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TRT 401 PHYSICAL CHEMISTRY. PART 1: INTRODUCTION TO PHYSICAL CHEMISTRY. What is physical chemistry? Physical chemistry is a study of the physical basis of phenomena related to the chemical composition and structure of substances . Or

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trt 401 physical chemistry




What is physical chemistry?

Physical chemistry is a study of the physical basis of phenomena related to the chemical composition and structure of substances.


Physical chemistry is quantitative and theoretical study of the

properties and structure of matter, and their relation to the interaction

of matter with energy.

  • This course serves as an introduction to chemical thermodynamics, giving you an understanding of basic principles, laws and theories of physical chemistry the are necessary for chemistry, biochemistry, pre-medical, general science and engineering
  • students.
  • You will develop the ability to solve quantitative problems, and learn to use original thought and logic in the solution of problems and derivation of equations.
  • You will learn to apply mathematics in chemistry in such a way that the equations paint a clear picture of the physical phenomena

Physical chemistry includes numerous disciplines:

  • Thermodynamics - relationship between energy interconversion
  • by materials, and the molecular properties
  • Kinetics - rates of chemical processes
  • Quantum Mechanics - phenomena at the molecular level
  • Statistical Mechanics - relationships between individual
  • molecules and bulk properties of matter
  • Spectroscopy - non-destructive interaction of light (energy) and
  • matter, in order to study chemical structure
  • Photochemistry - interaction of light and matter with the intent of
  • coherently altering molecular structure

Atoms and Molecules

Atoms are the submicroscopic particles that constitute the fundamental building

block of ordinary matter. They are most found in molecules, two or more atoms

joined in specific geometrical arrangement.

Carbon dioxide molecule

Carbon monoxide molecule





In the study of chemistry, atoms are often portrayed as colored spheres, with each

color representing a different kind of atom.

For example, a black sphere represents a carbon atom, a red sphere represents

an oxygen atom.




For more interactive periodic table please refer


Chemical Bonding


Sodium metal

Chlorine gas



Sodium Chloride

Oppositely charged ions are held together by ionic bonding,

forming a crystalline lattice.

Ionic compound

A compound that composed of cations and anions

bound together by electrostatic attraction


Covalent Compund

Compound that not contain a metallic element typically covalent compound consisting of discrete molecules.

Single bond


  • The shared pairs of electrons are bonding pairs.
  • The unshared pairs of electrons are lone pairs or non bonding pairs.

Polar Covalent Bond

  • In reality, fully ionic and covalent bonds represent the extremes of a spectrum
  • of bonding types.
  • Most of covalent bonds are polar covalent bonds, in which the electrons are
  • shared unequally, but are nit fully transfered from one atom to the other.
  • In polar bonds, one atom has a partial negative charge(δ-) and the other atom
  • has a partial positive charge(δ+).

Metallic Bonding

  • Metallic bonding, occurs in metals. Since metals have low ionization energies, they tend
  • to electron easily.
  • When metal atoms bonding together form a solid, each metal atom donates one or more
  • electron to an electron sea.

Example: sodium metal as an array of

positively charged Na+ ion immersed in

a sea of negatively charges electron (e-).

The table below summarize the three different types of bonding



The ability of an atom to attract electrons to itself in a chemical bond

(which results in polar bonds) is called electronegativity, χ (chi).

Fluorine is the most

Electronegativity element

Francium is the

least electronegative


Electronegativity generally increase as we move across a row in periodic table and

decrease as we move down a column.


Electronegativity and Bond Polarity

Bond polarity expressed numerically as dipole moment, μwhich occurs when

there is a separation between a positive and negative charge.

μ = qr

μ: dipole moment; q: separating a proton and an electron; r: distance



q = 1.6 x 10-19 C

r = 130 pm (the approximate length of a short chemical bond)

μ= q x r

= (1.6 x 10-19 C)(130 x 10-12m)

= 2.1 x 10-29 . M

= 6.2 D

The debye (D) is a common unit used for reporting dipole moment (1D = 3.34 x 10-30 C.m)

Table 1 Dipole moments of several molecules in the gas phase



Matter: anything that occupies space and has mass.

Example: book, desk, pen, pencil even your body is are all compose of matter.

