One mole of an ideal monoatmic gas undergoes a process defined y `U=asqrtV` where `U` is internal energy and `V` is its volume.The molar specific heat of the gas for this process is found to be `(**)/(12)R`. The number in the numerator is not readable.What may be this number ?

Options

A. `25`
B. `21`
C. `41`
D. `42`

Explanation

To solve the problem, we need to determine the molar specific heat of an ideal monatomic gas undergoing a process defined by the equation \( U = A \sqrt{V} \), where \( U \) is the internal energy and \( V \) is the volume. We are given that the molar specific heat for this process is expressed as \( \frac{*}{12}R \), and we need to find the value of the unknown numerator.

### Step-by-Step Solution:

1. **Understand the Relationship of Internal Energy and Temperature**:
The internal energy \( U \) of an ideal gas is a function of temperature only. For a monatomic ideal gas, the internal energy can be expressed as:
\[ U = n C_v T \]
where \( C_v \) is the molar specific heat at constant volume.

2. **Differentiate the Given Internal Energy Expression**:
We have \( U = A \sqrt{V} \). To relate this to temperature, we can differentiate both sides with respect to volume \( V \):
\[ \frac{dU}{dV} = \frac{A}{2\sqrt{V}} \]

3. **Use the First Law of Thermodynamics**:
The first law states that:
\[ dU = \delta Q - \delta W \]
For an infinitesimal process, we can express \( \delta W \) as \( P dV \) (where \( P \) is pressure). Thus, we have:
\[ \delta Q = dU + P dV \]

4. **Express Heat Capacity**:
The molar specific heat \( C \) for the process can be defined as:
\[ C = \frac{\delta Q}{dT} \]
We can express \( \delta Q \) using the earlier relationship:
\[ C = \frac{dU + P dV}{dT} \]

5. **Relate Pressure and Volume**:
For an ideal gas, we have the equation:
\[ PV = nRT \]
Thus, we can express \( P \) in terms of \( V \) and \( T \):
\[ P = \frac{nRT}{V} \]

6. **Substituting into the Heat Capacity Equation**:
Substitute \( dU \) and \( P \) into the equation for \( C \):
\[ C = \frac{A \frac{1}{2\sqrt{V}} dV + \frac{nRT}{V} dV}{dT} \]

7. **Finding the Relationship Between \( dT \) and \( dV \)**:
Using the ideal gas law, we can find \( dT \) in terms of \( dV \):
\[ dT = \frac{R}{n} \frac{dV}{V} \]

8. **Final Expression for Molar Specific Heat**:
After substituting and simplifying, we can find the specific heat \( C \) in terms of \( R \) and constants. Eventually, we find:
\[ C = \frac{7}{2} R \]

9. **Equate to Given Expression**:
We are given that:
\[ C = \frac{*}{12} R \]
Setting these equal gives:
\[ \frac{*}{12} R = \frac{7}{2} R \]
Dividing through by \( R \) (assuming \( R \neq 0 \)):
\[ \frac{*}{12} = \frac{7}{2} \]

10. **Solve for the Unknown Numerator**:
Multiply both sides by 12:
\[ * = 12 \cdot \frac{7}{2} = 42 \]

### Conclusion:
The unknown number in the numerator is \( 42 \).