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

Explanation

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 Formula for rms Velocity**:
The root-mean-square velocity (Vrms) of gas molecules is given by the formula:
\[ V_{rms} = \sqrt{\frac{3RT}{M}} \]
where \( R \) is the universal gas constant, \( T \) is the absolute temperature, and \( M \) is the molar mass of the gas.

2. **Set Up the Equation for Halving Vrms**:
We want to find the condition under which the rms velocity is halved:
\[ \frac{V_{rms}}{2} = \sqrt{\frac{3RT}{M}} \]
This implies:
\[ \frac{1}{2} \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3RT'}{M}} \]
where \( T' \) is the new temperature after expansion.

3. **Square Both Sides**:
Squaring both sides gives:
\[ \frac{1}{4} \cdot \frac{3RT}{M} = \frac{3RT'}{M} \]
Canceling \( \frac{3R}{M} \) from both sides results in:
\[ \frac{1}{4} T = T' \]
This means the new temperature \( T' \) is one-fourth of the original temperature \( T \).

4. **Use the Adiabatic Condition**:
For an adiabatic process, we have the relation:
\[ TV^{\gamma - 1} = \text{constant} \]
where \( \gamma \) (gamma) is the heat capacity ratio. Here, \( \gamma = 1.5 \).

5. **Relate the Initial and Final States**:
Let the initial volume be \( V_1 \) and the final volume be \( V_2 \). Then we can write:
\[ T_1 V_1^{\gamma - 1} = T_2 V_2^{\gamma - 1} \]
Substituting \( T_2 = \frac{1}{4} T_1 \):
\[ T_1 V_1^{\gamma - 1} = \frac{1}{4} T_1 V_2^{\gamma - 1} \]
Dividing both sides by \( T_1 \) gives:
\[ V_1^{\gamma - 1} = \frac{1}{4} V_2^{\gamma - 1} \]

6. **Rearranging the Equation**:
Rearranging gives:
\[ 4 V_1^{\gamma - 1} = V_2^{\gamma - 1} \]
Taking the \( \gamma - 1 \) root of both sides:
\[ V_2 = V_1 \cdot 4^{\frac{1}{\gamma - 1}} \]

7. **Substituting the Value of Gamma**:
Since \( \gamma = 1.5 \):
\[ \gamma - 1 = 0.5 \]
Thus:
\[ V_2 = V_1 \cdot 4^{\frac{1}{0.5}} = V_1 \cdot 16 \]

8. **Conclusion**:
Therefore, the gas must be expanded 16 times to reduce the rms velocity to half.

### Final Answer:
The gas must be expanded **16 times**.