Air also matter but it too occupies space and matter. These specific instance of matter- such as air, sand, water- a substance.

Matter can be classify to its state- solid, liquid, or gas according to its composition.



Glass, plastic &



Table salt, ice & diamond

Solid matter may be crystalline, in which case its atoms or molecules are

arranged in patterns with long range, repeating order.


Its may be amorphous in which case its atoms or molecules do not have any

long range order.


In Gaseous Matter

Gases can be compressed-squeezed into a small volume because there is so much

Empty space between atoms or molecules in the gaseous state.


The Composition of Matter

Matter can be classified as either pure substances, which have fixed composition,

or mixtures, which have variable composition.

Pure substance (element and compounds) are unique materials with their own

chemical and physical properties, and are composed of only one type of atom or



A pure substance that is composed

of atoms or two more different



A substance that cannot

be chemically broken down

Into simpler substance.






The Composition of Matter

Mixture are simply random combinations of two or more different types of atoms of

molecules, and retain the properties of the individual substances. They can therefore

be separated (although sometime with difficulty) by physical means (such as boiling,

distillation, melting, crystallizing, magnetism, etc.)

Heterogenus Mixture

One in which the composition varies

from one region to another.

Homogenus Mixture

One with the same composition



Summary Composition of Matter

Variable composition?







Pure water

Wet sand

Helium gas

Tea with sugar


Quantifying Matter

Amount of substance(n): a measure of a number of specified entities (atoms, molecules, or formula unit) present (unit; mole; mol).

1 mol of a substance contains as many entities as exactly 12 g of carbon-12 (ca. 6.02 x 1023 objects)

Avogadro’s Number: NA = 6.02 x 1023 mol-1

Extensive Property: Dependent upon the amount of matter in

the substance (e.g., mass & volume)

Intensive Property: Independent of the amount of matter in a

substance (e.g., mass density, pressure and temperature)

Molar Property: Xm, an extensive property divided by the

amount of substance, n: Xm = X/n

Molar Concentration:“Molarity” moles of solute dissolved in

litres of solvent: 1.0 M = 1.0 mol L-1



  • In science, the most commonly used set of units are those of the International System of Units (the SI System, for Système International d’Unités).
  • There are seven fundamental units in the SI system. The units for all other quantities (e.g., area, volume, energy) are derived from these base units.

Table 2 Example list of units

for more info:


SI Prefix – Small & Large Unit

Table 3 SI Prefix Multipliers


SI Prefix – Large Unit

Table 4 SI Prefix – large unit



Energy is define as the ability to do work.

Work is done when a force is exerted through a distance.

Force through distance; work is done.

Energy is measured in Joules (J) or Calories (cal).

1 J = 1 kg m2 s-2

Energy may be converted from one to another, but it is neither created nor destroyed (conversion of energy).

In generally, system tend to move from situations of high potential energy (less stable) to situations having lower energy (more stable).


Energy is the capacity to supply heat or to do work. Energy can be exchanged

between objects by some combination of either heat or work:

Energy = heat + work

∆E= q + w

  • work is done when a force is exerted through a distance
  • work = force x distance
  • heat is the flow of energy caused by a temperature
  • difference.

Example of a billiard ball rolling across the table

and colliding straight on with a second,

stationary billiard ball.


Potential and Kinetic Energy

  • Kinetic energy (EK) is the energy due to the motion of an object with mass m and velocity v:
  • EK = ½ mv2
  • – Thermal energy, the energy associated with the temperature of an object, is a form of kinetic energy, because it arises from the vibrations of the atoms and molecules within the object.
  • Potential energy (EP) is energy due to position, or any other form of “stored” energy. There are several forms of potential energy:
  • – Gravitational potential energy
  • – Mechanical potential energy
  • – Chemical potential energy (stored in chemical bonds)

Potential energy increases when things that attract each other are separated or when things that repel each other are moved closer.

• Potential energy decreases when things that attract each other are moved closer, or when things that repel each other are separated.

• According to the law of conservation of energy, energy cannot be created or destroyed, but kinetic and potential energy can be interconverted.


Energy transformation I

Energy transformation II


Water falling in a waterfall exchanges gravitational potential energy for kinetic energy as it falls faster and faster, but the energy is never destroyed.

