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 find the root mean square (RMS) speed of a molecule when the temperature is reduced from \( T \) to \( \frac{T}{2} \), we can use the relationship between RMS speed and temperature.

### Step-by-Step Solution:

1. **Understand the formula for RMS speed**: The RMS speed (\( v_{\text{rms}} \)) of a gas is given by the formula:
\[ v_{\text{rms}} = \sqrt{\frac{3RT}{M}} \]
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 the initial 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. **Apply the RMS speed formula at the new temperature**: Substitute \( \frac{T}{2} \) into the RMS speed formula:
\[ v_{\text{rms, new}} = \sqrt{\frac{3R\left(\frac{T}{2}\right)}{M}} \]

5. **Simplify the expression**:
\[ v_{\text{rms, new}} = \sqrt{\frac{3RT}{2M}} = \sqrt{\frac{1}{2}} \cdot \sqrt{\frac{3RT}{M}} = \frac{v}{\sqrt{2}} \]

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

### Summary:
The root mean square speed of the molecule when the temperature is reduced to \( \frac{T}{2} \) is \( \frac{v}{\sqrt{2}} \).