfer**:
The heat transferred to the gas can be expressed in terms of the molar specific heat capacity (C) as follows:
\[ dq = nC dT \]
where:
- \( n \) = number of moles of the gas
- \( C \) = molar specific heat capacity of the gas in this process
- \( dT \) = change in temperature

3. **Expressing Change in Internal Energy**:
For an ideal gas, the change in internal energy (du) is given by:
\[ du = nC_v dT \]
where:
- \( C_v \) = molar specific heat capacity at constant volume

4. **Setting Up the Equation**:
From the first step, we know that:
\[ dq = -du \]
Substituting the expressions for \( dq \) and \( du \):
\[ nC dT = -nC_v dT \]

5. **Cancelling Common Terms**:
Since \( n \) and \( dT \) are common on both sides of the equation, we can cancel them out (assuming \( n \neq 0 \) and \( dT \neq 0 \)):
\[ C = -C_v \]

6. **Using the Value of \( C_v \)**:
For a diatomic gas, the molar specific heat at constant volume \( C_v \) is given by:
\[ C_v = \frac{5}{2} R \]
where \( R \) is the universal gas constant.

7. **Finding the Value of \( C \)**:
Substituting the value of \( C_v \) into the equation \( C = -C_v \):
\[ C = -\frac{5}{2} R \]

8. **Conclusion**:
Therefore, the value of the molar specific heat \( C \) for the gas in this process is:
\[ C = -\frac{5}{2} R \]

### Final Answer:
The value of \( C \) is \( -\frac{5R}{2} \).