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 (dQ) and internal energy (dU)
Given that \( dQ = 3dU \), we can express the change in internal energy in terms of the number of moles and the molar heat capacity.

### Step 2: Write the expression for change in internal energy (dU)
For an ideal monoatomic gas, the change in internal energy is given by:
\[ dU = nC_v dT \]
where \( n \) is the number of moles, \( C_v \) is the molar specific heat at constant volume, and \( dT \) is the change in temperature.

### Step 3: Substitute the value of \( C_v \)
For a monoatomic ideal gas, the molar specific heat at constant volume \( C_v \) is:
\[ C_v = \frac{3}{2}R \]
Thus, we can rewrite the equation for \( dU \):
\[ dU = n \left(\frac{3}{2}R\right) dT \]

### Step 4: Write the expression for heat (dQ)
The heat added to the system can be expressed as:
\[ dQ = nC dT \]
where \( C \) is the molar heat capacity for the process we are trying to find.

### Step 5: Substitute \( dU \) into the equation for \( dQ \)
From the relationship \( dQ = 3dU \), we can substitute for \( dU \):
\[ dQ = 3 \left(n \left(\frac{3}{2}R\right) dT\right) \]
This simplifies to:
\[ dQ = n \left(\frac{9}{2}R\right) dT \]

### Step 6: Equate the two expressions for dQ
Now we have two expressions for \( dQ \):
1. \( dQ = nC dT \)
2. \( dQ = n \left(\frac{9}{2}R\right) dT \)

Setting these equal to each other gives:
\[ nC dT = n \left(\frac{9}{2}R\right) dT \]

### Step 7: Cancel out common terms
Since \( n \) and \( dT \) are common in both sides, we can cancel them out:
\[ C = \frac{9}{2}R \]

### Step 8: Final answer
Thus, the molar heat capacity for the process is:
\[ C = 4.5R \]

### Summary
The molar heat capacity for the process where \( dQ = 3dU \) for an ideal monoatomic gas is \( 4.5R \).

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