1549380323-Statistical Mechanics Theory and Molecular Simulation

72 Theoretical foundations

d. Finally, suppose that the volume changes fromV 1 toV 2 in an adiabatic

process (∆Q= 0). The pressure also changes fromP 1 toP 2 in the process.

Show that

P 1 V 1 γ=P 2 V 2 γ,

and find the numerical value of the exponentγ.

2.2. A substance has the following properties:

i. When its volume is increased fromV 1 toV 2 at constant temperatureT,

the work done in the expansion is

W=RTln

(

V 2

V 1

)

.

ii. When the volume changes fromV 1 toV 2 and the temperature changes

fromT 1 toT 2 , its entropy changes according to

∆S=k

(

V 1

V 2

)(

T 2

T 1

)α

,

whereαis a constant. Find the equation of state and Helmholtz free energy

of this substance.

2.3. Reformulate the Carnot cycle for an ideal gas as a thermodynamic cycle in

theT–Splane rather than theP–Vplane, and show that the area enclosed

by the cycle is equal to the net work done by the gas during the cycle.

2.4. Consider the thermodynamic cycle shown in Fig. 2.5. Compare theefficiency

of this engine to that of a Carnot engine operating between the highest and

lowest temperatures of the cycle in the figure. Which one is greater?

2.5. Consider an ensemble of one-particle systems, each evolving in one spatial

dimension according to an equation of motion of the form

x ̇=−αx,

wherex(t) is the position of the particle at timetandαis a constant. Since the

compressibility of this system is nonzero, the ensemble distribution function

f(x,t) satisfies a Liouville equation of the form

∂f

∂t

−αx

∂f

∂x

=αf

(see eqn. (2.5.7)). Suppose that att= 0, the ensemble distribution has a

Gaussian form

f(x,0) =

1

√

2 πσ^2

e−x

(^2) / 2 σ 2
.