The volume of one mode of an ideal gas with adiabatic exponent `gamma` is varied according to the law `V = a//T`, where a is constant . Find the amount of heat obtained by the gas in this process, if the temperature is increased by `Delta T`.

Explanation

To solve the problem, we need to find the amount of heat obtained by one mole of an ideal gas when its volume is varied according to the law \( V = \frac{a}{T} \), and the temperature is increased by \( \Delta T \).

### Step-by-Step Solution:

1. **Understand the Relationship Between Volume and Temperature**:
The volume \( V \) of the gas is given by the equation:
\[ V = \frac{a}{T} \]
This implies that \( VT = a \), which is a constant.

2. **Apply the Ideal Gas Law**:
The ideal gas law states:
\[ PV = nRT \]
For one mole of gas (\( n = 1 \)), this simplifies to:
\[ PV = RT \]

3. **Relate Pressure and Volume**:
From the ideal gas law, we can express pressure \( P \) in terms of volume \( V \) and temperature \( T \):
\[ P = \frac{RT}{V} \]

4. **Substituting for Volume**:
Substitute \( V = \frac{a}{T} \) into the pressure equation:
\[ P = \frac{RT}{\frac{a}{T}} = \frac{RT^2}{a} \]

5. **Identify the Type of Process**:
The relationship \( PV^x = \text{constant} \) can be compared to the polytropic process. Here, we have:
\[ PV^2 = \text{constant} \]
This implies \( x = 2 \).

6. **Calculate the Molar Specific Heat Capacity**:
The molar specific heat capacity \( C \) for a polytropic process is given by:
\[ C = C_v + \frac{R}{1 - x} \]
where \( C_v = \frac{R}{\gamma - 1} \) (for an ideal gas). Substituting \( x = 2 \):
\[ C = \frac{R}{\gamma - 1} + \frac{R}{1 - 2} = \frac{R}{\gamma - 1} - R \]
Simplifying this gives:
\[ C = \frac{R}{\gamma - 1} - \frac{R(\gamma - 1)}{\gamma - 1} = \frac{R(2 - \gamma)}{\gamma - 1} \]

7. **Calculate the Heat \( Q \)**:
The heat \( Q \) absorbed by the gas during the process when the temperature increases by \( \Delta T \) is given by:
\[ Q = nC\Delta T \]
For one mole (\( n = 1 \)):
\[ Q = C \Delta T = \left( \frac{R(2 - \gamma)}{\gamma - 1} \right) \Delta T \]

8. **Final Expression for Heat**:
Thus, the amount of heat obtained by the gas in this process is:
\[ Q = \frac{R(2 - \gamma)}{\gamma - 1} \Delta T \]

### Summary:
The amount of heat obtained by the gas when the temperature is increased by \( \Delta T \) is:
\[ Q = \frac{R(2 - \gamma)}{\gamma - 1} \Delta T \]