Concise Physical Chemistry

c01 JWBS043-Rogers September 13, 2010 11:20 Printer Name: Yet to Come

10 IDEAL GAS LAWS

x

y

z

FIGURE 1.3 The Gaussian Probability Density Distribution in 3-Space. The distribution

curve is in the fourth dimension of the space. The probability maximum is at the center of the

sphere.

along an axis defines aspace. For example, plottingxalong a horizontal axis defines

a one-dimensionalx-space. Space in thex,y, andzdimensions is the familiar3-space

often called a Cartesian space in honor of the seventeenth-century mathematician and

philosopher Rene Descartes. We usually plot functions along mutually perpendicular ́

ororthogonalaxes for mathematical convenience. If velocity is plotted along av

axis, we have a one-dimensional velocity space. If probability densityρis plotted

along one axis, and velocity is plotted along another axis, the result is aprobability

density–velocity spaceof two dimensions. Ifρ(v) is plotted invx,vyspace, the result

is a function in 3-space; or if it is thought of as a function of all three Cartesian

coordinates, the resulting function is in 4-space. That is,ρ(v) in 1-, 2-, or 3-space

gives a function in 2-, 3-, or 4-space, one dimension more. There should be nothing

terrifying about many-dimensionalspaceorhyperspace; it is merely an algebraic

generalization of the more commonplace use of the term.

The four-dimensional surface of the Gaussian distribution in Cartesian 3-space

cannot be precisely drawn but it can be imagined as a figure with spherical symmetry,

having a maximum at the center of the sphere. Imagine that the sphere in Fig. 1.3

can be rotated any amount in any angular direction, leaving the distribution curve

unchanged.

1.10 THE SUM-OVER-STATES OR PARTITION FUNCTION

Adding up all the particles in all the states of a system gives the total number of

particles in the system:

∑

Ni=N