1. For an ideal gas, the kinetic theory gives the relation between temperature and mean squared speed:

$\frac{1}{2}m\overline{v^2} = \frac{3}{2}kT$

where:

• $m$ = mass of one molecule,
• $\overline{v^2}$ = mean squared velocity,
• $k$ = Boltzmann constant,
• $T$ = absolute temperature.

2. Rearranging,

$m\overline{v^2} = 3kT$

$\overline{v^2} = \frac{3k}{m}T$

3. Since $\frac{3k}{m}$ is a constant for a particular ideal gas,

$\overline{v^2} \propto T$

So, mean squared velocity varies linearly with temperature and the graph must be a straight line passing through the origin.

4. Therefore, the correct graph is the one showing:

• $\overline{v^2}$ on y-axis,
• $T$ on x-axis,
• straight line through origin.

Hence the correct option is C.