An ideal gas has an adiabatic exponent `gamma`. In some process its molar heat capacity varies as `C = alpha//T`,where `alpha` is a constant Find :
(a) the work performed by one mole of the gas during its heating from the temperature `T_0` to the temperature `eta` times higher ,
(b) the equation of the process in the variables `p, V`.

Options

A. `a ln eta`
B. `(1)/(aln eta)`
C. `aln eta -((eta-1)/(gamma-1))RT_(0)`
D. `alneta -(gamma-1)RT_(0)`

Explanation

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

### Part (a): Work performed by one mole of the gas during its heating

1. **Identify the given parameters**:
- Molar heat capacity \( C = \frac{\alpha}{T} \)
- Initial temperature \( T_0 \)
- Final temperature \( T_f = \eta T_0 \)

2. **Write the expression for work done**:
The work done \( W \) during the heating process can be expressed as:
\[ W = \int_{T_0}^{\eta T_0} C \, dT \]

3. **Substitute the expression for \( C \)**:
Substitute \( C = \frac{\alpha}{T} \) into the work equation:
\[ W = \int_{T_0}^{\eta T_0} \frac{\alpha}{T} \, dT \]

4. **Factor out the constant \( \alpha \)**:
Since \( \alpha \) is a constant, it can be factored out of the integral:
\[ W = \alpha \int_{T_0}^{\eta T_0} \frac{1}{T} \, dT \]

5. **Evaluate the integral**:
The integral of \( \frac{1}{T} \) is \( \ln T \):
\[ W = \alpha \left[ \ln T \right]_{T_0}^{\eta T_0} \]

6. **Substitute the limits**:
Substitute the limits into the evaluated integral:
\[ W = \alpha \left( \ln(\eta T_0) - \ln(T_0) \right) \]

7. **Use properties of logarithms**:
Simplify the expression using properties of logarithms:
\[ W = \alpha \left( \ln \eta + \ln T_0 - \ln T_0 \right) = \alpha \ln \eta \]

Thus, the work performed by one mole of the gas during its heating is:
\[ \boxed{W = \alpha \ln \eta} \]

### Part (b): Equation of the process in the variables \( p \) and \( V \)

1. **Recall the adiabatic process equation**:
For an adiabatic process, the relationship between pressure \( p \) and volume \( V \) is given by:
\[ p V^\gamma = \text{constant} \]
where \( \gamma \) is the adiabatic exponent.

2. **Express \( \gamma \)**:
The adiabatic exponent \( \gamma \) is defined as:
\[ \gamma = \frac{C_p}{C_v} \]
where \( C_p \) is the molar heat capacity at constant pressure and \( C_v \) is the molar heat capacity at constant volume.

3. **Relate \( C \) to \( C_p \) and \( C_v \)**:
In the context of the problem, since \( C \) varies as \( \frac{\alpha}{T} \), we can use the relationship of \( C \) with \( C_p \) and \( C_v \) to find \( \gamma \).

4. **Final equation**:
The equation of the process in terms of \( p \) and \( V \) remains:
\[ p V^\gamma = \text{constant} \]

Thus, the equation of the process in the variables \( p \) and \( V \) is:
\[ \boxed{p V^\gamma = \text{constant}} \]