Ideal monoatomic gas is taken through a process `dQ = 2dU`. Find the molar heat capacity (in terms of `R)` for the process? (where `dQ` is heat supplied and `dU` is change in internla energy)

Answer

To solve the problem, we need to find the molar heat capacity \( C \) for an ideal monoatomic gas when the heat supplied \( dQ \) is twice the change in internal energy \( dU \).

### Step-by-step Solution:

1. **Understanding the relationship between \( dQ \) and \( dU \)**:
Given that \( dQ = 2 dU \), we can express \( dU \) in terms of the molar specific heat at constant volume \( C_v \):
\[ dU = n C_v dT \]
where \( n \) is the number of moles and \( dT \) is the change in temperature.

2. **Substituting \( dU \) into the equation for \( dQ \)**:
Substitute \( dU \) into the equation for \( dQ \):
\[ dQ = 2 dU = 2 n C_v dT \]

3. **Using the definition of heat for any process**:
The heat supplied in any process can also be expressed as:
\[ dQ = n C dT \]
where \( C \) is the molar heat capacity for the process.

4. **Equating the two expressions for \( dQ \)**:
Now we can set the two expressions for \( dQ \) equal to each other:
\[ n C dT = 2 n C_v dT \]

5. **Canceling common terms**:
Since \( n \) and \( dT \) are common in both sides, we can cancel them out (assuming \( n \neq 0 \) and \( dT \neq 0 \)):
\[ C = 2 C_v \]

6. **Finding \( C_v \) for a monoatomic gas**:
For a monoatomic ideal gas, the molar specific heat at constant volume \( C_v \) is given by:
\[ C_v = \frac{3R}{2} \]
where \( R \) is the universal gas constant.

7. **Substituting \( C_v \) into the equation for \( C \)**:
Now substituting the value of \( C_v \) into the equation for \( C \):
\[ C = 2 C_v = 2 \left(\frac{3R}{2}\right) = 3R \]

### Final Answer:
The molar heat capacity \( C \) for the process is:
\[ C = 3R \]