The root mean square speed of the molecule at constant pressure at temperature T is v, what is its rms speed, if temperature is reduced to `(T)/(2)?`

Explanation

To solve the problem, we need to determine the root mean square (RMS) speed of gas molecules when the temperature is reduced from \( T \) to \( \frac{T}{2} \).

### Step-by-Step Solution:

1. **Understand the formula for RMS speed**:
The root mean square speed (\( v_{rms} \)) 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. **Identify the initial conditions**:
At temperature \( T \), the RMS speed is given as \( v \):
\[ v = \sqrt{\frac{3RT}{M}} \]

3. **Determine the new temperature**:
The temperature is reduced to \( \frac{T}{2} \).

4. **Calculate the new RMS speed**:
Substitute \( \frac{T}{2} \) into the RMS speed formula:
\[ v_{rms, new} = \sqrt{\frac{3R \left(\frac{T}{2}\right)}{M}} = \sqrt{\frac{3RT}{2M}} \]

5. **Relate the new speed to the original speed**:
We can express the new RMS speed in terms of the original speed \( v \):
\[ v_{rms, new} = \sqrt{\frac{1}{2}} \cdot \sqrt{\frac{3RT}{M}} = \sqrt{\frac{1}{2}} \cdot v \]
Therefore:
\[ v_{rms, new} = \frac{v}{\sqrt{2}} \]

6. **Conclusion**:
The new RMS speed when the temperature is reduced to \( \frac{T}{2} \) is:
\[ v_{rms, new} = \frac{v}{\sqrt{2}} \]

### Final Answer:
The RMS speed of the molecules when the temperature is reduced to \( \frac{T}{2} \) is \( \frac{v}{\sqrt{2}} \).