One mole of an ideal gas is taken through an adiabatic process where the temperature rises from `27^(@)C` to `37^(@)C`. If the ideal gas is composed of polyatomic molecule that has 4 vibrational modes, which of the following is true ?
`[R=8.314" J mol"^(-1)k^(-1)]`

Options

A. work done by the gas is close to 332 J
B. work done on the gas is close to 582 J
C. Work done by the gas is close to 582 J
D. work done on the gas close to 332 J

Answer

To solve the problem, we need to follow these steps:

### Step 1: Understand the Given Data
We have 1 mole of an ideal gas undergoing an adiabatic process, with an initial temperature \( T_1 = 27^\circ C \) and a final temperature \( T_2 = 37^\circ C \). We also know that the gas is polyatomic with 4 vibrational modes.

### Step 2: Convert Temperatures to Kelvin
Convert the temperatures from Celsius to Kelvin:
\[ T_1 = 27 + 273.15 = 300.15 \, K \]
\[ T_2 = 37 + 273.15 = 310.15 \, K \]

### Step 3: Calculate the Degrees of Freedom (F)
For a polyatomic gas with \( f_v = 4 \) vibrational modes, the total degrees of freedom \( F \) can be calculated as:
\[ F = 3 \, (\text{translational}) + 3 \, (\text{rotational}) + 4 \times 2 \, (\text{vibrational}) = 3 + 3 + 8 = 14 \]

### Step 4: Calculate the Heat Capacity Ratio (γ)
The heat capacity ratio \( \gamma \) is given by:
\[ \gamma = 1 + \frac{2}{F} \]
Substituting \( F = 14 \):
\[ \gamma = 1 + \frac{2}{14} = 1 + \frac{1}{7} = \frac{8}{7} \]

### Step 5: Calculate Work Done in Adiabatic Process
In an adiabatic process, the work done \( W \) can be expressed as:
\[ W = nR \frac{(T_1 - T_2)}{\gamma - 1} \]
Substituting the values:
- \( n = 1 \, \text{mol} \)
- \( R = 8.314 \, \text{J/mol·K} \)
- \( T_1 = 300.15 \, K \)
- \( T_2 = 310.15 \, K \)
- \( \gamma = \frac{8}{7} \)

Calculating \( W \):
\[ W = 1 \times 8.314 \times \frac{(300.15 - 310.15)}{\frac{8}{7} - 1} \]
\[ = 8.314 \times \frac{-10}{\frac{1}{7}} = 8.314 \times -70 = -582.98 \, \text{J} \]

### Step 6: Interpret the Result
The negative sign indicates that work is done on the gas. Thus, the magnitude of the work done is approximately \( 582 \, \text{J} \).

### Conclusion
The work done on the gas is approximately \( 582 \, \text{J} \).

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