The molecules of a given mass of a gas have rms velocity of `200 m//s at 27^(@)C and 1.0 xx 10^(5) N//m_(2)` pressure. When the temperature and pressure of the gas are respectively `127^(@)C and 0.05 xx 10^(5) Nm^(-2)`, the rms velocity of its molecules in `ms^(-1)` is

Options

A. `(400)/(sqrt(3))`
B. `(100sqrt(2))/3`
C. `100/3`
D. `100 sqrt(2)`

Explanation

To find the root mean square (rms) velocity of the gas molecules when the temperature and pressure change, we can follow these steps:

### Step 1: Understand the relationship between rms velocity, temperature, and pressure
The rms velocity \( v \) of gas molecules is given by the formula:
\[ v_{\text{rms}} = \sqrt{\frac{3RT}{M}} \]
where \( R \) is the universal gas constant, \( T \) is the absolute temperature in Kelvin, and \( M \) is the molar mass of the gas.

### Step 2: Note that rms velocity depends on temperature only
From the formula, we can see that the rms velocity is directly proportional to the square root of the absolute temperature \( T \). It does not depend on pressure \( P \).

### Step 3: Convert temperatures from Celsius to Kelvin
- Initial temperature \( T_1 = 27^\circ C = 27 + 273 = 300 \, K \)
- Final temperature \( T_2 = 127^\circ C = 127 + 273 = 400 \, K \)

### Step 4: Set up the ratio of the rms velocities
Using the relationship of rms velocities with temperature:
\[ \frac{v_2}{v_1} = \sqrt{\frac{T_2}{T_1}} \]
Given that the initial rms velocity \( v_1 = 200 \, m/s \), we can substitute the values:
\[ \frac{v_2}{200} = \sqrt{\frac{400}{300}} \]

### Step 5: Simplify the ratio
Calculating the right-hand side:
\[ \sqrt{\frac{400}{300}} = \sqrt{\frac{4}{3}} = \frac{2}{\sqrt{3}} \]
Now substituting this back into the equation:
\[ \frac{v_2}{200} = \frac{2}{\sqrt{3}} \]

### Step 6: Solve for \( v_2 \)
Multiplying both sides by 200:
\[ v_2 = 200 \cdot \frac{2}{\sqrt{3}} = \frac{400}{\sqrt{3}} \, m/s \]

### Step 7: Final answer
Thus, the rms velocity of the gas molecules at the new temperature and pressure is:
\[ v_2 = \frac{400}{\sqrt{3}} \, m/s \]