The molecules of a given mass of a gas have root mean square speeds of $100 \ m/s$ at $27^{\circ} \ C$ and $1.00 \ \text{atm}$ pressure. What will be the root mean square speeds of the molecules of the gas at $127^{\circ} \ C$ and $2.0 \ \text{atm}$ pressure?

The temperature of a gas is doubled $(i)$ on the absolute scale and $(ii)$ on the centigrade scale. The increase in the root mean square velocity of the gas will be:

Given below are two statements:
Statement $I$: The average momentum of a molecule in a sample of an ideal gas depends on temperature.
Statement $II$: The rms speed of oxygen molecules in a gas is $v$. If the temperature is doubled and the oxygen molecules dissociate into oxygen atoms,the rms speed will become $2v$.
In the light of the above statements,choose the correct answer from the options given below.

In two vessels of the same volume,atomic hydrogen and helium at pressures of $1\, atm$ and $2\, atm$ are filled,respectively. If the temperature of both samples is the same,then the average speed of hydrogen atoms $\langle C_H \rangle$ will be related to that of helium $\langle C_{He} \rangle$ as:

The temperature of an ideal gas is increased from $200 \ K$ to $800 \ K$. If the r.m.s. speed of the gas at $200 \ K$ is $v_0$,then the r.m.s. speed of the gas at $800 \ K$ will be:

How does the $rms$ velocity of an ideal gas vary with density at constant pressure?