Distribution of molecular speeds

Click Coached Problems for a self-study module on distribution of molecular speeds. 

Distribution of molecular speeds of oxygen molecules at 25°C and 1000°C. At the higher temperature the fraction of molecules moving at very high speeds is greater. 

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. 

J I I Describe the effect of molar mass and temperature on the Maxwell distribution of molecular speeds (Section 4.11). 

Maxwell distribution of molecular speeds The formula for calculating the percentage of molecules that move at any given speed in a gas at a specified temperature. 

Therefore the three-dimensional Maxwell-Boltzmann distribution of molecular speeds is... 

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 ... 

This result was given in Eq. (2.28). The well-known Maxwell-Boltzmann distribution of molecular speeds, Eq. (2.27), is obtained after substitution of E = mv 2/2, dE = mvdv. 

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... 

A simple example of a system of this kind is a mixture of perfect gases capable of reacting chemically, in which there is a Maxwellian distribution of molecular speeds, but where the concentrations are not those corresponding to chemical equilibrium among the components. 

It is useful to have a complete picture of the entire distribution of molecular speeds. This turns out to be important when we study chemical kinetics (see Chapter 18), where we will need to know what fraction of a sample of molecules has kinetic energy above the minimum necessary for a chemical reaction. In particular, we would like to know what fraction of molecules, AN/N, have speeds between u and u + Am. This fraction gives the speed distribution function f u) ... 

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). 

I See the Saunders Interactive General Chemistry CD-ROM, Screen 12.11, Distribution of Molecular Speeds. 

Figure 5.12 Distribution of molecular speeds at three temperatures. At a...

This is the result for the 7-diffusion model. In the M-diffusion model the molecules have a distribution of molecular speeds, p(coo), but each molecule retains the same speed through all collisions. Letting Cj<°>(/o wo), Ci(n,t a>o) denote the free particle and n-free step-correlation... 

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