EK converted to

thermal energy

and sound

high EPdecreasing EP low EP

low EK increasing EK high EP


Chemical Energy

The chemical potential energy of a substance results from the relative positions and

the attractions and repulsions among all its particles. Under some circumstances, this

energy can be released, and can be used to do work:

Using chemical energy to do work – The compound produced when gasoline burns have

Less chemical potential energy than the gasoline molecules.


Law of Conservation of Energy

  • A law stating that energy can neither be created nor destroyed, only converted from one form to another, and it can assume in different forms.
  • E.g.:

The cycle held the

gravitational energy

The energy transformed

into kinetic energy of



Contributions to Energy

Kinetic Energy, EK: Energy an object possesses as a result of its motion.

KE = ½mv2

Potential Energy, V: Energy an object possesses as a result of its position. Zero of potential energy is relative:

1. Gravitational Potential Energy:

zero when object at surface (V = 0 when h = 0)

VG = mgh, m = mass, g = 9.81 m s-2, h = height

2. Electrical Potential Energy:

zero when 2 charged particles infinitely separated

qi = charge on particle i, r = distance

ε0 = 8.85 x 10-12 C2 J-1 m-1

(vacuum permittivity)


Equipartition of Energy

Molecules have a certain number of degrees of freedom: they can vibrate, rotate and translate - many properties depend on these degrees of freedom:

Equipartition theorem:

All degrees of freedom have the same average energy at temperature T: total energy is partitioned over all possible degrees of freedom

Quadratic energy terms:

½mvx2 + ½mvy2 + ½mvz 2

Average energy associated with each quadratic term is ½kT, where k = 1.38 x 10-23 J K-1 (Boltzmann constant), where k is related to the gas constant, R = 8.314 J K-1 mol-1 by R = NAk

However: this theorem is derived by classical physics, and can only be applied to translational motion.


Relationship Between Molecular and Bulk Properties

The energy of a molecule, atom, or subatomic particle that is confined to a region of space

Is quantized, or districted to certain discrete values. These permitted energies are called

energy level.


Populations of States

At temperatures > 0, molecules are distributed over available energy levels according to the Boltzmann Distribution, which gives the ratio of particles in each energy state:

Boltzmann constant k=1.381 x 10-23 JK-1

At the lowest temperature T = 0, only the lowest energy state is occupied. At infinite temperature, all states are equally occupied.

In real life, the population of states is described by an exponential function, with the highest energy states being the least populated.

Degenerate states: States which have the same energy

These will be equally populated!


Boltzmann Distributions

Populations for (a) low

& (b) high temperatures

Boltzmann predicts an

exponential decrease in

population with

increasing temperature

At room T, only the

ground electronic state

is populated. However,

many rotational states

are populated, since the

energy levels are so

closely spaced.

More states are

significantly populated if

energy level spacing

are near kT!


The Electromagnetic Field

Light is a form of electromagnetic radiation. In classical physics, electromagnetic

radiation is understood in term of the electromagnetic field.

Electric field – charged particles (whether stationary or moving)

Magnetic field – acts only on moving charged particles.


Wavelength ,λ is the distance between the neighboring peaks of two wave, and is

frequency, v (nu) the number of times in a given interval at which its displacement

at a fixed point returns to its original value divided by the length of the time intervals.

Frequency is measure in Hertz, 1 Hz= 1 s-1.

Wavelength, λ = c/ v

Wavenumber, ṽ (nu tilde) the number of complete wavelengths in a given length.

Wavenumber, ṽ = v/c =1/λ

e.g.: A wave number of 5 cm-1 indicates there are 5 complete wavelength in 1 cm.


Matter: Substance, intensive and extensive properties, molarity and molality


  • A substance is a distinct, pure form of matter.
  • The amount of a substance, n, in a sample is reported in terms of the unit called a mole (mol). In 1 mol are NA=6.0221023 objects (atoms, molecules, ions, or other specified entities). NA is the Avogadro constant.

Extensive and intensive properties

 An extensive property is a property that depends on the amount of substance in the sample.

Examples: mass, volume…

  • An intensive property is a property that is independent on the amount of substance in the sample.

