MOLECULAR DISTRIBUTION OF SPEEDS: In a sample of a gas the molecules are in a continuous and random motion as we already know. This is the reason why it is expected that the speeds of the molecules will lie in widespread ranges. Maxwell and Boltzmann studied the speed distribution in a sample of an ideal gas at a given temperature. We will not study the exact equation that they gave for the distribution but will study the distribution curve of speeds which is very important.(MAXWELL-BOLTZMANN DISTRIBUTION).
If we plot the fraction of molecules versus velocity graph for a given sample of an ideal gas at a particular temperature we obtain the following curve:

To calculate the probability that a molecule will have a speed in the range v1 to v2, we integrate the distribution between those two limits; the integral is equal to the area of the curve between the limits, as shown shaded here.
This is called probabilistic distribution curve. The area covered between any two velocities as shown represents the molecules present in that range of speeds. The total area under the curve represents total number of molecules in the sample. And therefore the ratio of these two gives us the fraction of molecules lying in the specified range of speeds. There are some very vital conclusions that can be drawn from the above curve:
1.    In the sample, molecules with very low velocity and those with very high velocities both are less in number as the curve is narrow at both the extremes.
2.    Most of the molecules lie around the central point of the curve which is called Most Probable point of the curve. Maximum area is covered around this point. The speed at this point is called “Most Probable Speed” (uMP).
3.    The area under the curve gives the total number of gas molecules.
4.    There are two more molecular speeds defined for a sample called “Average speed (uAVG)” and “Root mean square speed (uRMS)”.
These speeds are given by the following expressions:
uMP =   2RT /M    uAVG =   8RT / пM    uRMS =   3RT /M
These speeds are represented on the curve. It can be observed that
                            uRMS > uAVG > uMP
The ratio of these speeds is:
              uMP: uAVG : uRMS = 1: 1.128: 1.224
5.    On increasing the temperature of the sample, the velocities of the molecules increases and the curve takes a shift to the right, the area under the curve remaining constant. Now, the fraction in the lower velocity range will be lesser and that in the higher velocity range will be higher. The molecular speeds also increase with temperature as shown.

Fig. The Maxwell Boltzmann distribution of speeds and its variation with the temperature. Note the broadening of the distribution and the shift of the rms speed (denoted by the locations of the vertical lines) to higher values as the temperature is increased.

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