Calculate `gamma` ( ratio of `C_(p) ` & `C_(v)` ) for
Triatomic nonlinear gas at low temperature.

Answer

To calculate the ratio of specific heats \( \gamma \) (gamma) for a triatomic nonlinear gas at low temperature, we will follow these steps:

### Step 1: Identify Degrees of Freedom
For a triatomic nonlinear gas, we need to consider the translational and rotational degrees of freedom. At low temperatures, we will not consider vibrational degrees of freedom.

- **Translational Degrees of Freedom (f_t)**: A molecule can move in three dimensions (x, y, z), which gives it 3 translational degrees of freedom.

- **Rotational Degrees of Freedom (f_r)**: A nonlinear molecule can rotate about three axes, which gives it 3 rotational degrees of freedom.

### Step 2: Calculate \( C_v \)
The molar heat capacity at constant volume \( C_v \) can be calculated using the formula:
\[ C_v = \left( \frac{f_t}{2} + \frac{f_r}{2} \right) R \]
Substituting the values:
\[ C_v = \left( \frac{3}{2} + \frac{3}{2} \right) R = 3R \]

### Step 3: Calculate \( C_p \)
The molar heat capacity at constant pressure \( C_p \) is related to \( C_v \) by the equation:
\[ C_p = C_v + R \]
Substituting the value of \( C_v \):
\[ C_p = 3R + R = 4R \]

### Step 4: Calculate \( \gamma \)
Now, we can calculate \( \gamma \) using the formula:
\[ \gamma = \frac{C_p}{C_v} \]
Substituting the values of \( C_p \) and \( C_v \):
\[ \gamma = \frac{4R}{3R} = \frac{4}{3} \approx 1.33 \]

### Final Answer
Thus, the value of \( \gamma \) for a triatomic nonlinear gas at low temperature is approximately \( 1.33 \).

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