Consider one mole of an ideal gas whose volume changes with temeperature as `V=(alpha)/(T)`, where `alpha` is a constant. Heat is supplied to the gas to raise its temperature by `DeltaT`. If `gamma` adiabatic constant then choose the correct options.

Options

A. The work done in the process is `RDeltaT`
B. The work done in the process is `-RDeltaT`
C. Change in internal energy of gas is `(RDeltaT)/(gamma-1)`
D. Change in internal energy of gas is `-(RDeltaT)/(gamma-1)`

Explanation

To solve the problem, we will analyze the given information step by step.

### Step 1: Understand the relationship between volume and temperature
We are given that the volume \( V \) of one mole of an ideal gas changes with temperature \( T \) as:
\[ V = \frac{\alpha}{T} \]
where \( \alpha \) is a constant.

### Step 2: Determine the change in internal energy
The change in internal energy \( \Delta U \) for one mole of an ideal gas can be expressed as:
\[ \Delta U = n \cdot C_V \cdot \Delta T \]
For one mole of gas, \( n = 1 \). The molar heat capacity at constant volume \( C_V \) can be expressed in terms of degrees of freedom \( F \) as:
\[ C_V = \frac{F}{2} R \]
For a monatomic ideal gas, \( F = 3 \), and for a diatomic gas, \( F = 5 \). However, in general, we can express it as:
\[ C_V = \frac{R}{\gamma - 1} \]
where \( \gamma \) is the adiabatic constant. Thus, the change in internal energy becomes:
\[ \Delta U = \frac{R}{\gamma - 1} \Delta T \]

### Step 3: Calculate the work done
Using the ideal gas equation \( PV = nRT \), we can express \( T \) in terms of \( V \):
\[ T = \frac{PV}{R} \]
Substituting \( V = \frac{\alpha}{T} \) into the ideal gas equation gives:
\[ P \left(\frac{\alpha}{T}\right) = R \cdot T \]
Rearranging this, we find:
\[ PV^2 = \alpha R \]
This indicates that \( PV^2 \) is a constant.

### Step 4: Determine the work done during the process
For a process described by \( PV^\eta = \text{constant} \), the work done \( W \) can be calculated using:
\[ W = \frac{nR\Delta T}{1 - \eta} \]
In our case, \( \eta = 2 \) (since \( PV^2 = \text{constant} \)):
\[ W = \frac{R\Delta T}{1 - 2} = -R\Delta T \]

### Step 5: Summary of results
1. The change in internal energy is:
\[ \Delta U = \frac{R}{\gamma - 1} \Delta T \]
2. The work done on the gas is:
\[ W = -R\Delta T \]

### Conclusion
Based on the calculations:
- The change in internal energy \( \Delta U \) is directly related to \( \Delta T \) and the adiabatic constant \( \gamma \).
- The work done \( W \) is negative, indicating work is done by the gas.