Key Insight — Ignore the mass ratio!
The 2:1 mass ratio of Ar to Cl₂ is a classic NEET distractor. RMS speed does not depend on the amount of gas present — only on temperature and molar mass.
Formula for RMS Speed:
$$v_{rms} = \sqrt{\frac{3RT}{M}}$$
Since both gases are in the same flask at the same temperature T = 27°C = 300 K, T is the same for both.
Take the ratio:
$$\frac{v^{Ar}_{rms}}{v^{Cl}_{rms}} = \sqrt{\frac{3RT/M_{Ar}}{3RT/M_{Cl_2}}} = \sqrt{\frac{M_{Cl_2}}{M_{Ar}}}$$
Note: 3RT cancels out completely!
Substitute values:
$$\frac{v^{Ar}_{rms}}{v^{Cl}_{rms}} = \sqrt{\frac{70}{40}} = \sqrt{\frac{7}{4}} = \frac{\sqrt{7}}{2}$$
Answer: \(\dfrac{\sqrt{7}}{2}\) ✓
The kinetic theory of gases is built on a set of assumptions that simplify the behaviour of real gases into an elegant model. The key postulates are: (1) A gas consists of a very large number of tiny molecules in constant, random motion. (2) The volume of individual molecules is negligible compared to the total volume of the gas. (3) Molecules exert no forces on each other except during collisions — there are no intermolecular attractions or repulsions. (4) Collisions between molecules and the walls of the container are perfectly elastic (kinetic energy is conserved). (5) The average kinetic energy of molecules is directly proportional to the absolute temperature of the gas.
These postulates describe an ideal gas perfectly. Real gases deviate from this model at high pressure (where molecular volume matters) and low temperature (where intermolecular attractions become significant). The Van der Waals equation corrects for these deviations.
Not all molecules in a gas move at the same speed — there is a distribution of speeds called the Maxwell-Boltzmann distribution. Three characteristic speeds are defined:
\(v_{mp} = \sqrt{\dfrac{2RT}{M}}\) (Most Probable Speed)
\(v_{avg} = \sqrt{\dfrac{8RT}{\pi M}}\) (Average Speed)
\(v_{rms} = \sqrt{\dfrac{3RT}{M}}\) (Root Mean Square Speed)
📌 Ratio: \(v_{mp} : v_{avg} : v_{rms} = \sqrt{2} : \sqrt{\dfrac{8}{\pi}} : \sqrt{3}\)
📌 Numerically: 1 : 1.128 : 1.225
📌 Always: \(v_{mp} < v_{avg} < v_{rms}\)
The root mean square speed is defined as \(v_{rms} = \sqrt{\langle v^2 \rangle}\), i.e., the square root of the mean of the squares of the individual molecular speeds. It is the most important of the three speeds because it directly relates to the kinetic energy and pressure of the gas.
The pressure of an ideal gas is P = (1/3)ρv²_rms, where ρ is the density. This connects macroscopic pressure to the microscopic molecular speed. The higher the rms speed, the higher the pressure for the same density.
The rms speed \(v_{rms} = \sqrt{3RT/M}\) depends on two factors: absolute temperature T and molar mass M. Specifically:
📌 \(v_{rms} \propto \sqrt{T}\) — double the absolute temperature → rms speed increases by √2
📌 \(v_{rms} \propto \dfrac{1}{\sqrt{M}}\) — heavier molecules move slower at the same T
📌 Amount of gas does NOT affect rms speed
📌 Volume of container does NOT affect rms speed (at constant T)
This is why in a gas mixture at thermal equilibrium, lighter gas molecules always have higher rms speed than heavier ones. Hydrogen molecules move about 4 times faster than oxygen molecules at the same temperature (√32/√2 = 4).
The Maxwell-Boltzmann speed distribution describes the fraction of molecules having speeds in a given range. The distribution is asymmetric — it has a long tail towards higher speeds. As temperature increases: (1) the peak shifts to the right (higher most probable speed), (2) the peak height decreases (distribution flattens), (3) the distribution broadens. At a given temperature, heavier gases have narrower distributions peaked at lower speeds.
Important NEET point: the area under the Maxwell-Boltzmann curve always equals 1 (total fraction of molecules = 100%). The curve never reaches zero at very high speeds — there are always some molecules with very high speeds, however few.
The average translational kinetic energy per molecule is given by:
\(KE = \dfrac{3}{2}k_BT\)
where k_B = 1.38 × 10⁻²³ J/K is Boltzmann's constant and T is absolute temperature. This is a profound result: the average kinetic energy depends ONLY on temperature, regardless of the type of gas. At 300 K, every ideal gas molecule — whether it is a hydrogen molecule or a uranium hexafluoride molecule — has the same average translational KE of (3/2) × 1.38 × 10⁻²³ × 300 = 6.21 × 10⁻²¹ J.
Graham's law states that the rate of effusion (or diffusion) of a gas is inversely proportional to the square root of its molar mass:
\(\dfrac{r_1}{r_2} = \sqrt{\dfrac{M_2}{M_1}}\)
This is a direct consequence of the rms speed formula. Lighter gases effuse faster. This principle is used in separating isotopes (e.g., uranium-235 from uranium-238 using UF₆ gas) and in understanding the smell spreading across a room.
⚠️ Mass ratio is a trap: The ratio of gases in a mixture does NOT affect rms speed. Always ignore it when finding speed ratios.
⚠️ Celsius vs Kelvin: Always convert to Kelvin (T = °C + 273) before using any kinetic theory formula.
⚠️ Atomic vs Molecular mass: Use molar mass of the molecule, not the atom. Cl₂ has M = 70, not 35.
⚠️ Speed vs KE: At the same temperature, all gases have the same average KE per molecule. But they have different speeds — heavier gas has lower speed to have the same KE as lighter gas.