What is the specific heat ratio of monoatomic gases ?

Answer

To find the specific heat ratio (γ) of a monoatomic gas, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definition of Specific Heat Ratio**:
The specific heat ratio (γ) is defined as the ratio of the specific heat at constant pressure (C_p) to the specific heat at constant volume (C_v):
\[ \gamma = \frac{C_p}{C_v} \]

2. **Use the Relation for Specific Heat Ratio**:
There is a known relation for the specific heat ratio in terms of the degrees of freedom (F) of the gas:
\[ \gamma = 1 + \frac{2}{F} \]

3. **Identify the Degrees of Freedom for Monoatomic Gas**:
For a monoatomic gas, the degrees of freedom (F) is 3. This is because monoatomic gases have three translational degrees of freedom (movement in x, y, and z directions).

4. **Substitute the Value of F into the Equation**:
Now, substitute F = 3 into the equation for γ:
\[ \gamma = 1 + \frac{2}{3} \]

5. **Perform the Calculation**:
Calculate the value:
\[ \gamma = 1 + \frac{2}{3} = 1 + 0.6667 = \frac{5}{3} \]

6. **Convert to Decimal Form (if needed)**:
The fraction \(\frac{5}{3}\) can also be expressed in decimal form:
\[ \frac{5}{3} \approx 1.6667 \]

### Final Answer:
Thus, the specific heat ratio (γ) of a monoatomic gas is:
\[ \gamma = \frac{5}{3} \approx 1.6667 \]