If M and T are the molar mass and the absolute temperature of a gas, then RMS speed of a gas molecule is equal to the:

CONCEPT:

Dalton’s law of partial pressures:

• For a mixture of non-reactive ideal gases, the total pressure gets contribution from each gas in the mixture.

\(⇒ P=\frac{1}{3}[n_1m_1\overline{v_1^2}+n_2m_2\overline{v_2^2}+...+n_nm_n\overline{v_n^2}]\)

• In equilibrium, the average kinetic energy of the molecules of different gases will be equal.

\(⇒ \frac{1}{2}m_1\overline{v_1^2}=\frac{1}{2}m_2\overline{v_2^2}=...=\frac{1}{2}m_n\overline{v_n^2}=k_bT\)

So,

⇒ P = (n1 + n2 + ... + nn)kBT

• The mean of the square speed is given as,

\(⇒ \overline{v^2}=\frac{3k_BT}{m}\)

• The square root of \(\overline{v^2}\) is known as root mean square (RMS) speed and is denoted by vrms.
• At the same temperature, lighter molecules have a greater RMS speed.

EXPLANATION:

• We know that for an ideal gas the root mean square (RMS) speed is given as,

\(⇒ v_{rms}=\sqrt{\frac{3k_BT}{m}}\)     -----(1)

Where kB = Boltzmann constant, m = mass of the molecule and T = absolute temperature

• If M and T are the molar mass and the absolute temperature of a gas, then the RMS speed of a gas molecule is given as,

\(⇒ v_{rms}=\sqrt{\frac{3RT}{M}}\)

• Hence, option 3 is correct.

Additional Information

Kinetic interpretation of temperature:

• We know that pressure P of an ideal gas is given as,

\(⇒ P=\frac{1}{3}nm\overline{v^2}\)

• If the volume of the gas is V, then,

\(⇒ PV=\frac{1}{3}nVm\overline{v^2}\)

∵ N = nV

\(\therefore PV=\frac{1}{3}Nm\overline{v^2}\)

• The internal energy of an ideal gas is purely kinetic.
• So the total internal energy E of an ideal gas is given as,

\(⇒ E=N\times\frac{1}{2}m\overline{v^2}\)

• So we can say,

\(⇒ PV=\frac{2}{3}E\)

Where n = number of molecules per unit volume, m = mass of the molecule, N = total number of molecules, and \(\overline{v^2}\) = mean of the squared speed

• If the absolute temperature of an ideal gas is T, then the total internal energy is given as,

\(⇒ E=\frac{3}{2}k_BNT\)

Where kB = Boltzmann constant

• So the average kinetic energy of a molecule is given as,

\(⇒ \frac{E}{N}=\frac{1}{2}m\overline{v^2}=\frac{3}{2}k_BT\)

• The average kinetic energy of a molecule is proportional to the absolute temperature of the gas.
• The average kinetic energy of a molecule is independent of pressure, volume, or the nature of the ideal gas.
• So we can say that the internal energy of an ideal gas depends only on temperature, not on pressure or volume.