Chapter 1. The Properties of Gases

2011 Fall Semester Physical Chemistry 1

(CHM2201)

Contents

The Perfect Gas 1.1 The states of gases 1.2 The gas laws

Real Gases

1.3 Molecular interactions 1.4 The van der Waals equation

States of Matter

Gases – fill container, random rapid motion, never coming to rest or clumping together. Motion is mainly translational

Liquids – fixed volume, flow and assume shape of container, only slightly compressible, stronger forces hold molecules together. Motion is mainly translational

Solids – fixed volume, definite shape, generally less compressible than liquids, forces hold particles in a fixed shape. Motion is mainly vibrational

1.1 The states of gases

Key points 1.  Pressure provides a criterion of mechanical

equilibrium for systems free to change their volume

2.  Pressure is measured with a barometer 3.  The Zeroth Law of thermodynamics

1.1 The states of gases

Physical State 1.  Two samples of a substance that have the same

physical properties are in the same state 2.  Physical Properties : V, n, p, and T 3.  It is sufficient to specify only three of these

variables 4.  p = f(T,V,n) : an equation of state

(e.g.) p = nRT/V for a perfect gas

1.1 The states of gases

(a) Pressure •  Defined as force, F, divided by the area, A •  p = F/A

Mechanical equilibrium

1.1 The states of gases

(b) The measurement of pressure •  barometer •  Hydrostatic pressure : P = ρgh

ü  ρ : the mass density of liquid ü  g : gravitational acceleration ü  h : height

1.1 The states of gases

(c) Temperature •  Indicates the direction of the flow of energy

through a thermally conducting rigid wall •  Types of boundary between two objects at

different temperatures ü  Diathermic : a change of state is observed ü  Adiabatic : no change occur

•  Thermal equilibrium : No change of state occurs when two objects are in contact through a diathermic boundary

1.1 The states of gases

(c) Temperature

Thermal equilibrium

1.1 The states of gases The Zeroth Law of thermodynamics •  If A is in thermal equilibrium with B, and B is in

thermal equilibrium with C, then C is also in thermal equilibrium with A

•  Justifies the concept of temperature and the use of a thermometer

•  Celcius scale : °C •  Thermodynamic temperature scale : K (kelvins)

T/K = θ/°C + 273.15

The perfect gas : 1.2 The gas laws

Key points 1.  The perfect gas law summerizes Boyle’s

and Charles’s laws and Avogadro’s principle

2.  The kinetic theory of gases provides a model for the gas laws

3.  A mixture of perfect gases behaves like a single perfect gas

1.2 The gas laws

(a) The perfect gas law •  Boyle’s law : pV = constant at constant n, T •  Charles’s law : V = constant × T at constant n, P

P = constant × T at constant n, V •  Avogadro’s principle : V = constant × n at

constant p, T

isotherm isobars isochores

1.2 The gas laws

(a) The perfect gas law •  pV = nRT : perfect gas law •  R : gas constant •  Perfect (ideal) gas vs. Real gas

1.2 The gas laws (a) The perfect gas law •  pV = nRT : perfect gas law •  R : gas constant •  Perfect (ideal) gas vs. Real gas •  The perfect gas law is

important because it is used to derive a wide range of relations that are used throughout thermodynamics

1.2 The gas laws

(b) The kinetic model of gases 1.  The gas consists of molecules of mass m in

ceaseless random motion 2.  The size of the molecules is negligible, in the

sense that their diameters are much smaller than the average distance travelled between collisions

3.  The molecules interact only through brief, infrequent, and elastic collisions

1.2 The gas laws

(b) The kinetic model of gases

pV =13nMc2

M : the molar mass of the molecules c : the root mean-square speed of the molecules

c = v21/2=3RTM

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pV = nRT

1.2 The gas laws

(c) Mixtures of gases •  Dalton’s law : The pressure exerted by a

mixture of gases is the sum of the pressures that each one would exert if it occupied the container alone.

pJ = xJ p : partial pressure

xJ =nJN

: mole fraction

pA + pB += (xA + xB +)p = p

The real gas : 1.3 Molecular Interactions

Key points 1.  The extent of deviations from perfect

behavior is summarized by the compression factor

2.  The virial equation is an empirical extension of the perfect gas equation

3.  The isotherms of a real gas introduce the concept of vapor pressure and critical behavior

4.  A gas can be liquefied by pressure alone only if T ≤ TC.

1.3 Molecular interactions

•  Repulsive forces and attractive forces assist expansion and compression, respectively

1.  At low pressure, the intermolecular forces play no significant role

2.  At moderate pressures, the attractive forces dominate

3.  At high pressures, the repulsive forces dominate

1.3 Molecular interactions

1.  At low pressure, Z ≈ 1 2.  At moderate pressures, Z < 1 3.  At high pressures, Z > 1

Z = VmVm

o =pVmRT

: Compression factor

(a) Compression factor

1.3 Molecular interactions

pVm = RT (1+B ' p+C ' p2 +)

= RT 1+ BVm

+CVm2 +

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= RTZ

(b) Virial coefficients

dZdp

= B '+ 2pC '+→ B as p→ 0

The virial equation of state

The virial coefficients depend on T dZ/dp⟶0 as p⟶0 at Boyle temperature

1.3 Molecular interactions (c) Condensation

Relatively ideal

Deviate from the perfect gas law

•  A liquid appears •  The pressure is

constant •  Two phases exist

vapor pressure

Experimental Isotherms of carbon dioxide

1.3 Molecular interactions (c) Critical constants

•  At TC, a surface separating two phases does not appear

•  Critical Point : the volumes at each end of the horizontal part of the isotherm merge to a critical point

•  TC : critical temperature •  pC : critical pressure •  VC : critical molar volume •  At T > TC, the supercritical fluid

Experimental Isotherms of carbon dioxide

1.4 The van der Waals equation

Key points 1.  The van der Waals equation is a model

equation of state with two parameters 2.  The van der Waals equation captures the

general features including the critical behavior

3.  The properties of real gases are coordinated by expressing their equations of state in terms of reduced variables

1.4 The van der Waals equation (a) Formulation of the equation

p = nRTV − nb

− a n2

V 2 =RT

Vm − b−aVm2

•  a : the strength of attractive interactions •  b : the volume taken by molecules

1.4 The van der Waals equation (b) The features

p = nRTV − nb

− a n2

V 2 =RT

Vm − b−aVm2

•  Perfect gas isotherms are obtained at high T and large molar volumes

•  Liquid and gases coexist when cohesive and dispersing effects are in balance

•  The critical constants are related to a and b

The unrealistic Van der Waals loops need be treated by the Maxwell construction

1.4 The van der Waals equation (b) The features

p = nRTV − nb

− a n2

V 2 =RT

Vm − b−aVm2

•  The critical constants are related to a and b

dpdVm

= −RT

(Vm − b)2 +2aVm3 = 0

d 2pdVm

2 =2RT

(Vm − b)3 −6aVm4 = 0

VC = 3b, pC =a

27b2 , TC =8a

27Rb

ZC =pCVCRTC

=38

1.4 The van der Waals equation (b) The principle of corresponding states

p = nRTV − nb

− a n2

V 2 =RT

Vm − b−aVm2

•  Reduced variables

Vr =VmVC

, pr =ppC

, Tr =TTC

pr =8Tr3Vr −1

−3Vr2

•  Real gases at the same reduced volume and reduced temperature exert the same reduced pressure