An ideal diatomic gas `(gamma=7/5)` undergoes a process in which its internal energy relates to the volume as `U=alphasqrtV`, where `alpha` is a constant.
(a) Find the work performed by the gas to increase its internal energy by 100J.
(b) Find the molar specific heat of the gas.

Explanation

To solve the problem step by step, we will address both parts (a) and (b) of the question.

### Part (a): Find the work performed by the gas to increase its internal energy by 100J.

1. **Understanding the Relationship**:
The internal energy \( U \) of the gas is given by the relation:
\[ U = \alpha \sqrt{V} \]
where \( \alpha \) is a constant.

2. **Change in Internal Energy**:
We need to find the work done when the internal energy increases by \( \Delta U = 100 \, \text{J} \).

3. **Using the First Law of Thermodynamics**:
The first law states:
\[ \Delta U = Q - W \]
where \( Q \) is the heat added to the system and \( W \) is the work done by the system. Rearranging gives:
\[ W = Q - \Delta U \]

4. **Finding Molar Specific Heat**:
From the earlier analysis, we find that the process can be modeled as a polytropic process. The molar specific heat \( C \) for a polytropic process is given by:
\[ C = C_v + \frac{R}{1 - x} \]
where \( x = \frac{1}{2} \) (as derived from the relation \( PV^{1/2} = \text{constant} \)).

5. **Calculating Specific Heat**:
For a diatomic gas, the molar specific heat at constant volume \( C_v \) is:
\[ C_v = \frac{5}{2} R \]
Therefore,
\[ C = \frac{5}{2} R + \frac{R}{1 - \frac{1}{2}} = \frac{5}{2} R + 2R = \frac{9}{2} R \]

6. **Finding Work Done**:
Now we can find the ratio of work done to the change in internal energy:
\[ \frac{W}{\Delta U} = \frac{C - C_v}{C_v} = \frac{\frac{9}{2} R - \frac{5}{2} R}{\frac{5}{2} R} = \frac{4/2 R}{5/2 R} = \frac{4}{5} \]
Thus, we have:
\[ W = \frac{4}{5} \Delta U \]
Substituting \( \Delta U = 100 \, \text{J} \):
\[ W = \frac{4}{5} \times 100 \, \text{J} = 80 \, \text{J} \]

### Part (b): Find the molar specific heat of the gas.

From the previous calculations, we found that:
\[ C = \frac{9}{2} R \]
Thus, the molar specific heat of the gas is:
\[ C = \frac{9}{2} R \]

### Summary of Results:
- (a) The work performed by the gas to increase its internal energy by 100J is \( W = 80 \, \text{J} \).
- (b) The molar specific heat of the gas is \( C = \frac{9}{2} R \).