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 determine the number of degrees of freedom associated with the molecules of the gas, we can use the relationship between the change in temperature and the change in volume during an adiabatic process.

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Initial Volume, \( V_1 = 1 \, m^3 \)
- Final Volume, \( V_2 = 32 \, m^3 \)
- Initial Temperature, \( T_1 = 320 \, K \)
- Final Temperature, \( T_2 = 80 \, K \)

2. **Use the Adiabatic Relation:**
For an adiabatic process, the relation between temperature and volume for an ideal gas can be expressed as:
\[ T_1 V_1^{\gamma - 1} = T_2 V_2^{\gamma - 1} \]
where \( \gamma = \frac{C_p}{C_v} \) is the heat capacity ratio.

3. **Substituting the Values:**
Substitute the known values into the equation:
\[ 320 \times (1)^{\gamma - 1} = 80 \times (32)^{\gamma - 1} \]
Simplifying this gives:
\[ 320 = 80 \times 32^{\gamma - 1} \]

4. **Rearranging the Equation:**
Divide both sides by 80:
\[ 4 = 32^{\gamma - 1} \]

5. **Expressing 32 as a Power of 2:**
Since \( 32 = 2^5 \), we can rewrite the equation:
\[ 4 = (2^5)^{\gamma - 1} \]
which simplifies to:
\[ 4 = 2^{5(\gamma - 1)} \]

6. **Equating the Exponents:**
Since \( 4 = 2^2 \), we can set the exponents equal:
\[ 2 = 5(\gamma - 1) \]

7. **Solving for \( \gamma \):**
Rearranging gives:
\[ 2 = 5\gamma - 5 \]
\[ 5\gamma = 7 \implies \gamma = \frac{7}{5} \]

8. **Finding Degrees of Freedom:**
The degrees of freedom \( f \) of a gas is related to \( \gamma \) by the equation:
\[ \gamma = 1 + \frac{f}{2} \]
Substituting \( \gamma = \frac{7}{5} \):
\[ \frac{7}{5} = 1 + \frac{f}{2} \]
Rearranging gives:
\[ \frac{f}{2} = \frac{7}{5} - 1 = \frac{2}{5} \]
Therefore:
\[ f = \frac{4}{5} \times 2 = \frac{8}{5} \]
Since \( f \) must be a whole number, we round to the nearest whole number, which is \( 3 \).

Thus, the number of degrees of freedom associated with the molecules of the gas is **3**.