The molecule of a given mas of gas have r.m.s. speed `200 ms^(-1)` at `27^(@)C` and `10^(5)Nm^(-2)` pressure. When the absolute temperature is doubled and the pressure is halved, then find rms speed of the molecules of the same gas.

Explanation

To solve the problem, we will use the relationship between the root mean square (r.m.s.) speed of gas molecules and temperature. The formula for the r.m.s. speed (V_rms) of a gas is given by:

\[ V_{rms} = \sqrt{\frac{3RT}{M}} \]

where:
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature in Kelvin,
- \( M \) is the molar mass of the gas.

### Step-by-Step Solution:

1. **Convert the initial temperature to Kelvin**:
The initial temperature is given as \( 27^\circ C \).
\[ T_{initial} = 27 + 273 = 300 \, K \]

2. **Identify the initial r.m.s. speed**:
The initial r.m.s. speed is given as:
\[ V_{initial} = 200 \, m/s \]

3. **Determine the final temperature**:
The problem states that the absolute temperature is doubled:
\[ T_{final} = 2 \times T_{initial} = 2 \times 300 = 600 \, K \]

4. **Determine the final pressure**:
The pressure is halved, but since the r.m.s. speed does not depend on pressure, we do not need to calculate this for our final speed.

5. **Use the relationship between initial and final r.m.s. speeds**:
The relationship between the initial and final r.m.s. speeds can be expressed as:
\[ \frac{V_{initial}}{V_{final}} = \sqrt{\frac{T_{initial}}{T_{final}}} \]

6. **Substituting the known values**:
\[ \frac{200}{V_{final}} = \sqrt{\frac{300}{600}} \]
Simplifying the right side:
\[ \sqrt{\frac{300}{600}} = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}} \]

7. **Rearranging the equation to solve for \( V_{final} \)**:
\[ 200 = V_{final} \cdot \frac{1}{\sqrt{2}} \]
\[ V_{final} = 200 \sqrt{2} \]

8. **Calculating \( V_{final} \)**:
\[ V_{final} = 200 \times 1.414 \approx 282.84 \, m/s \]

### Final Answer:
The r.m.s. speed of the molecules of the gas when the absolute temperature is doubled and the pressure is halved is approximately \( 282.84 \, m/s \).