Ideal mono-atomic gas is taken through process such that dQ = 3dU. The molar heat capacity for process is:

Options

A. `3 R`
B. `4.5 R`
C. `4 R`
D. ` 2 R`

Explanation

To find the molar heat capacity for the process where \( dQ = 3dU \) for an ideal monoatomic gas, we can follow these steps:

### Step 1: Understand the relationship between heat, internal energy, and temperature change.
The molar heat capacity \( C \) is defined as:
\[ C = \frac{dQ}{dT} \]
where \( dQ \) is the heat added to the system and \( dT \) is the change in temperature.

### Step 2: Use the given relationship \( dQ = 3dU \).
From the problem, we know:
\[ dQ = 3dU \]
This implies that the heat added to the system is three times the change in internal energy.

### Step 3: Relate internal energy change to temperature change.
For an ideal monoatomic gas, the change in internal energy \( dU \) is given by:
\[ dU = \frac{f}{2} nR dT \]
where \( f \) is the degrees of freedom. For a monoatomic gas, \( f = 3 \). Therefore:
\[ dU = \frac{3}{2} nR dT \]

### Step 4: Substitute \( dU \) into the equation for \( dQ \).
Substituting \( dU \) into the equation for \( dQ \):
\[ dQ = 3dU = 3 \left( \frac{3}{2} nR dT \right) = \frac{9}{2} nR dT \]

### Step 5: Substitute \( dQ \) into the molar heat capacity equation.
Now, substituting \( dQ \) into the molar heat capacity equation:
\[ C = \frac{dQ}{dT} = \frac{\frac{9}{2} nR dT}{dT} = \frac{9}{2} nR \]
Since we are considering one mole of gas (\( n = 1 \)):
\[ C = \frac{9}{2} R \]

### Step 6: Simplify the result.
Thus, the molar heat capacity for the process is:
\[ C = 4.5 R \]

### Final Answer:
The molar heat capacity for the process is \( \frac{9}{2} R \) or \( 4.5 R \).

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