A sample of gas is allowed to expand adiabatically. As a consequence its volume increases from `1m^3` to `32m^3` and temperature drops from 320 K to 80 K. How many degree of freedom are associated with the molecules of gas?

Explanation

To find the number of degrees of freedom associated with the molecules of the gas, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Adiabatic Process**: In an adiabatic process, the relationship between pressure (P), volume (V), and temperature (T) can be described using the equation:
\[ PV^\gamma = \text{constant} \]
or in terms of temperature and volume:
\[ TV^{\gamma - 1} = \text{constant} \]

2. **Identify Given Values**: From the problem, we have:
- Initial volume \( V_1 = 1 \, m^3 \)
- Final volume \( V_2 = 32 \, m^3 \)
- Initial temperature \( T_1 = 320 \, K \)
- Final temperature \( T_2 = 80 \, K \)

3. **Set Up the Equation**: Using the relationship \( TV^{\gamma - 1} = \text{constant} \), we can write:
\[ T_1 V_1^{\gamma - 1} = T_2 V_2^{\gamma - 1} \]
Substituting the known values:
\[ 320 \cdot 1^{\gamma - 1} = 80 \cdot 32^{\gamma - 1} \]

4. **Simplify the Equation**: Since \( 1^{\gamma - 1} = 1 \), the equation simplifies to:
\[ 320 = 80 \cdot 32^{\gamma - 1} \]
Dividing both sides by 80 gives:
\[ 4 = 32^{\gamma - 1} \]

5. **Express 4 and 32 in Powers of 2**: We can express 4 and 32 as powers of 2:
\[ 4 = 2^2 \quad \text{and} \quad 32 = 2^5 \]
Thus, we can rewrite the equation:
\[ 2^2 = (2^5)^{\gamma - 1} \]

6. **Equate the Exponents**: This leads to:
\[ 2 = 5(\gamma - 1) \]
Simplifying gives:
\[ 2 = 5\gamma - 5 \]
Rearranging gives:
\[ 5\gamma = 7 \quad \Rightarrow \quad \gamma = \frac{7}{5} \]

7. **Relate Gamma to Degrees of Freedom**: The relationship between \( \gamma \) and the degrees of freedom \( f \) is given by:
\[ \gamma = 1 + \frac{2}{f} \]
Substituting \( \gamma = \frac{7}{5} \):
\[ \frac{7}{5} = 1 + \frac{2}{f} \]

8. **Solve for Degrees of Freedom**: Rearranging gives:
\[ \frac{2}{f} = \frac{7}{5} - 1 = \frac{2}{5} \]
Therefore:
\[ f = 5 \]

### Final Answer:
The number of degrees of freedom associated with the molecules of the gas is \( f = 5 \).