A mixture of two gases is contained in a vessel. The Gas 1 is monoatomic and gas 2 is diatomic and the ratio of root mean square speeds of the molecules of two gases is

Options

A. 2
B. 4
C. 8
D. 16

Answer

To find the ratio of the root mean square (RMS) speeds of two gases, we can follow these steps:

### Step 1: Understand the formula for RMS speed
The root mean square speed (Vrms) of a gas is given by the formula:
\[ V_{rms} = \sqrt{\frac{3RT}{M}} \]
where:
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature in Kelvin,
- \( M \) is the molecular mass of the gas.

### Step 2: Identify the gases and their properties
In this problem:
- Gas 1 is monoatomic,
- Gas 2 is diatomic.

Let:
- \( M_1 \) be the molecular mass of gas 1 (monoatomic),
- \( M_2 \) be the molecular mass of gas 2 (diatomic).

### Step 3: Write the expression for the ratio of RMS speeds
The ratio of the RMS speeds of the two gases can be expressed as:
\[ \frac{V_{rms1}}{V_{rms2}} = \frac{\sqrt{\frac{3RT}{M_1}}}{\sqrt{\frac{3RT}{M_2}}} \]
Since \( R \) and \( T \) are constant for both gases, they cancel out:
\[ \frac{V_{rms1}}{V_{rms2}} = \sqrt{\frac{M_2}{M_1}} \]

### Step 4: Use the given ratio of molecular masses
According to the problem, the ratio of the molecular masses is given as:
\[ \frac{M_1}{M_2} = \frac{1}{4} \]
This means:
\[ M_2 = 4M_1 \]

### Step 5: Substitute the molecular mass ratio into the RMS speed ratio
Now substituting \( M_2 \) in the RMS speed ratio:
\[ \frac{V_{rms1}}{V_{rms2}} = \sqrt{\frac{M_2}{M_1}} = \sqrt{\frac{4M_1}{M_1}} = \sqrt{4} = 2 \]

### Step 6: State the final result
Thus, the ratio of the root mean square speeds of the two gases is:
\[ \frac{V_{rms1}}{V_{rms2}} = 2:1 \]

### Summary
The ratio of the root mean square speeds of the monoatomic gas (Gas 1) to the diatomic gas (Gas 2) is \( 2:1 \).

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