Maxwell-Boltzmann speed theory

An alternative interpretation of the Maxwell-Boltzmann speed distribution is helpful in statistical analysis of the experiment. Experimentally, the probability that a molecule selected from the gas will have speed in the range Au is defined as the fraction AN/N discussed earlier. Because AN/N is equal to f u) Au, we interpret this product as the probability predicted from theory that any molecule selected from the gas will have speed between u and u + Au. In this way we think of the Maxwell-Boltzmann speed distribntion f(u) as a probability distribution. It is necessary to restrict Au to very small ranges compared with u to make sure the probability distribution is a continuous function of u. An elementary introdnction to probability distributions and their applications is given in Appendix C.6. We suggest you review that material now. 

To obtain Eqs (1.203) and (1.206) we need to assume that P vanishes asx - 00 faster than Physically this must be so because a particle that starts at x = 0 cannot reach beyond some finite distance at any finite time if only because its speed cannot exceed the speed of light. Of course, the diffusion equation does not know the restrictions imposed by the Einstein relativity theory (similarly, the Maxwell-Boltzmann distribution assigns finite probabilities to find particles with speeds that exceed the speed of light). The real mathematical reason why P has to vanish faster than jg that in... 

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... 

The distribution function that describes the speeds of a collection of gas particles is known as the Maxwell-Boltzmann distribution of speeds. This function, which was originally derived from a detailed consideration of the postulates of the kinetic theory, predicts the fraction of the molecules in a gas that travel at a particular speed. It has since been verified by a variety of clever experiments that allow measuring the speed distribution in the laboratory. 

FIGURE 4.21 Distribution of speeds in a gas as predicted by the kinetic theory of gases (Maxwell-Boltzmann distribution). As T increases or M decreases, the distribution flattens and spreads to higher overall speeds. 

The 19th century saw numerous extensions of the atomic theory, one of the most important being the kinetic theory of gases. The macroscopic behavior of gases may be explained by considering them to be composed of molecules in rapid random motion, with characteristic speeds of the order of several hundred meters per second (m/s). The continual collisions of the highspeed molecules with the walls constitute the pressure exerted by the gas on a container. The molecules exhibit a characteristic statistical spread of velocities, known as the Maxwell-Boltzmann distribution, which will be discussed in detail in a later chapter because of its relevance to the physics of neutrons in a scattering medium such as a reactor lattice. 

Since the molecules in a gas move at great speeds, they collide with one another billions of times per second at room temperature and pressure. An individual molecule frequently speeds up and slows down as it undergoes these elastic collisions. However, within a short period of time the distribution of speeds of all the molecules in a given system becomes constant and well defined. It is termed the Maxwell-Boltzmann distribution and can be derived using the kinetic theory of gases. 

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