To solve the problem, we need to find the root mean square (RMS) speed of gas molecules at different pressures and temperatures. The formula for RMS speed is given by:
\[
V_{rms} = \sqrt{\frac{3RT}{M}}
\]
where:
- \( V_{rms} \) is the root mean square speed,
- \( R \) 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 Temperatures to Kelvin:**
- The initial temperature \( T_1 = 127^\circ C = 127 + 273 = 400 \, K \)
- The final temperature \( T_2 = 227^\circ C = 227 + 273 = 500 \, K \)
2. **Write the RMS Speed Formula for Both Conditions:**
- For the first condition (1 atm, 400 K):
\[
V_1 = \sqrt{\frac{3R \cdot 400}{M}} = 200 \, m/s
\]
- For the second condition (2 atm, 500 K):
\[
V_2 = \sqrt{\frac{3R \cdot 500}{M}}
\]
3. **Relate the Two Speeds:**
- Since \( V_{rms} \) is proportional to the square root of temperature and inversely proportional to the square root of pressure, we can write:
\[
\frac{V_2}{V_1} = \sqrt{\frac{T_2 \cdot P_1}{T_1 \cdot P_2}}
\]
- Here, \( P_1 = 1 \, atm \) and \( P_2 = 2 \, atm \).
4. **Substituting the Values:**
- Substitute \( T_1 = 400 \, K \), \( T_2 = 500 \, K \), \( P_1 = 1 \, atm \), and \( P_2 = 2 \, atm \):
\[
\frac{V_2}{200} = \sqrt{\frac{500 \cdot 1}{400 \cdot 2}}
\]
5. **Calculating the Right Side:**
- Calculate the fraction:
\[
\frac{500}{800} = \frac{5}{8}
\]
- Therefore:
\[
\frac{V_2}{200} = \sqrt{\frac{5}{8}}
\]
6. **Finding \( V_2 \):**
- Multiply both sides by 200:
\[
V_2 = 200 \cdot \sqrt{\frac{5}{8}} = 200 \cdot \frac{\sqrt{5}}{2\sqrt{2}} = 100\sqrt{5}
\]
### Final Answer:
\[
V_2 = 100\sqrt{5} \, m/s
\]