Examples: temperature, pressure, mass density…

 A molar property Xm is the value of an extensive property X divided by the amount of substance, n: Xm=X/n. A molar property is intensive. It is usually denoted by the index m, or by the use of small letters. The one exemption of this notation is the molar mass, which is denoted simply M.

 A specific property Xs is the value of an extensive property X divided by the mass m of the substance: Xs=X/m. A specific property is intensive, and usually denoted by the index s.

Measures of concentration: molarity and molality

 The molar concentration(‘molarity’) of a solute in a solution refers to the amount of substance of the solute divided by the volume of the solution. Molar concentration is usually expressed in moles per litre (mol L-1 or mol dm-3). A molar concentration of x mol L-1 is widely called ‘x molar’ and denoted x M.

  • The term molalityrefers to the amount of substance of the solute divided by the mass of the solvent used to prepare the solution. Its units are typically moles of solute per kilogram of solvent (mol kg-1).

Some fundamental terms:

System and surroundings:

For the purposes of Physical Chemistry, the universe is divided into two parts, the system and its surroundings.

 The system is the part of the world, in which we have special interest.

 The surroundings is where we make our measurements.

The type of system depends on the characteristics of the boundary which divides it from the surroundings:

(a) An open system can exchange matter and energy with its

(b) A closed system can exchange energy with its surroundings, but it cannot exchange matter.

(c) An isolated system can exchange neither energy nor matter with its surroundings.

Except for the open system, which has no walls at all, the walls in the two other have certain characteristics, and are given special names:

 An adiabatic (isolated) system is one that does not permit the passage of energy as heat through its boundary even if there is a temperature difference between the system and its surroundings. It has adiabatic walls.

 A diathermic (closed) system is one that allows energy to escape as heat through its boundary if there is a difference in temperature between the system and its surroundings. It has diathermic walls.


H2O (water)


1 bar

H2 + ½ O2


1 bar






Homogeneous system:

The macroscopic properties are identical in all parts of the system.

Heterogeneous system:

The macroscopic properties jump at the phase boundaries.


Homogeneous part of a (possibly) heterogeneous system.

Equilibrium condition:

 The macroscopic properties do not change without external influence.

 The system returns to equilibrium after a transient perturbation.

 In general exists only a single true equilibrium state.

Equilibrium in Mechanics:

Equilibrium in Thermodynamics:


The concept of “Temperature”:

 Temperature is a thermodynamic quantity, and not known in mechanics.

 The concept of temperature springs from the observation that a change in physical state (for example, a change of volume) may occur when two objects are in contact with one another (as when a red-hot metal is plunged into water):

If, upon contact of A and B, a change in any physical property of these systems is found, we know that they have not been in thermal equilibrium.






The Zeroth Law of thermodynamics:

If A is in thermal equilibrium with B, and B is in thermal equilibrium with C, than C is also in thermal equilibrium with A. All these systems have a common property: the same temperature.

Energy flows as heat from a region at a higher temperature to one at a lower temperature if the two are in contact through a diathermic wall, as in (a) and (c). However, if the two regions have identical temperatures, there is no net transfer of energy as heat even though the two regions are separated by a diathermic wall (b). The latter condition corresponds to the two regions being at thermal equilibrium.



Linear relation between the Celsius temperature  and an observable quantity x, like the length of a Hg column, the pressure p of a gas at constant volume V, or the volume V of the gas for constant pressure p:

Left: The variation of the volume of a fixed amount of gas with the temperature constant. Note that in each case they extrapolate to zero volume at -273.15 C.

Right: The pressure also varies linearly with the temperature, and extrapolates to zero at

T= 0 (-273.15 C).

For the pressure p, this transforms to:


For all (ideal) gases one finds

 Introduction of the thermodynamic temperature scale (in ‘Kelvin’):


The thermodynamic temperature scale:

In the early days of thermometry (and still in laboratory practice today), temperatures were related to the length of a column of liquid (e.g. Mercury, Hg), and the difference in lengths shown when the thermometer was first in contact with melting ice and then with boiling water was divided into 100 steps called ‘degrees’, the lower point being labelled 0. This procedure led to the Celsius scale of temperature with the two reference points at 0 °C and 100 °C, respectively.


