(533) |

(534) |

(535) |

Suppose that we now want to determine
: *i.e.*,
the mean number of molecules with positions between and
, and velocities in the range and
.
Since
, it is easily seen that

(536) |

(537) |

(538) |

(539) |

Here, is the number density of the molecules. We have omitted the variable in the argument of , since clearly does not depend on position. In other words, the distribution of molecular velocities is uniform in space. This is hardly surprising, since there is nothing to distinguish one region of space from another in our calculation. The above distribution is called the

Let us consider the distribution of a given component of velocity: the -component,
say. Suppose that is the average number of molecules per unit volume
with the -component of velocity in the range to , irrespective
of the values of their other velocity components. It is fairly obvious that
this distribution is obtained from the Maxwell distribution by summing (integrating
actually) over all possible values of and , with in the specified
range. Thus,

(541) |

(542) |

or

(543) |

(544) |

It is clear that each component (since there is nothing special about the -component) of the velocity is distributed with a Gaussian probability distribution
(see Sect. 2),
centred on a mean value

Equation (545) implies that each molecule is just as likely to be moving in the plus -direction as in the minus -direction. Equation (546) can be rearranged to give

(547) |

Note that Eq. (540) can be rewritten

(548) |

Suppose that we now want to calculate :
*i.e.*, the average number of molecules
per unit volume with a speed in the range to . It is
obvious that we can obtain this quantity by adding up all molecules with speeds
in this range, irrespective of the *direction* of their velocities. Thus,

(549) |

(550) |

(551) |

(552) |

(553) |

The mean molecular speed is given by

(554) |

(555) |

(556) |

(557) |

(558) |

However, this result can also be obtained from the equipartition theorem. Since

(560) |

(561) |

(562) |

(563) |

Figure 7 shows the Maxwell velocity distribution as a function of molecular speed in units of the most probable speed. Also shown are the mean speed and the root mean square speed.

It is difficult to directly verify the Maxwell velocity distribution. However, this distribution can be verified indirectly by measuring the velocity distribution of atoms exiting from a small hole in an oven. The velocity distribution of the escaping atoms is closely related to, but slightly different from, the velocity distribution inside the oven, since high velocity atoms escape more readily than low velocity atoms. In fact, the predicted velocity distribution of the escaping atoms varies like , in contrast to the variation of the velocity distribution inside the oven. Figure 8 compares the measured and theoretically predicted velocity distributions of potassium atoms escaping from an oven at C. There is clearly very good agreement between the two.