PHYSICAL	PROPERTIES		5.169
TABLE 5.30 Triple Points of Various Materials (	Continued	)
Substance	Triple point, K		Pressure, mmHg
Phosphorus, white	863		32 760
Plutonium hexafluoride	324.74		533.0
Propene	103.95		500
Radon	202
Rhenium dioxide trifluoride	363
Rhenium heptafluoride	321.4
Rhenium oxide pentafluoride	313.9
Rhenium pentafluoride	321
Succinonitrile (NIST standard)	331.23
Sulfur dioxide	197.68		1.256
Tantalum pentabromide	553
Tantalum pentachloride	489.0
Tungsten oxide tetrafluoride	377.8
Uranium hexafluoride	337.20		1 139.6
Water	273.16
Xenon	161.37		612

5.9.1Some Physical Chemistry Equations for Gases

A number of physical chemistry relationships, not enumerated in other sections (							see	Index), will be
discussed in this section.
Boyle’s law	states that the volume of a given quantity of a gas varies inversely as the pressure,
the temperature remaining constant. That is,
	V		constant		or	PV	constant
			P
A convenient form of the law, true strictly for ideal gases, is
				P V1	1 P	V2 2
Charles’ law,	also known as	Gay-Lussac’s law,				states	that the volume of a given mass	of gas

varies directly as the absolute temperature if the pressure remains constant, that is,

V constant

T

Combining the laws of Boyle and Charles into one expression gives

				P V1	1	P V 2	2
				T	1	T	2
In	terms of moles,	Avogadro’s	hypothesis		can be stated: The same volume is occupied by one
mole of any gas at a given temperature and pressure. The number of molecules in one mole is known
as the	Avogadro number constant	N	A .
The behavior of all gases that obey the laws of Boyle and Charles, and Avogadro’s hypothesis,
can be expressed by the ideal gas equation:
				PV		nRT

5.170

SECTION

5

where

R is called the

gas

constant

and

n

is the

number

of

moles of

gas. If pressure is written as

force per unit area and the volume as area times length, then

R

has the dimensions of energy per

degree per mole— 8.314 J · K

1 · mol 1 or 1.987 cal · K 1 · mol 1.

Dalton’s law of partial pressures

states

that

the total

pressure exerted

by a mixture of gases is

equal to the sum of the pressures which each component would exert if placed separately into the

container:

P total

p1

p2

p3

· · ·

There are two ways to express the fraction which one gaseous component contributes to the total

mixture: (1) the pressure fraction,

pand/i P(2)totalthe, mole fraction,

n i /n total .

5.9.1.1

Equations

of State

(PVT

Relations

for Real Gases

)

1.

Virial equation

represents the experimental compressibility of a gas by an empirical equation

of state:

PV

A p B pP

C p P

2 · · ·

or

PV

A v B vV

C v

· · ·

V

2

where

A , B , C

, . . .

are called the virial coefficients and are a function of the nature of the gas and

the temperature.

2.

Van der Waals’ equation:

P

an 2

(V

nb

)

nRT

V 2

where the term

an 2 /V

2 is

the correction for intermolecular attraction among the gas molecules and

the nb

term is the correction for the volume occupied by the gas molecules. The constants

a

and b

must be fitted for each gas from experimental data (Table 5.28); consequently the equation is

semiempirical. The constants are related to the critical-point constants (Table 6.5) as follows:

a

3P

Vc

2

b

V

c

3

R

8P Vc

c

3T c

Substitution into van der Waals’ equation and rearrangement leads to only the terms

P /P c , V

/V

c , and

T /T c , which are called the reduced variables

and P R , ForV R

,1 moleTRof .gas,

V R2

3

3

3

1

8

3.

P R

V

R

T

R

Berthelot’s equation of

state,

used by many thermodynamicists, is

2

128

P Tc

T

9

PT

c

T

2

c

PV

nRT

1

1

6

PHYSICAL PROPERTIES

5.171

This equation requires only knowledge of the critical temperature and pressure for its use and gives accurate results in the vicinity of room temperature for unassociated substances at moderate pressures.

5.9.1.2 Properties of Gas Molecules

Vapor Density.

Substitution of the Antoine vapor-pressure equation for its equivalent log

P in

the ideal gas equation gives

log vap log M

log

R

log ( t 273.15) A

B

t

C

where

vap is the vapor density in g · mL

1 at

t C,

M

is the molecular weight,

R

is the gas constant,

and A

, B , and

C are the constants of the Antoine equation for vapor pressure. Since this equation is

based on the ideal gas law, it is accurate only at temperatures at which the vapor of any specific

compound follows this law. This condition prevails at reduced temperatures (

T

R ) of about 0.5 K.

Velocities of

Molecules.

The mean square velocity of gas molecules is given by

3kT

3RT

u 2

m

M

where

k is Boltzmann’s constant and

m is the mass of the molecule.

The mean velocity is given by

3

8u 2

1/2

u

Viscosity. On the assumption that molecules interact like hard spheres, the viscosity of a gas is

165 2 mkT 1/2

where is the molecular diameter.
Mean Free Path.	The mean free path of a gas molecule									l and the mean time between collisions
are given by
	l				m
		2 p
							2
		1				4
						5P
			u

Graham’s Law of Diffusion. The rates at which gases diffuse under the same conditions of temperature and pressure are inversely proportional to the square roots of their densities:

		r 1		2			1/2
		r 2		1
Since MP /RTfor an ideal gas, it follows that			M
	r 2				1
	r 1		M		2		1/2

5.172	SECTION	5
Henry’s Law.	The solubility of a gas is directly proportional to the partial pressure exerted by
the gas:
	p i kx i
Joule-Thompson	Coefficient for Real Gases.		This expresses the change in temperature with
respect to change in pressure at constant enthalpy:
			T	H
			P