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 find the molar heat capacity for the process where \( dQ = 2 dU \) for an ideal monoatomic gas, we can follow these steps:

### Step 1: Understand the relationship between heat, internal energy, and molar heat capacity
We know that the change in internal energy \( dU \) for an ideal gas can be expressed as:
\[ dU = N C_v dT \]
where \( N \) is the number of moles, \( C_v \) is the molar heat capacity at constant volume, and \( dT \) is the change in temperature.

### Step 2: Substitute \( dU \) into the equation for \( dQ \)
Given the relationship \( dQ = 2 dU \), we can substitute \( dU \):
\[ dQ = 2 (N C_v dT) = 2N C_v dT \]

### Step 3: Express \( dQ \) in terms of molar heat capacity
The heat supplied \( dQ \) can also be expressed in terms of the molar heat capacity \( C \):
\[ dQ = N C dT \]
where \( C \) is the molar heat capacity for the process.

### Step 4: Set the two expressions for \( dQ \) equal to each other
Now we can set the two expressions for \( dQ \) equal to each other:
\[ N C dT = 2N C_v dT \]

### Step 5: Cancel \( N \) and \( dT \) from both sides
Assuming \( N \) and \( dT \) are not zero, we can cancel them:
\[ C = 2 C_v \]

### Step 6: Find \( C_v \) for a monoatomic gas
For a monoatomic ideal gas, the molar heat capacity at constant volume \( C_v \) is given by:
\[ C_v = \frac{f R}{2} \]
where \( f \) is the degrees of freedom. For a monoatomic gas, \( f = 3 \):
\[ C_v = \frac{3R}{2} \]

### Step 7: Substitute \( C_v \) back into the equation for \( C \)
Now substituting \( C_v \) into the equation for \( C \):
\[ C = 2 C_v = 2 \left(\frac{3R}{2}\right) = 3R \]

### Conclusion
Thus, the molar heat capacity for the process is:
\[ \boxed{3R} \]

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