Physics · Kinetic Theory · NEET
Molecules do not all move at the same speed. They have a spread of speeds (the Maxwell-Boltzmann distribution). So we describe the whole gas with three summary numbers. Most probable speed Vp = sqrt(2RT/M) is the single speed the largest number of molecules have (the peak of the curve). Average speed Vavg = sqrt(8RT/piM) is the simple arithmetic mean of all speeds. RMS speed Vrms = sqrt(3RT/M) is the square root of the mean of the squares, and it connects to energy because kinetic energy uses v squared. Same gas, three honest ways to summarise.
Always Vp less than Vavg less than Vrms. Compare the numbers under the root: 2 for most probable, 8/pi = 2.55 for average, 3 for RMS. Since 2 less than 2.55 less than 3, RMS is largest. In numbers the ratio is 1 : 1.128 : 1.224. RMS is biggest because squaring gives extra weight to the fast molecules, pulling that average up.
Cancel RT/M because it is common to all three. You are left with sqrt(2) : sqrt(8/pi) : sqrt(3) = 1.414 : 1.596 : 1.732. Divide every term by 1.414 (the smallest) to get 1 : 1.128 : 1.224. Memorise this final ratio directly, it saves time in the exam.
Use the fixed ratio. Vp = Vrms times (1/1.224) = 0.816 times Vrms. Vavg = Vrms times (1.128/1.224) = 0.921 times Vrms. Or go from most probable: Vavg = 1.128 times Vp and Vrms = 1.224 times Vp. All three change with temperature and molar mass in the same way (proportional to sqrt(T/M)), so the ratio never changes.
Yes, every one of them is proportional to sqrt(T/M). Heat the gas (raise T) and all three speeds rise together. Use a heavier gas (larger molar mass M) and all three drop together. But the ratio 1 : 1.128 : 1.224 stays exactly the same, because T and M cancel out when you take ratios.
Try the real previous-year questions from this chapter — each with the answer and a full solution.
Vp : Vavg : Vrms = sqrt(2) : sqrt(8/pi) : sqrt(3) = 1 : 1.128 : 1.224. Most probable speed is smallest and RMS speed is largest.
RMS squares each speed before averaging, so fast molecules count more heavily and push the value up. That is why sqrt(3) beats sqrt(8/pi), giving Vrms greater than Vavg.
Average (mean) speed Vavg = sqrt(8RT/piM) = sqrt(8 kB T / pi m). It sits between most probable and RMS speed.
No. Because sqrt(2), sqrt(8/pi) and sqrt(3) are three different numbers, the three speeds are always distinct for any real gas at any temperature above absolute zero.
RMS speed. Average kinetic energy per molecule is (1/2) m Vrms squared = (3/2) kB T, so RMS connects directly to temperature and energy.
