The average translational energy and the rms speed of molecules in a sample of oxygen gas at 300 K are 6.21x 10⁻²¹ J and 484 m/s respectively. The corresponding values at 600 K are nearly (assuming ideal gas behaviour)

The correct option is 12.42x10⁻²¹ J, 684 m/s.

To find the average translational kinetic energy and the root mean square (rms) speed at the new temperature, we can use the principles of the kinetic theory of gases.

1. Average Translational Kinetic Energy:
The average translational kinetic energy ($E$) of a gas molecule at an absolute temperature $T$ is given by the formula:
$E=\frac{3}{2}{k}_{B}T$
where $k_B$ is the Boltzmann constant. From this formula, we can see that the average translational kinetic energy is directly proportional to the absolute temperature:
$E\propto T$
Let $E_1$ be the energy at temperature $T_1=300\text{K}$ and $E_2$ be the energy at temperature $T_2=600\text{K}$. We can set up the ratio:
$\frac{E_2}{E_1}=\frac{T_2}{T_1}$
Substituting the given values:
$E_2=E_1\times \frac{T_2}{T_1}$
$E_2=(6.21\times {10}^{-21}\text{J})\times \frac{600\text{K}}{300\text{K}}$
$E_2=6.21\times {10}^{-21}\times 2$
$E_2=12.42\times {10}^{-21}\text{J}$

2. Root Mean Square (rms) Speed:
The rms speed ($v_{\text{rms}}$) of gas molecules is given by the formula:
$v_{\text{rms}}=\sqrt{\frac{3k_{B}T}{m}}$
where $m$ is the mass of a single molecule. This shows that the rms speed is directly proportional to the square root of the absolute temperature:
$v_{\text{rms}}\propto \sqrt{T}$
Let $v_1$ be the rms speed at $T_1=300\text{K}$ and $v_2$ be the rms speed at $T_2=600\text{K}$. We can express the ratio as:
$\frac{v_2}{v_1}=\sqrt{\frac{T_2}{T_1}}$
Substituting the given values:
$v_2=v_1\times \sqrt{\frac{T_2}{T_1}}$
$v_2=484\text{m/s}\times \sqrt{\frac{600\text{K}}{300\text{K}}}$
$v_2=484\times \sqrt{2}\text{m/s}$
Using the approximation $\sqrt{2}\approx 1.414$:
$v_2\approx 484\times 1.414\text{m/s}\approx 684.38\text{m/s}$
Thus, the rms speed at 600 K is approximately 684 m/s.

Combining both results, the values at 600 K are 12.42x10⁻²¹ J and 684 m/s respectively.