For a given gas at 1 atm pressure, rms speed of the molecules is 200 m/s at `127^(@)C`. At 2 atm pressure and at `227^(@)C`, the rms speed of the molecules will be:

Options

A. 80 m/s
B. 100 m/s
C. `100 sqrt(5)` m/s
D. `80 sqrt(5)` m/s

Explanation

To solve the problem, we need to find the root mean square (rms) speed of the gas molecules at a different pressure and temperature using the relationship between rms speed, temperature, and molar mass.

### Step-by-Step Solution:

1. **Understand the Formula for RMS Speed**:
The formula for the root mean square speed (v_rms) of gas molecules is given by:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
where \( R \) is the universal gas constant, \( T \) is the absolute temperature in Kelvin, and \( M \) is the molar mass of the gas.

2. **Convert Temperatures to Kelvin**:
- For the first condition (127°C):
\[ T_1 = 127 + 273 = 400 \, K \]
- For the second condition (227°C):
\[ T_2 = 227 + 273 = 500 \, K \]

3. **Calculate the RMS Speed at the First Condition**:
Given that the rms speed at 1 atm and 127°C is 200 m/s, we can express this as:
\[ v_{1} = 200 \, m/s \]
Using the rms speed formula:
\[ v_{1} = \sqrt{\frac{3R \cdot 400}{M}} \]
Squaring both sides gives:
\[ v_{1}^2 = \frac{3R \cdot 400}{M} \]
\[ 200^2 = \frac{3R \cdot 400}{M} \]
\[ 40000 = \frac{1200R}{M} \quad (1) \]

4. **Calculate the RMS Speed at the Second Condition**:
For the second condition at 2 atm and 227°C, we express the rms speed as:
\[ v_{2} = \sqrt{\frac{3R \cdot 500}{M}} \]
Squaring both sides gives:
\[ v_{2}^2 = \frac{3R \cdot 500}{M} \]
\[ v_{2}^2 = \frac{1500R}{M} \quad (2) \]

5. **Relate the Two Conditions**:
We can divide equation (2) by equation (1):
\[ \frac{v_{2}^2}{v_{1}^2} = \frac{\frac{1500R}{M}}{\frac{1200R}{M}} \]
The \( R \) and \( M \) cancel out:
\[ \frac{v_{2}^2}{200^2} = \frac{1500}{1200} \]
Simplifying gives:
\[ \frac{v_{2}^2}{40000} = \frac{5}{4} \]
Therefore:
\[ v_{2}^2 = 40000 \cdot \frac{5}{4} = 50000 \]
Taking the square root:
\[ v_{2} = \sqrt{50000} = 100\sqrt{5} \, m/s \]

### Final Answer:
The rms speed of the molecules at 2 atm pressure and 227°C is:
\[ v_{2} = 100\sqrt{5} \, m/s \]