Work, heat, and energy:

 The fundamental physical propertyin thermodynamics is work: work is done when an object is moved against an opposing force.

(Examples: change of the height of a weight, expansion of a gas that pushes a piston and raises the weight, or a chemical reaction which e.g. drives an electrical current)

 The energy of a system is its capacity to do work. When work is done on an otherwise isolated system (e.g. by compressing a gas or winding a spring), its energy is increased. When a system does work (e.g. by moving a piston or unwinding the spring), its energy is reduced.

 When the energy of a system is changed as a consequence of a temperature difference between it and the surroundings, the energy has been transferred as heat. When, for example, a heater is immersed in a beaker with water (the system), the capacity of the water to do work increases because hot water can be used to do more work than cold water.

Heat transfer requires diathermic walls.

 A process that releases energy as heat is called exothermic, a process that absorbs energy as heat endothermic.

(a) When an endothermic process occurs in an adiabatic system, the temperature falls; (b) if the process is exothermic, then the temperature rises. (c) When an endothermic process occurs in a diathermic container, energy enters as heat from the surroundings, and the system remains at the same temperature; (d) if the process is exothermic, then energy leaves as heat, and the process is isothermal.


When energy is transferred to the surroundings as heat, the transfer stimulates disordered motion of the atoms in the surroundings. Transfer of energy from the surroundings to the system makes use of disordered motion (thermal motion) in the surroundings.

When a system does work, it stimulates orderly motion in the surroundings. For instance, the atoms shown here may be part of a weight that is being raised. The ordered motion of the atoms in a falling weight does work on the system.

Work, heat, and energy (continued):

Molecular interpretation

  • In molecular terms, heat is the transfer of energy that makes use of chaotic molecular motion (thermal motion).
  • In contrast, work is the transfer of energy that makes use of organized motion.
  • The distinction between work and heat is made in the surroundings.

State functions and state variables


  • If only two intensive properties of a phase of a pure substance are known, all intensive properties of this phase of the substance are known, or
  • If three properties of a phase of a pure substance are known, all properties of this phase of the substance are known.

example: - p and T as independent variables means: Vm (=v) = f(p,T),

i.e. the resulting molar volume is pinned down, or

- p, T, n as independent variables means: V = f(p,T,n)

  • The resulting function is termed a state function.
  • The variables which describe the system state, are termed

- state variables, and are related to each other via the

- state functions.





The thermal equation of state and the perfect gas equation

Thermal equation of state:

  • The thermal equation of state combines volumeV, temperatureT, pressurep, and the amount of substancen:

V = f(p,T,n)orVm = v = f(p,T)

The “perfect gas” (or “ideal gas”):

  • mass points without expansion
  • no interactions between the particles
  • a real gas, an actual gas, behaves more and more like a perfect gas the lower the pressure, and the higher the temperature

Some empirical gas laws:

V = f(T) for p=const.:


p = f(T) for V=const.:


p = f(V) for T=const.:


V = const.  ( + 273.15°C)

= const.’  T

p = const.  ( + 273.15°C)

= const.’  T

p  V = const.

Boyle’s Law

Charles’s Law


Step 1: Isobaric change

Step 2: Isothermal change


= const. !

  • Combination of 1 and 3 for:
  •  1 mol gas at
  •  p0 = 1.013 bar
  •  T0 = 273.15 K
  •  v0 = 22.42 l

A region of the p,V,T surface of a fixed amount of perfect gas. The points forming the surface represent the only states of the gas that can exist.

Sections through the surface shown in the figure at constant temperature give the isotherms shown for the Boyle-Mariotte law and the isobars shown for the Gay-Lussac law.

p  v = R  T

p  V = n  R  T

‘perfect gas equation’

R : ‘gas constant’

(= 8.31434 J K-1 mol-1)


Step 1: Isothermal change

Step 2: Isobaric change


= const. !

  • Swap the changes:
  • a combination of 3 and 1 for
  •  1 mol gas at
  •  p0 = 1.013 bar
  •  T0 = 273.15 K
  •  v0 = 22.42 l

The change of a state variable is independent of the path, on which the change of the state has been made, as long as initial and final state are identical.

