CHEM 155: BASIC PHYSICAL CHEMISTRY I

Paradigm Shift in our conception

Physical Chemistry and its Problems of Primary Concern

Structure of Science and its Classification

Theory Development: Concepts, Construct, Relationship, Proposition, Laws, Hypothesis and Models.

States of Matter I: Classification, Structure and Properties of Matter, System & State Variables and Equation of State

Thermodynamics I: First Law, Heat capacity, Enthalpy and thermochemistry.

Chemical Kinetics I: Elementary Chemical Kinetics, Basic Laws, Effects of Temperature and the Arrhenius equation.

MATTER

States of Matter, Structure (the way something is put

together) and Property.

State and Variables of a System

Equation of State.

Gas Laws

Kinetic theory of Gases

Real Gases

Evans Adei CHEM 155

States of Matter

Matter – Definition:

Aggregation of atoms/ions/molecules which come within the scope of human experience i.e. any substance that has mass and occupies space.

States of Matter:

Matter exists in one of the following six states due to the energy of its particles: Gaseous, Liquid, Solid, Plasma, Bose-Einstein and Filament. The states of matter are also known as phases of matter or states of aggregation.

The division of matter into states is not always simple. How, for example, should chocolate spread be considered. Somescientists beleive that the so called colloids should beconsidered.

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0 1 2 3 4 5

Name : Bose-Einstein Solid Liquid Gaseous Plasma Filament

Energy : ̃~zero <crystal <attract. <attract. <105 eV <1026 eV

Character : non-thermal thermal Thermal thermal thermal non-

thermal

Particle

motion

no motion Particles move in all three dimensions motion in

only one

direction

Six States of Matter

Evans Adei CHEM 155

State Property determined by Structure

Gas (simplest

state of matter)

Kinetic energy is more important

than potential energy.

Intermolecularforce are weak.

Collection of particles in continuous

random chaotic motion.

Has no well-defined surface.

No limit on extent of space or

volume it can occupy.

Liquid Potential energy is more important

than kinetic energy.

Intermolecular forces strong

No long-range order but vestiges of

crystal lattice structure at short

range.

Has a surface and place a limit on

the extent of space or volume it can

occupy.

Solid Potential energy is more important

than kinetic energy.

Intermolecular force strong

Generally possesses crystal lattice

structure –well defined shape

NB. Glass and solids exists as amorphous

solids.

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Thermodynamic System and Surrounding

A ( thermodynamic) system is that part of the physicaluniverse that is separated from the rest of the universeby a real or imaginary boundaries for consideration or discussion.

The part of the universe outside the boundary of the system is referred to as surroundings i.e. where wemake our observations.

Boundaries are of four types: fixed, moveable, real, and imaginary.

For closed systems, boundaries are real while for open system boundaries are often imaginary

Evans Adei CHEM 155

Exchange with the surrounding

Matter Heat Work

Isolated No No No

Closed No Yesa

Nob

Yes

Open Yes Yes Yes

Possible exchange between a system and its surroundingsa: system with diathermal walls.

b: System with adiabatic walls.

Evans Adei CHEM 155

PROPERTY

A property is an essential or distinctive attribute

(quality) of something or a quality or characteristics

that something has.

A property cannot be a function of the past history

(previous conditions under which it has existed) of a

system ; it depends only on the conditions at the

time of consideration

Evans Adei CHEM 155

Description of State of matter

Evans Adei CHEM 155

Extensive – Dependent on Size

Thermodynamics or e.g., m, n V,

Macrostate State Variables

Macroscopic Bulk

Intensive – Independent on Size

e.g., P, ρ, T

States of Matter

Nuclei

Micro or Atomic Variables

Microscopic State

Electronic

Thermodynamic & Steady State Equilibria

Thermodynamic equilibrium

Properties of state are independent of time and there is no flow of mass or energy.

Steady state equilibria

When there is a flow of matter or energy through a system, and yet no change of properties

Homogenous and Heterogenous systems

A system is homogeneous if its properties are uniformthroughout and consists of a single phase or physical state.

A system is heterogeneous if it contains more than one phase

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State function, Thermodynamic Property or

Property of State, State Variables

State Function : Any system property (measurable

physical characteristic of a system, independent of how)

determined exclusively by the values of the initial and

final states. Examples are U, H, A, T, P, V etc.

Path Function : Relate to the preparation of the state.

Examples, energy transfered as heat and work that is

done in preparing a state.

Thermodynamic leads to the definition of additional

properties that can also be used to describe the states

of a system, and are themselves state variables. A, G, ..

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The internal energy is symbolically expressed as U(S, V, N), where S, V and N are

referred to as the natural variables of internal energy U.

