The Distribution of Molecular Speeds

The graph below shows the distribution of molecular speeds for helium and carbon dioxide at the same temperature. 

The distribution of molecular speeds in gas can be determined experimentally. To du so, the gas is heated to the required temperature in an oven. The molecules then stream out of the oven through a small hole into an evacuated region. To ensure that the molecules form a narrow Iteam, they may also pass through a senes of slits, and the pressure must Ik kept very low so that collisions within the beam do not cause spreading. 

Max Planck once said, Experiments are the only means of knowing that we have at our disposal the rest is theory and imagination. How do we know. . . signals a box that introduces either a crucial experiment or experimental procedure. How do we know the distribution of molecular speeds How do we know that an electron has spin ... 

Even though an increase in temperature results in an increase in the average kinetic energy, not all of the gas molecules will have the same amount of kinetic energy. This is why the term average kinetic energy is used. The two curves in Figure 2.5 show the distribution of molecular speeds of a gas at two... 

The schematic form of the distribution of molecular speeds in a gas at two different temperatures. The blue line corresponds to the lower temperature, and the red line to the higher temperature. 

We are now in a position to solve the original problem, namely to find the distribution of molecular speeds and to evaluate the constants A and P that appear in the distribution function. 

Second, suppose we keep the volume fixed but increase the temperature. How does this change affect the entropy of the system Recall the distribution of molecular speeds presented in Figure 10.17(a). An increase in temperature increases the most probable speed of the molecules and also broadens the distribution of speeds. Hence, the molecules have a greater number of possible kinetic energies, and the number of microstates increases. Thus, the entropy of the sj stem increases with increasing temperature. 

A Figure 10.14 shows the distribution of molecular speeds for several gases at 25 °C. Notice how the distributions are shifted toward higher speeds for gases of lower molar masses. The most probable speed of a gas molecule can also be derived ... 

Figure 5.6 I The distributions of molecular speeds for CO2 molecules at three different temperatures are shown. The quantity on the y axis is the fraction of gas molecules moving at a particular speed. Notice that as the temperature increases, the fraction of molecules moving at higher speed increases.

The distribution of molecular speeds can be danonstrated with the apparatus shown in Figure 5.18. A beam of atoms (or molecules) exits from an oven at a known temperature and passes through a pinhole (to collimate the beam). Two circular plates mounted on the same shaft are rotated by a motor. The first plate is called the chopper and the second is the detector. The purpose of the chopper is to allow small bursts of atoms (or molecules) to pass through it whenever the slit is aligned with the beam. Within each burst, the faster-moving molecules will reach the detector earlier... 

The distribution of molecular speeds in a gas was first studied by Maxwell in 1860 and later refined by Boltzmann. For a gas sample containing N molecules of mass m at a temperature T, they showed that the number of molecules (7V ) in the sample with molecular speeds in the range m to m-i- du is given by... 

The distribution of molecular speeds can be demonstrated with the apparatus shown in Figure 5.20. A beam of atoms (or molecules) exits from an oven at a known temperature and passes through a pinhole (to collimate the beam). Two circular plates... 

Maxwell-Boltzmann distribution of molecular speeds. The distribution of molecular speeds in a gas, given by Equation 5.36. (5.4)... 

Figure 5.12 depicts the distribution of molecular speeds ( ) for N2 gas at three temperatures. The curves flatten and spread at higher temperatures. Note especially that the most probable speed (the peak of each curve) increases as the temperature increases. This increase occurs because the average kinetic energy of the molecules (EiZ the overbar indicates the average value of a quantity), which incorporates the most probable speed, is proportional to the absolute temperature T, or... 

At the boiling point of the liquid another abrupt increase in entropy occurs. We can understand this increase as resulting from the increased volume in which the molecules may be found. The increase in volume means an increase in randomness. As the gas is heated, the entropy increases steadily as more eneigy is stored in the translational motion of the gas molecules. At W er temperatures the distribution of molecular speeds is spread out toward hi er values (Figure 10.18). The expansion of the range of speeds of the gas molecules leads to the increased kinetic energy and increased disorder, and hence increased entropy. 

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