  • Some mathematical consequences:
  • The change can be described as an ‘exact differential’, i.e. the variables can be varied independently; e.g. for z=f(x,y):
  • The mixed derivatives are identical (Schwarz’s theorem):
  • Upon variation of x, y for z=const (Euler’s theorem):

 same result !!!


A more general approach to thermal expansion and compression:

 V = f(T) for p=const.:V = V0 (1 + )


 : (thermal) expansion coefficient

 p = f(T) for V=const.:p = p0 (1 + )

 p = f(V) for T=const.:pV = const. and d(pV) = pdV + Vdp = 0


 : (isothermal) compressibility

generally valid!

 Due to generally valid!

Exact differential of V=f(p,T):


The mole fraction, xJ, is the amount of J expressed as a fraction of the total amount of molecules, n, in the sample:

When no J molecules are present, xJ=0; when only J molecules are present, xJ=1. Thus the partial pressure can be defined as:

The partial pressure of a gas is the pressure that it would exert if it occupied the container alone. If the partial pressure of a gas A is pA, that of a perfect gas B is pB, and so on, then the partial pressure when all the gases occupy the same container at the same temperature is

where, for each substance J,


The partial pressures pA and pB of a binary mixture of (real or perfect) gases of total pressure p as the composition changes from pure A to pure B. The sum of the partial pressures is equal to the total pressure. If the gases are perfect, then the partial pressure is also the pressure that each gas would exert if it were present alone in the container.

Mixtures of gases: Partial pressure and mole fractions

Dalton’s law:

The pressure exerted by a mixture of perfect gases is the sum of the partial pressures of the gases.


Compression factor

For a perfect gas, Z=1 under all conditions. Deviation of Z from 1 is a measure of departure from perfect behaviour.

Molecular interactions

Real gases show deviations from the perfect gas law because molecules (and atoms) interact with each other: Repulsive forces (short-range interactions) assist expansion, attractive forces (operative at intermediate distances) assist compression.

At very low pressures, all the gases have Z1 and behave nearly perfect. At high pressure, all gases have Z>1, signifying that they are more difficult to compress than a perfect gas, and repulsion is dominant. At intermediate pressure, most gases have Z<1, indicating that the attractive forces are dominant and favor compression.

The variation of the potential energy of two molecules on their separation. High positive potential energy (at very small separations) indicates that the interactions between them are strongly repulsive at these distances. At intermediate separations, where the potential energy is negative, the attractive interactions dominate. At large separations (on the right) the potential energy is zero and there is no interaction between the molecules.

The variation of the compression factor Z = pv/RT with pressure for several gases at 0C. A perfect gas has Z = 1 at all pressures. Notice that, although the curves approach 1 as p  0, they do so with different slopes.

Real gases: An introduction


Below, some experimental isotherms of carbon dioxide are shown. At large molar volumes v and high temperatures the real isotherms do not differ greatly from ideal isotherms. The small differences suggest an expansion in a series of powers either of p or v, the so-called virial equations of state:

Experimental isotherms of carbon dioxide at several temperatures. The `critical isotherm', the isotherm at the critical temperature, is at 31.04 C. The critical point is marked with a star.

The virial equation can be used to demonstrate the point that, although the equation of state of a real gas may coincide with the perfect gas law as p  0, not all of its properties necessarily coincide. For example, for a perfect gas dZ/dp = 0 (because Z=1 for all pressures), but for a real gas

as p  0, and

as v  , corresponding to p  0.

Real gases: The virial equation of state

The third virial coefficient, C, is usually less important than the second one, B, in the sense that at typical molar volumes C/v2<<B/v. In simple models, and for p  0, higher terms than B are therefore often neglected.


Because the virial coefficients depend on the temperature (see table above), there may be a temperature at which Z1 with zero slope at low pressure p or high molar volume v. At this temperature, which is called the Boyle temperature, TB, the properties of a real gas coinicide with those of a perfect gas as p 0, and B=0. It then follows that pvRTB over a more extended range of pressures than at other temperatures.

Real gases: The Boyle temperature

The compression factor approaches 1 at low pressures, but does so with different slopes. For a perfect gas, the slope is zero, but real gases may have either positive or negative slopes, and the slope may vary with temperature. At the Boyle temperature, the slope is zero and the gas behaves perfectly over a wider range of conditions than at other temperatures.


