Physics · Kinetic Theory · NEET
You build the value in three steps, and the name lists them backwards. First you SQUARE every molecule's speed (mean square = average of v^2), then take the square ROOT. We do this because pressure and kinetic energy depend on v^2, not on v. So v_rms = sqrt(average of v^2). This is not the same as the ordinary average speed, which is just the sum of speeds divided by the number of molecules. RMS speed is always a little larger than the average speed.
Both give the same answer; just keep the pair together. Use v_rms = sqrt(3RT/M) when you are given the MOLAR mass M (mass of one mole, in kg/mol) and R = 8.314 J/mol/K. Use v_rms = sqrt(3kT/m) when you are given the mass of ONE molecule m (in kg) and k = 1.38 x 10^-23 J/K. Mixing them (R with m, or k with M) is the most common NEET slip. Remember R = k times Avogadro number, and M = m times Avogadro number, so the two forms are identical.
No. From v_rms = sqrt(3RT/M), the speed depends only on absolute temperature T and molar mass M. Pressure and volume do not appear. In many NEET problems the pressure is given only as a distractor. If the temperature is unchanged, v_rms is unchanged, no matter how you change the pressure or volume. This exact trap appeared in NEET 2016.
Molecules move in all directions, so if you averaged the velocity vectors you would get zero (equal numbers going left and right). Squaring makes every term positive and, more importantly, kinetic energy is (1/2)m v^2, so v^2 is the quantity that links directly to temperature and pressure. The mean of v^2 connects cleanly to (3/2)kT per molecule, which is why RMS speed, not plain average speed, comes out of the pressure derivation.
Order is v_mp < v_avg < v_rms. Roughly the ratio is about 1 : 1.13 : 1.22. All three are proportional to sqrt(T/M), so they rise and fall together, but RMS is always the largest. This ordering is a favourite NEET one-liner; the next concept page covers average and most probable speed in detail.
The molecules of a given mass of a gas have r.m.s. velocity of 200 m/s at 27 C and 1.0 x 10^5 N/m^2 pressure. When the temperature and pressure of the gas are respectively 127 C and 0.05 x 10^5 N/m^2, the r.m.s. velocity of its molecules in m/s is:
A flask contains argon and chlorine in the ratio of 2 : 1 by mass. The temperature of the mixture is 27 C. The ratio of root mean square speeds of the molecules of the two gases (v_rms of Ar / v_rms of Cl) is: (Atomic mass of argon = 40.0 u, molecular mass of chlorine = 70.0 u)
Match Column-I with Column-II. Column-I: (A) Root mean square speed of gas molecules (B) Pressure exerted by ideal gas (C) Average kinetic energy of a molecule (D) Total internal energy of 1 mole of a diatomic gas. Column-II: (i) (1/3) n m v^2_rms (ii) sqrt(3RT/M) (iii) (5/2) RT (iv) (3/2) k_B T.
Try the real previous-year questions from this chapter — each with the answer and a full solution.
v_rms = sqrt(3RT/M) = sqrt(3kT/m). Here T is absolute temperature in kelvin, M is molar mass in kg/mol with R = 8.314 J/mol/K, or m is the mass of one molecule in kg with k = 1.38 x 10^-23 J/K. Both forms give the same value.
Use v_rms = sqrt(3RT/M) with T = 273 K. For example, for oxygen (M = 0.032 kg/mol): v_rms = sqrt(3 x 8.314 x 273 / 0.032) which is about 461 m/s. Lighter gases like hydrogen move much faster at the same temperature.
Yes. v_rms is proportional to sqrt(T) where T is in kelvin. Doubling the absolute temperature multiplies the RMS speed by sqrt(2), about 1.41 times, not by 2. Always convert Celsius to kelvin first.
At a fixed temperature, v_rms is proportional to 1/sqrt(M). A heavier gas moves slower. For two gases at the same T, v_rms(1)/v_rms(2) = sqrt(M_2/M_1).
No. RMS speed is the root of the mean of squared speeds and is always slightly larger. The ordering is most probable < average < RMS, with a ratio of about 1 : 1.13 : 1.22.
