Mechanical Engineering Principles

23 Ideal gas laws

23.1 Introduction

The relationships that exist between pressure, vol-
ume and temperature in a gas are given in a set of
laws calledthe gas laws.

23.2 Boyle’s law

Boyle’s lawstates:

the volume V of a fixed mass of gas is inversely pro-

portional to its absolute pressure p at constant temper-

ature.

i.e.p∝

1

V

orp=

k

V

or pV=k at constant

temperature, wherep=absolute pressure in Pas-
cal’s (Pa),V=volume in m^3 ,andk=a constant.
Changes that occur at constant temperature are
calledisothermalchanges. When a fixed mass of
gas at constant temperature changes from pressure
p 1 and volumeV 1 to pressurep 2 and volumeV 2
then:

p 1 V 1 =p 2 V 2

Problem 1. A gas occupies a volume of

0.10 m^3 at a pressure of 1.8 MPa. Determine

(a) the pressure if the volume is changed to

0.06 m^3 at constant temperature, and (b) the

volume if the pressure is changed to 2.4 MPa

at constant temperature.

(a) Since the change occurs at constant tempera-
ture (i.e. an isothermal change), Boyle’s law
applies, i.e. p 1 V 1 = p 2 V 2 ,wherep 1 =
1 .8MPa,V 1 = 0 .10 m^3 andV 2 = 0 .06 m^3.
Hence( 1. 8 )( 0. 10 )=p 2 ( 0. 06 )from which,

pressurep 2 =

1. 8 × 0. 10

0. 06

=3MPa

(b) p 1 V 1 =p 2 V 2 wherep 1 = 1 .8MPa,

V 1 =0.10m^3 andp 2 = 2 .4 MPa.

Hence ( 1. 8 )( 0. 10 )=( 2. 4 )V 2 from which

volumeV 2 =

1. 8 × 0. 10

2. 4

= 0 .075 m^3

Problem 2. In an isothermal process, a

mass of gas has its volume reduced from

3200 mm^3 to 2000 mm^3. If the initial

pressure of the gas is 110 kPa, determine the

final pressure.

Since the process is isothermal, it takes place at

constant temperature and hence Boyle’s law applies,

i.e.p 1 V 1 =p 2 V 2 ,wherep 1 =110 kPa,

V 1 =3200 mm^3 and V 2 = 2000 mm^3.

Hence ( 110 )( 3200 )=p 2 ( 2000 ),from which,

final pressure, p 2 =

110 × 3200

2000

=176 kPa

Problem 3. Some gas occupies a volume of

1.5 m^3 in a cylinder at a pressure of

250 kPa. A piston, sliding in the cylinder,

compresses the gas isothermally until the

volume is 0.5 m^3. If the area of the piston is

300 cm^2 , calculate the force on the piston

when the gas is compressed.

An isothermal process means constant temperature

and thus Boyle’s law applies, i.e.p 1 V 1 = p 2 V 2 ,

whereV 1 = 1 .5m^3 ,V 2 = 0 .5m^3 andp 1 =250 kPa.

Hence, ( 250 )( 1. 5 )=p 2 ( 0. 5 ),from which,

pressure,p 2 =

250 × 1. 5

0. 5

=750 kPa

Pressure=

force

area

, from which,

force=pressure×area.