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 root mean square (rms) speed of molecules in a sample of oxygen gas at `600K`, we can use the following relationships for an ideal gas:

1. **Average Translational Energy (E_avg)**:
The average translational energy of a gas molecule is given by the formula:
\[ E_{\text{avg}} = \frac{3}{2} k T \]
where \( k \) is the Boltzmann constant (\( k \approx 1.38 \times 10^{-23} \, \text{J/K} \)) and \( T \) is the temperature in Kelvin.

2. **Root Mean Square Speed (v_{rms})**:
The rms speed of gas molecules is given by the formula:
\[ v_{\text{rms}} = \sqrt{\frac{3kT}{m}} \]
where \( m \) is the mass of a gas molecule.

### Step 1: Calculate Average Translational Energy at 600K
Using the average translational energy formula:
\[ E_{\text{avg}}(600K) = \frac{3}{2} k (600) \]
Substituting \( k \):
\[ E_{\text{avg}}(600K) = \frac{3}{2} (1.38 \times 10^{-23}) (600) \]
Calculating this gives:
\[ E_{\text{avg}}(600K) = \frac{3}{2} (8.28 \times 10^{-21}) \approx 1.24 \times 10^{-20} \, \text{J} \]

### Step 2: Calculate RMS Speed at 600K
Using the rms speed formula:
\[ v_{\text{rms}}(600K) = \sqrt{\frac{3k(600)}{m}} \]
We know that the rms speed at 300K is 484 m/s. The rms speed is proportional to the square root of the temperature:
\[ v_{\text{rms}}(600K) = v_{\text{rms}}(300K) \sqrt{\frac{600}{300}} = 484 \sqrt{2} \]
Calculating this gives:
\[ v_{\text{rms}}(600K) = 484 \times 1.414 \approx 684 \, \text{m/s} \]

### Final Results
- Average Translational Energy at 600K: \( \approx 1.24 \times 10^{-20} \, \text{J} \)
- RMS Speed at 600K: \( \approx 684 \, \text{m/s} \)