The average translational energy and the rms speed of molecules in a sample of oxygen gas at `300K` are `6.21xx10^(-21)J` and `484m//s`, respectively. The corresponding values at `600K` are nearly (assuming ideal gas behaviour)

Options

A. `12.42xx10^(-21)J, 968 m//s`
B. `8.78xx10^(-21)J, 684 m//s`
C. `6.21xx10^(-21)J, 968m//s`
D. `12.42xx10^(-21)J, 684m//s`

Explanation

To find the average translational energy and the RMS speed of molecules in a sample of oxygen gas at 600K, we can follow these steps:

### Step 1: Calculate the Average Translational Kinetic Energy at 600K

The average translational kinetic energy (KE) of a gas is given by the formula:
\[ KE = \frac{3}{2} k T \]
where:
- \( k \) is the Boltzmann constant,
- \( T \) is the absolute temperature in Kelvin.

Since we know that the average translational kinetic energy is directly proportional to the temperature, we can set up the ratio:
\[ \frac{KE_2}{KE_1} = \frac{T_2}{T_1} \]
Given:
- \( KE_1 = 6.21 \times 10^{-21} \, J \) at \( T_1 = 300 \, K \)
- \( T_2 = 600 \, K \)

Substituting the values:
\[ \frac{KE_2}{6.21 \times 10^{-21}} = \frac{600}{300} = 2 \]

Thus, we find:
\[ KE_2 = 2 \times 6.21 \times 10^{-21} = 12.42 \times 10^{-21} \, J \]

### Step 2: Calculate the RMS Speed at 600K

The RMS speed (\( V_{rms} \)) is given by the formula:
\[ V_{rms} = \sqrt{\frac{3RT}{M}} \]
where:
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature,
- \( M \) is the molar mass of the gas.

Since \( V_{rms} \) is proportional to the square root of the temperature, we can set up the ratio:
\[ \frac{V_{rms2}}{V_{rms1}} = \sqrt{\frac{T_2}{T_1}} \]
Given:
- \( V_{rms1} = 484 \, m/s \) at \( T_1 = 300 \, K \)
- \( T_2 = 600 \, K \)

Substituting the values:
\[ \frac{V_{rms2}}{484} = \sqrt{\frac{600}{300}} = \sqrt{2} \]

Thus, we find:
\[ V_{rms2} = \sqrt{2} \times 484 \approx 1.414 \times 484 \approx 684 \, m/s \]

### Final Values

At 600K:
- Average Translational Kinetic Energy \( KE_2 \approx 12.42 \times 10^{-21} \, J \)
- RMS Speed \( V_{rms2} \approx 684 \, m/s \)

### Summary

The corresponding values at 600K are:
- Average Translational Energy: \( 12.42 \times 10^{-21} \, J \)
- RMS Speed: \( 684 \, m/s \)

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