The intensive properties of a system ; T, P and µ can be obtained by taking derivatives of

the extensive property U with respect to other extensive properties

T = 𝜕𝑈

𝜕𝑆 V,Nj P =

𝜕𝑈

𝜕𝑉 S,Nj μ =

𝜕𝑈

𝜕𝑁𝑗 S,V, Ni

The internal energy (U) and other thermodynamic properties defined (using Legendre

transformation) starting with the internal energy are referred to as Thermodynamic

Potentials : Enthalpy, H(S, P, N) ; Helmholz, A(T, V, N); and Gibbs, G(T,P,N). Where U,

H, A, and G contain the same information.

Evans Adei CHEM 155

EQUATIONS OF STATE

The important idea that a system’s behaviour can bedescribed and predicted mathematically is expressed for gaseous (simplest form of matter), liquid and solid systemsby saying that P is a function of T, V and n, denoted by

P = f(T,V, n) (1)

The pressure P, is the dependent variable and there are three independent variables, T, V and n. The letter f stands for the functional relationship.

Eqn (1) – Phenomenogically summarizes empiricalobservation called laws or rules i.e. reflect some aspect of the behaviour of nature and must therefore be correct (within limits of experimental error).

Evans Adei CHEM 155

EQUATIONS OF STATE

A state function is a concept related to an equationof state ; it is a mathematical expression for a property in terms of values of their properties usedto specify the state of a system.

The equations of state for real systems ordinarilyobtained are approximate.

The equation of state of most substances are not known, so in general mathematical relation betweenthe four properties that defines a state cannot bewritten down.

Evans Adei CHEM 155

Since Boyle’s law is applicable only at constant temperature and since Charles law is applicable

only at constant pressure, it is not immediately clear that the combination of the two laws is

justified. It follows from these laws that for a given mass of gas the volume is a function of the

pressure and the absolute temperature or

𝑉 = 𝑓 𝑃,𝑇 (1)

𝑑𝑉 = 𝜕𝑉

𝜕𝑃 𝑇𝑑𝑃 +

𝜕𝑉

𝜕𝑇 𝑃𝑑𝑇 (2)

𝑉 𝛼 1

𝑃 and 𝑉 𝛼

𝑘1

𝑃 ( T is constant) Boyles Law

𝜕𝑉

𝜕𝑃 𝑇

= - 𝑘1

𝑃2 = - 𝑃𝑉

𝑃 = -

𝑉

𝑃 (3)

Also

𝑉 𝛼 𝑇 and 𝑉 = 𝑘2 𝑇 ( P is constant) Charles Law

𝜕𝑉

𝜕𝑇 𝑃

= 𝑘2 = 𝑉

𝑇 (4)

Evans Adei CHEM 155

Substituting (3) and (4) into (2)

𝑑𝑉 = − 𝑉

𝑃 𝑑𝑃 +

𝑉

𝑇 𝑑𝑇

𝑑𝑉

𝑉+

𝑑𝑃

𝑃=

𝑑𝑇

𝑇 (5)

Integrating (5)

ln𝑉 + ln𝑃 = ln𝑇 + ln 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

PV = kT

For the special case in which only one mole of gas is considered,

k = R; R is called the molar gas constant defined by R = NAk = 8.31145Jk-1mol-1

Then, for n moles of gas

PV = nNAkT = nRT (Ideal or perfect gas law)

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REPORTING DATA

Two sets of conditions are currently used as standard values for reporting data:

T P Vm

STP (Standard temp. and pressure) 0oC 1 atm 22.414L/mo l

SATP (Standard ambient T and P 25oC 105Pa or 1 bar 24.789 L/mol

R = PV/nT = [(1.00 atm)(22.4 L)]/[(1.00 mol.)(273 K)

= [(0.0821 L.atm)(101.32J)]/[(mol. K)(1 L. atm)]

= 8.314 J/mol. K

Evans Adei CHEM 155

PARTIAL PRESSURES AND MOLE FRACTIONS The mole fraction (XJ) of J in a mixture is the amounts of J molecules present nJ,

expressed as a fraction of the total amount of molecules (n) in the sample.

XJ = nJ/n (where n = nA + nB +…) (1)

Whatever the composition of the mixture,

XA + XB +…………. = 1 (2)

nA/n + nB/n + nC/n +…….. = 1 The partial pressure of a gas is the pressure, which a gas would exert at the same

temperature if it were alone in the container. The partial pressure PJ of a gas in a

mixture (any gas, not just a perfect gas) is:

PJ = XJP ………………………… (3)

It follows from (2)

XAP + XBP +……. (XA + XB+…….)P = P

Therefore P = PA + PB + ……….. (Dalton’s Law)

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The internal energy is symbolically expressed as U(S, V, N), where S, V and N are

referred to as the natural variables of internal energy U.

