Molecular speeds, probability distribution

Figure 1-7 A Molecular Speed Distribution. The probability density is the expected number of speeds within an infinitesimal speed interval dv.

Find a formula for the most probable molecular speed, cmp. Sketch the Maxwell-Boltzmann velocity distribution and show the relative positions of (c), cmp, and cms on your sketch. 

PROBLEM 4.20.2. Show for a Maxwell-Boltzmann distribution of Eq. (4.20.1) that the most probable molecular speed vmp is given by Eq. (4.20.6). 

FIG U R E 9.14 The Maxwell-Boltzmann distribution of molecular speeds in nitrogen at three temperatures. The peak in each curve gives the most probable speed, u p, which is slightly smaller than the root-mean-square speed, Urms The average speed Uav (obtained simply by adding the speeds and dividing by the number of molecules in the sample) lies in between. All three measures give comparable estimates of typical molecular speeds and show how these speeds increase with temperature. 

Use the Maxwell-Boltzmann distribution of molecular speeds to calculate root-mean-square, most probable, and average speeds of molecules in a gas (Section 9.5, Problems 41-44). 

Thus, although the individual molecular motions are chaotically unpredictable, their average behavior is entirely predictable and satisfies a particular probability distribution (the Maxwell-Boltzmann distribution). Quantities, such as the temperature that appear in Boyle s Law are measures of the average speed of the molecules. If we reran the tape of the history of our gas, we would find essentially the same average behavior, in accord with Boyle s Law, even though the individual trajectories of the molecules would be quite different. 

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 10.13 Distribution of molecular speeds for nitrogen gas. (a) The effect of temperature on molecular speed. The relative area under the curve for a range of speeds gives the relative fraction of molecules that have those speeds, (b) Position of most probable (Ump), average (Uav), and root-mean-square (u,rns) speeds of gas molecules. The data shown here are for nitrogen gas at 0°C. 

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 ... 

The kinetic theory of gases assumes that molecules have negligible size compared to their separation, are in continuous random motion, and interact only via elastic scattering. These postulates permit the calculation of molecular speed and velocity distributions. The probability that a molecule has a speed between v and u - - du is found to be... 

Daive Equation 5.37 for the most probable speed (m) from the MaxweU-Boltzmann distribution of molecular speeds (Equation 5.36). [Hint The daivative of a function is zero at its maximum (or minimum).]... 

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... 

Boltzmann made many other significant contributions to science, particularly in the area oi statistical mechanics, which is the derivation of bulk thermodynamic properties for large collections of atoms or molecules by using the laws of probability. For example, the molecular speed distributions shown in Figure 10.19 are derived by using statistical mechanics such plots are known as Maxu eU—Boltzmann distributions. 

In the sample as a whole, each particle has a molecular speed (m) most are moving near the most probable speed, but some are much faster and others much slower. Figure 5.13 depicts this distribution of molecnlar speeds for N2 gas at three temperatures. 

Rgure 9.12 The Probability Distribution of Molecular Speeds for Oxygen Molecules at 298 K. 

The probability distribution for molecular speeds is The mean speed of molecules in our model gas is given by... 

The probability of a gas molecule having a speed c lying in a range between c and c + dc is given by the Maxwell distribution law for molecular speeds ... 

It is probably obvious at this point that the molecular weight distribution of a resole is an extremely important characteristic. It has major influence on such important performance capabilities as cure speed, viscosity, green strength development (or prepress), assembly time tolerance, required application rates. 

A pattern emerges when this molecular beam experiment is repeated for various gases at a common temperature Molecules with small masses move faster than those with large masses. Figure 5 shows this for H2, CH4, and CO2. Of these molecules, H2 has the smallest mass and CO2 the largest. The vertical line drawn for each gas shows the speed at which the distribution reaches its maximum height. More molecules have this speed than any other, so this is the most probable speed for molecules of that gas. The most probable speed for a molecule of hydrogen at 300 K is 1.57 X 10 m/s, which is 3.41 X 10 mi/hr. 

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