By what factor the rms velocity will change, if the temperature is raised from `27^(@)C` to `327^(@)C`?

Options

A. `sqrt2`
B. `2`
C. `sqrt3`
D. `1`

Explanation

To find the factor by which the root mean square (RMS) velocity changes when the temperature is raised from \(27^\circ C\) to \(327^\circ C\), we can follow these steps:

### Step 1: Convert Celsius to Kelvin
First, we need to convert the temperatures from Celsius to Kelvin because the RMS velocity formula requires absolute temperatures.

- Temperature \(T_1 = 27^\circ C = 27 + 273 = 300 \, K\)
- Temperature \(T_2 = 327^\circ C = 327 + 273 = 600 \, K\)

### Step 2: Use the RMS Velocity Formula
The formula for the RMS velocity (\(V_{rms}\)) of a gas is given by:

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

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

### Step 3: Understand the Proportionality
Since \(R\) and \(M\) are constants for a given gas, we can say that the RMS velocity is directly proportional to the square root of the absolute temperature:

\[ V_{rms} \propto \sqrt{T} \]

### Step 4: Set Up the Ratio of RMS Velocities
We can express the ratio of the RMS velocities at the two temperatures:

\[ \frac{V_{rms2}}{V_{rms1}} = \sqrt{\frac{T_2}{T_1}} \]

### Step 5: Substitute the Values
Now, substituting the values of \(T_1\) and \(T_2\):

\[ \frac{V_{rms2}}{V_{rms1}} = \sqrt{\frac{600}{300}} = \sqrt{2} \]

### Step 6: Conclusion
Thus, the factor by which the RMS velocity changes when the temperature is raised from \(27^\circ C\) to \(327^\circ C\) is:

\[ V_{rms2} = \sqrt{2} \times V_{rms1} \]

### Final Answer
The factor is \(\sqrt{2}\).

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