Aim of the Experiment: To plot for Maxwell's speed distribution formula (in speed and translational energy form) and calculation of the fraction of molecules in a given speed range by numeral integration.
Theory: The distribution of molecular speed as described by Maxwell is a deserving case for computer use because of the large computational work involved. The molecules of the given gas system (with a definite mass, say m, for each of its molecules) obey a definite distribution for their speed depending on the temperature T. The distribution is expressed in terms of the number of molecules dN (out of the total N molecules, where N must be very large, say 1018 ) with their speeds lying between V and V + dV, expressed as a function of V and dV. Relations used are (p = 3.14159, T is absolute temperature, K is Boltzmann constant):
mass of the molecule, m = Molar mass (M) / (6.022.1023 mol-1)
Number dN = 4p.N [m /(2 pkT)] 1.5 .v2 exp {–mv2 /(2KT)} dV
Number N12 lying between V1 and V2 = Integration result (from V1 to V2) of dN
The molecular speeds have a distribution that is centered around the most probable speed (mp), such that for a small speed-range around the speed mp, a large fraction of the molecule reside (have speeds in that range) as is clear from the figure below:
Procedure: Using the MolSpeed program available as a part of the Computers in Chemistry Experiments Set from www.geocities.com/riturajkalita/compu_chemi.htm, the no. of molecules in different speed ranges within a definite range of speeds are investigated for two different gases (viz. O2 and H2 ) at two different temperatures ( viz. 250C and 1000C ). In the program the molar mass of the gas, system temperature, total no. of molecules and the speed range are to be entered, and then on pressing the 'Continue' button the computational investigation proceeds and the computational results are obtained in the file 'MolSpeed.txt'
Results: The results for the molecular speeds of the two gases system and the two different temperatures are attached herewith as photocopies of print-out.
The no. of molecules lying within the speed range out of an Avogadro no. of molecules (6.022.1023) for the two gas system at two different temperatures are as tabulated below:
| Molar mass of gas (g/ mol) | Total number of molecules | System temperature (0C) | Speed range (ms-1) | Most probable speed (ms-1) | Number of molecules in speed range | |
| Lower | Upper | |||||
| 2.016( H2) | 6.022.1023 | 298.15 | 1555.0 | 1585.0 | 1568.166424 | 95.6249.1020 |
| 373.15 | 1740.0 | 1770.0 | 1754.350912 | 85.4777.1020 | ||
| 32.0 (O2) | 6.022.1023 | 298.15 | 380.0 | 410.0 | 393.606649 | 380.6267.1020 |
| 373.15 | 425.0 | 455.0 | 440.338584 | 340.3039.1020 | ||
Conclusion: As is expected, a large fraction of molecules lies within a small speed range around the most probable speed. Also the most probable speed is observed to shift towards large values as the system temperature is increased (as is known from the speed-distribution theory)