Molecular speed Maxwell distribution

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

According to kinetic theory, the speeds of molecules in a gas vary over a range of values. The British physicist James Clerk Maxwell (1831-1879) showed theoretically—and it has since been demonstrated experimentally—that molecular speeds are distributed as shown in Figure 5.25. This distribution of speeds depends on the temperatnre. At any temperature, the molecular speeds vary widely, but most are close to the average speed, which is close to the speed corresponding to the maximum in the distribution cnrve. As the temperature increases, the average speed increases. 

Maxwell distribution A relation describing the way in which molecular speeds or energies are shared among gas molecules, 121... 

A plot of the Maxwell distribution for the same gas at several different temperatures shows that the average speed increases as the temperature is raised (Fig 4.27). We knew that already (Section 4.9) but the curves also show that the spread of speeds widens as the temperature increases. At low temperatures, most molecules of a gas have speeds close to the average speed. At high temperatures, a high proportion have speeds widely different from their average speed. Because the kinetic energy of a molecule in a gas is proportional to the square of its speed, the distribution of molecular kinetic energies follows the same trends. 

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. 

The distribution function (24) for an ideal gas, shown in figure 6 is known as the Maxwell-Boltzmann distribution and is specified more commonly [118] in terms of molecular speed, as... 

D) Whether you can answer this question depends on whether you are acquainted with what is known as the Maxwell-Boltzmann distribution. This distribution describes the way that molecular speeds or energies are shared among the molecules of a gas. If you missed this question, examine the following figure and refer to your textbook for a complete description of the Maxwell-Boltzmann distribution. 

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

FIGURE 4.27 The range of molecular speeds for several gases, as given by the Maxwell distribution. All the curves correspond to the same temperature. The greater the molar mass, the lower the average speed and the narrower the spread of speeds. 

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. 

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. 

In Section 5.2, we will derive the three-dimensional Maxwell-Boltzmann distribution n(v)dv of molecular speeds between v and v + dv in the gas phase ... 

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). 

The apparently random stepwise or zig-zag movement of colloidal particles (Figure 6.1) was first observed by the botanist Robert Brown in 1827, and named after him. It provided early evidence for the molecular kinetic theory and was interpreted as arising from the random buffeting or jostling of the particles by molecules of the surrounding medium. The directions of movement of the molecules of the medium immediately adjacent to the particles are randomly oriented, while their speeds are distributed according to the Maxwell-Boltzmann law. The force acting upon the surface of a colloidal particle is proportional both to the frequency with which molecules collide with it and to the velocity of these molecules. The former is proportional to the local density of the molecules within one free path of the surface. Since the local... 

In Eq. (20.32) we have the result that the expectation value of the square of the energy is equal to the square of the expectation value of the energy. This could not be correct if the energy were in some way distributed. The reader will recall that in dealing with the Maxwell distribution of molecular speeds we found that... 

The main aspect of kinetic theory is that the molecules in a gas are in a state of continuous random motion. The speed of molecules depends on the temperature, and can have a remge of values. Maxwell s distribution curve is perfect to analyze this fact. In the Maxwell s distribution curve, the relative number of molecules are plotted against the molecular speed on the x-axis. Take a look at the curve in Figure 6-2. 

I describe the Maxwell-Boltzmann distribution of speeds and the effects of temperature and molar mass on molecular speed. 

Maxwell-Boltzmann distribution tells us the overall collection of molecular speeds but does not specify the speed of any individual particle. Energy exchange during molecular collisions can change the speed of individual molecules without disrupting the overall distribution. 

The Maxwell distribution of molecular speeds permits the evaluation of such important quantities as the pressure p exerted by a dilute gas and the collision frequency Z in the gas under given conditions. The pressure is then given by... 

FIGURE 5 Maxwell distributions of speeds for molecular nitrogen at 25°C (298 K) and 500°C (773 K). Arrows indicate v and vms for each case. The most probable velocity has been arbitrarily scaled to unity in each case. 

---