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 internal energy)

Options

A. 5 R
B. 3 R
C. R
D. None of these

Answer

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

### Step 1: Understand the relationship between heat, internal energy, and molar heat capacity
The heat supplied \( dQ \) can be expressed in terms of the molar heat capacity \( C \) and the change in temperature \( dT \):
\[ dQ = nC dT \]
where \( n \) is the number of moles of the gas.

### Step 2: Relate the change in internal energy to the molar heat capacity at constant volume
For an ideal gas, the change in internal energy \( dU \) is given by:
\[ dU = nC_V dT \]
where \( C_V \) is the molar heat capacity at constant volume.

### Step 3: Substitute the expression for \( dU \) into the equation for \( dQ \)
Given that \( dQ = 2dU \), we can substitute \( dU \):
\[ dQ = 2(nC_V dT) \]
Thus, we have:
\[ nC dT = 2nC_V dT \]

### Step 4: Cancel \( n \) and \( dT \) from both sides
Assuming \( n \) and \( dT \) are not zero, we can simplify the equation:
\[ C = 2C_V \]

### Step 5: Determine \( C_V \) for a monatomic ideal gas
For a monatomic ideal gas, the molar heat capacity at constant volume is given by:
\[ C_V = \frac{3}{2}R \]
where \( R \) is the universal gas constant.

### Step 6: Substitute \( C_V \) into the equation for \( C \)
Now substituting \( C_V \) into the equation for \( C \):
\[ C = 2\left(\frac{3}{2}R\right) = 3R \]

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