Physical Chemistry Third Edition

392 9 Gas Kinetic Theory: The Molecular Theory of Dilute Gases at Equilibrium

c.The kinetic energy of a particle of massmis defined by

K 

1

2

mv^2

wherevis its speed. If the particle of part b is a neon atom, find its kinetic energy.

There is a vast difference between the amount of information in the specification of

the microscopic state and the specification of the macroscopic state of our model gas.

Specification of the equilibrium macrostate requires the values of only three variables

such asT,V, andn. The microstate of the model system is specified by giving the

values of three coordinates and three velocity components for each of theNparticles.

If the model system contains roughly 1 mol of particles, specification of the microstate

requires approximately 4× 1024 values, which makes it impossible to list them, even

if we would determine the values. To apply our postulate, we need a way to average

mechanical variables over the microstates without being able to list all of them.

Mean Values of Mechanical Variables

Any variable that depends on the positions and velocities of the particles is determined

by the state of the system and is called amicroscopic state functionor amechanical

state function. The most important mechanical state function of our model system is

the energy, which is the sum of thekinetic energyand thepotential energy:

EK +V (9.2-12)

The kinetic energy of the system,K, is the sum of kinetic energies of all of the particles,

K 

m

2

(

v^21 +v^22 +v^23 + ··· +v^2 N

)



m

2

∑N

i 1

v^2 i (9.2-13)

where we use the fact that all of the particles in our model system have the same

mass. The potential energy of the system,V, is a state function of the positions of the

particles:

VV(r 1 ,r 2 ,...,rn) (9.2-14)

The component of the force on particle numberiin thexdirection is

Fix−

∂V

∂xi

(9.2-15)

with other force components given by analogous equations.

We assume for now that the gravitational forces on the particles of our system can

be neglected. The molecules of our model system do not interact with each other and

have no forces exerted on them except by the walls of the box confining the system.

Their potential energy is constant so long as they remain within the box. We set this

potential energy equal to zero (you can always set a constant potential energy equal to

zero or to any other constant so long as you do it consistently). In order to represent

the confinement of the particles in the box, we assign an infinite value to the potential

energy if any particle is outside of the box. Since the total energy of the system must

be finite, no particle can escape from the box, and the potential energy remains equal

to zero.