Reconsider the experimental isotherms of carbon dioxide. What happens, when gas initially in the state A is compressed at constant temperature (by pushing a piston)?

Real gases: Condensation and critical point

  • Near A, the pressure rises in approximate agreement with Boyle’s law.
  • Serious deviations from the law begin to appear when the volume has been reduced to B.
  • At C (about 60 bar for CO2), the piston suddenly slides in without any further rise in pressure. Just to the left of C a liquid appears, and there are two phases separated by a sharply defined surface.
  • As the volume is decreased from C through D to E, the amount of liquid increases. There is no additional resistance to the piston because the gas can respond by condensation. The corresponding pressure is the vapour pressure of the liquid at this temperature.
  • At E, the sample is entirely liquid and the piston rests on its surface. Further reduction of volume requires the exertion of a considerable amount of pressure, as indicated by the sharply rising line from E to F. This is due to the low compressibility of condensed phases.
  • The isotherm at the temperature Tc plays a special role :
  • Isotherms below Tc behave as described above.
  • If the compression takes place at Tc itself, a surface separating two phases does not appear, and the volumes at each end of the horizontal part of the isotherm have merged to a single point, the critical point of the gas. The corresponding parameters are the critical temperature, Tc, critical pressure, pc, and critical molar volume, vc, of the substance.
  • The liquid phase of a substance does not form above Tc.

Real gases: Critical constants, compression factors, Boyle temperatures, and the supercritical phase

A gas can not be liquefied if the temperature is above its critical temperature. To liquefy it - to obtain a fluid phase which does not occupy the entire volume - the temperature must first be lowered to below Tc, and then the gas compressed isothermally.

The single phase that fills the entire volume at T>Tc may be much denser then is normally considered typical of gases. It is often called the supercritical phase, or a supercritical fluid.


Comparison to the virial equation of state:

The surface of possible states allowed by the van der Waals equation.

The van der Waals equation of gases: A model

P= nRT – a n2

V- nb V2

Starting point: The perfect gas lawpv = nRT


Correction 1: Attractive forces lower the pressure

 replace p by (p+∏), where  is the ‘internal pressure’. More detailed analysis shows that =a/v2.

P= RT – a

Vm- b Vm2

Van der Waals equation

Correction 2: Repulsive forces are taken into account by supposing that the molecules (atoms) behave as small but impenetrable spheres

 replace v by (v-b), where b is the ‘exclusion volume’. More detailed analysis shows that b is approximately the volume of one mole of the particles.

a, b: van der Waals coefficients


(3) The critical constants are related to the van der Waals constants.

At the critical point the isotherm has a flat inflexion. An inflexion of this type occurs if both the first and second derivative are zero:

at the critical point. The solution is

and the critical compression factor, Zc, is predicted to be equal to

for all gases.

Van der Waals isotherms at several values of T/Tc. The van der Waals loops are normally replaced by horizontal straight lines. The critical isotherm is the isotherm for T/Tc = 1.

(2) Liquids and gases coexist when cohesive and dispersing effects are in balance. The ‘van der Waals loops’ are unrealistic because they suggest that under some conditions an increase in presure results in an increase of volume. Therefore they are replaced by horizontal lines

drawn so the loops define

equal areas above and below the

lines (‘Maxwell construction’)

Analysis of the van der Waals equation of gases

(1) Perfect gas isotherms are obtained athigh enough temperatures and large molar volumes.



If the critical constants are characteristic properties of gases, than characteristic points, like melting or boiling point, should be unitary defined states. We therefore introduce reduced variables

and obtain the reduced van der Waals equation:

b) Reduced melting temperature

c) Reduced boiling temperature


a) Compression factors

The compression factors of four gases, plotted for three reduced temperatures as a function of reduced pressure. The use of reduced variables organizes the data on to single curves.

The principle of corresponding states

d) Trouton’s rule (or: Pictet-Troutons’s rule)

A wide range of liquids gives approximately the same standard entropy of vaporization of

S  85 J K-1 mol-1


 approximation!

 works best for gases composed of spherical particles

 fails, sometimes badly, when the particles are non-spherical or polar