The ratio of average speed of an oxygen molcule to the RMS speed of a nitrogen molecule at the same temperature is

Options

A. `((3pi)/(7))^(1//2)`
B. `((7)/(3pi))^(1//2)`
C. `((3)/(7pi))^(1//2)`
D. `((7pi)/(3))^(1//2)`

Answer

To find the ratio of the average speed of an oxygen molecule to the RMS speed of a nitrogen molecule at the same temperature, we will use the formulas for average speed and RMS speed from the kinetic theory of gases.

### Step 1: Write the formulas for average speed and RMS speed

The average speed (\( V_{avg} \)) of a gas molecule is given by the formula:
\[ V_{avg} = \sqrt{\frac{8RT}{\pi m}} \]
where:
- \( R \) is the gas constant,
- \( T \) is the temperature,
- \( m \) is the molar mass of the gas.

The RMS speed (\( V_{rms} \)) of a gas molecule is given by the formula:
\[ V_{rms} = \sqrt{\frac{3RT}{m}} \]

### Step 2: Substitute the molar masses for oxygen and nitrogen

For oxygen (\( O_2 \)):
- Molar mass \( m_{O_2} = 32 \, \text{g/mol} \)

For nitrogen (\( N_2 \)):
- Molar mass \( m_{N_2} = 28 \, \text{g/mol} \)

### Step 3: Calculate the average speed of oxygen

Substituting the molar mass of oxygen into the average speed formula:
\[ V_{avg, O_2} = \sqrt{\frac{8RT}{\pi \times 32}} \]

### Step 4: Calculate the RMS speed of nitrogen

Substituting the molar mass of nitrogen into the RMS speed formula:
\[ V_{rms, N_2} = \sqrt{\frac{3RT}{28}} \]

### Step 5: Find the ratio of average speed of oxygen to RMS speed of nitrogen

Now, we find the ratio:
\[ \text{Ratio} = \frac{V_{avg, O_2}}{V_{rms, N_2}} = \frac{\sqrt{\frac{8RT}{\pi \times 32}}}{\sqrt{\frac{3RT}{28}}} \]

### Step 6: Simplify the ratio

We can simplify this ratio:
\[ = \frac{\sqrt{8RT}}{\sqrt{\pi \times 32}} \times \frac{\sqrt{28}}{\sqrt{3RT}} \]
\[ = \frac{\sqrt{8 \times 28}}{\sqrt{32 \times 3\pi}} \]
\[ = \frac{\sqrt{224}}{\sqrt{96\pi}} \]

### Step 7: Further simplify the square roots

Calculating the square roots:
\[ = \frac{4\sqrt{14}}{4\sqrt{6\pi}} = \frac{\sqrt{14}}{\sqrt{6\pi}} \]

### Step 8: Final expression

Thus, the final ratio of the average speed of an oxygen molecule to the RMS speed of a nitrogen molecule is:
\[ \frac{\sqrt{14}}{\sqrt{6\pi}} \]