An ideal gas `(gamma = 1.5)` is expanded adiabatically. How many times has the gas to be expanded to reduce the roo-mean-square velocity of molecules becomes half ?

Options

A. 4 times
B. 16 times
C. 8 times
D. 2 times

Answer

To solve the problem of how many times an ideal gas must be expanded adiabatically to reduce the root-mean-square (RMS) velocity of its molecules to half, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship of RMS Velocity and Temperature:**
The root-mean-square velocity (VRMS) of gas molecules is given by the formula:
\[ V_{\text{RMS}} = \sqrt{\frac{3RT}{m}} \]
where \( R \) is the universal gas constant, \( T \) is the temperature, and \( m \) is the molar mass of the gas.

2. **Express the Condition for VRMS:**
We want to find the condition when the RMS velocity is halved:
\[ V_2 = \frac{1}{2} V_1 \]
Squaring both sides gives:
\[ V_2^2 = \frac{1}{4} V_1^2 \]

3. **Relate RMS Velocity to Temperature:**
Since \( V \) is proportional to the square root of temperature:
\[ \frac{V_2}{V_1} = \sqrt{\frac{T_2}{T_1}} \]
Substituting \( V_2 = \frac{1}{2} V_1 \) into the equation:
\[ \frac{1}{2} = \sqrt{\frac{T_2}{T_1}} \]
Squaring both sides gives:
\[ \frac{1}{4} = \frac{T_2}{T_1} \quad \Rightarrow \quad T_1 = 4T_2 \]

4. **Apply the Adiabatic Condition:**
For an adiabatic process, the relationship between temperature and volume is given by:
\[ T_1 V_1^{\gamma - 1} = T_2 V_2^{\gamma - 1} \]
where \( \gamma \) (gamma) is the heat capacity ratio, given as \( \gamma = 1.5 \).

5. **Substitute Known Values:**
Substitute \( T_1 = 4T_2 \) into the adiabatic equation:
\[ 4T_2 V_1^{\gamma - 1} = T_2 V_2^{\gamma - 1} \]
Dividing both sides by \( T_2 \) (assuming \( T_2 \neq 0 \)):
\[ 4 V_1^{\gamma - 1} = V_2^{\gamma - 1} \]

6. **Substituting the Value of Gamma:**
With \( \gamma = 1.5 \):
\[ \gamma - 1 = 0.5 \]
Thus, we can rewrite the equation:
\[ 4 V_1^{0.5} = V_2^{0.5} \]
Squaring both sides gives:
\[ 16 V_1 = V_2 \]

7. **Conclusion:**
This means that the gas must be expanded by a factor of 16:
\[ V_2 = 16 V_1 \]

### Final Answer:
The gas must be expanded **16 times** to reduce the root-mean-square velocity of the molecules to half.