Physical Chemistry Third Edition

50 2 Work, Heat, and Energy: The First Law of Thermodynamics

∫

dP

nR

V 1

∫T 2

T 1

dT−nRT 2

∫V 2

V 1

1

V^2

+

nR

V 2

∫T 1

T 2

dT



nR

V 1

(T 2 −T 1 )+nRT 2

(

1

V 2

−

1

V 1

)

+

nR

V 2

(T 1 −T 2 )

( 1 .2516 mol)(8.3145 J K−^1 mol−^1 )

(

− 98 .15 K

0 .00500 m^3

)

+( 1 .2516 mol)(8.3145 J K−^1 mol−^1 )(200.00 K)

(

1

0 .0100 m^3

−

1

0 .0050 m^3

)

+( 1 .2516 mol)(8.3145 J K−^1 mol−^1 )

(

98 .15 K

0 .01000 m^3

)

− 2. 0428 × 105 Jm^3 − 2. 0813 × 105 Jm^3 + 1. 0214 × 105 Jm^3

− 3. 103 × 105 Jm^3  3 .062 atm

This value is the same as in the previous example, showing that the integral ofdPis

path-independent so far as these two paths are concerned.

Exercise 2.4

a.Calculate the amount of work done on the surroundings if the isothermal expansion of

Example 2.2 is carried out at a constant transmitted pressure of 1.000 atm instead of rever-

sibly, but with the same initial and final states as in Example 2.2. Why is less work done

on the surroundings in the irreversible process than in the reversible process?

b.What is the change in the pressure of the system for the irreversible process?

PROBLEMS

Section 2.1: Work and the State of a System

2.1Calculate the work done on the surroundings if 1.000 mol of

neon (assumed ideal) is heated from 0.0◦C to 250.0◦Cata

constant pressure of 1.00 atm.

2.2Calculate the work done on the surroundings if 100.0 g

of water freezes at 0.0◦C and a constant pressure of

1.00 atm. The density of ice is 0.916 g cm−^3 and that of

liquid water is 1.00 g cm−^3.

2.3Calculate the work done on 100.0 g of benzene if it is

pressurized reversibly from 1.00 atm to 50.00 atm at a

constant temperature of 293.15 K.

2.4Calculate the work done on the surroundings if 1.000 kg of

water is heated from 25.0◦C to 100.0◦C at a constant

pressure of 1.00 atm.

2.5 a.The tension force for a spring that obeys Hooke’s law is

given by

τ−k(x−x 0 )

wherexis the length of the spring,x 0 is its equilibrium

length, andkis a constant called the spring constant.

Obtain a formula for the work done on the spring if its

length is changed reversibly fromx 0 tox′at constant

volume.

b.Show that the force can be derived from a potential

energy

V

1

2

k(x−x 0 )^2

and that the work done on the spring in part a is equal to

the change in the potential energy.