The intensive properties of a system ; T, P and µ can be obtained by taking derivatives of

the extensive property U with respect to other extensive properties

T = 𝜕𝑈

𝜕𝑆 V,Nj P =

𝜕𝑈

𝜕𝑉 S,Nj μ =

𝜕𝑈

𝜕𝑁𝑗 S,V, Ni

The internal energy (U) and other thermodynamic properties defined (using Legendre

transformation) starting with the internal energy are referred to as Thermodynamic

Potentials : Enthalpy, H(S, P, N) ; Helmholz, A(T, V, N); and Gibbs, G(T,P,N). Where U,

H, A, and G contain the same information.

Evans Adei CHEM 155

The internal energy is symbolically expressed as U(S, V, N), where S, V and N are

referred to as the natural variables of internal energy U.

The intensive properties of a system ; T, P and µ can be obtained by taking derivatives of

the extensive property U with respect to other extensive properties

T = V,Nj P = S,Nj μ = S,V, Ni

The internal energy (U) and other thermodynamic properties defined (using Legendre

transformation) starting with the internal energy are referred to as Thermodynamic

Potentials : Enthalpy, H(S, P, N) ; Helmholz, A(T, V, N); and Gibbs, G(T,P,N). Where U,

H, A, and G contain the same information.

Evans Adei CHEM 155

PARTIAL PRESSURES AND MOLE FRACTIONS

The pressure exerted by a mixture of perfect gases is the sum of the pressure exerted by the individual gases occupying the same volume alone.

P = nRT/V and XJ = nJ/n = PJ/P

PJ = (nJ/n) (nRT/V) = nJRT/V

It is important to remember that the partial pressure is defined by PJ = XJP for any gas but is equal to the pressure that it would exert alone only in the case of a perfect gas (no interaction between particles is a perfect gas)

Evans Adei CHEM 155

THE KINETIC THEORY OF GASES

The kinetic theory is concerned with understanding the behavior of gases in terms of molecular motion. Thus after describing the phenomenological equation of gases there is the need for a quantitative theory which permits the various laws of gaseous behaviour to be coordinated.

The kinetic theory of gases is based on five assumptions.

The gas consists of molecules of mass m and diameter d in ceaseless random motion.

The size of the molecules is negligible (their diameters are much smaller than the average distance traveled between collisions)

The molecules do not interact except that they make perfectly elastic collisions (i.e. the translation KE of a molecule is the same before and after a collision, no energy is transferred to its internal moles of motion) when the separation of their centers is equal to d. Thus there is no potential energy of interaction between the particles, i.e. their energy is independent of their separation.

ET = EKE + EPE

= EKE, (EPE = 0 ; only translational energy is possessed by gas)

The KE of the gas is directly proportional to the temperature.

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Qualitative Conclusions of Continuous Chaotic Motion

The continuous chaotic motion is in agreement with

observations:

Elastic impacts on the vessel supposed to be

responsible for the pressure exerted by the gas.

At constant volume an increase in temperature

results in more vigorous movement with consequent

increase in pressure.

Decrease in volume increases elastic impacts on the

vessel and increase the pressure.

Evans Adei CHEM 155

Quantitative Conclusions

It is more important however, to see if the theory can predict quantitatively the behaviour of gases:

To calculate the pressure of the gas, we need to calculate the force exerted by the molecules per unit area of wall as they collide with it.

That force is calculated using Newton’s 2nd Law of motion i.e. Force = rate of change of linear momentum.

Computation of change in momentum per unit area leads to

PV = (1/3) nMc2 = (1/3)Nmc2 (1)Where M is the molar mass and c is the root- mean square speed (rms speed) of the molecules.

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Quantitative Conclusions

PV = (1/3) nMc2 = (1/3)Nmc2 (1)

Equation (1) has relation similar to PV = nRT.

In equation (1) if the rms speed of the molecules depend only on the temperature, then at constant temperature then,

PV = constant (which is the Boyle’s law) (2)

This conclusion is a major success of the kinetic model, for from model a result has been successfully derived which is valid experimentally.

From equation (1), if a gas is at constant temperature and pressure, (V/n)T,Pis constant, which is Avogadro’s law; equal volume of gases at same temperature and pressure contain the same number of molecules.

For a gas at constant pressure,

(c2)1/2 [V/(Nm)]1/2,

or the rate of effusion R α 1/D1/2 which is Graham’s law.

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EQUIPARTITION

The total kinetic energy (transitional energy) EKE, of 1 mole of molecules is NA times the

average energy per molecule, that is

EKE = NAεKE = ( ½)NAmc2

Hence EKE = (½)Mc2 (3)

But PV = (1/3)nMc2 (4)

From (3) 2EKE = Mc2 (5)

Substituting (5) into (4)

PV = (1/3) n.2EkE = (2/3)nEKE = nRT (6)

therefore EKE = (3/2)RT (7)

Equation (7) shows that the Kinetic energy of an ideal gas is directly proportional to the

Kelvin temperature. Interpretation of absolute zero by kinetic theory is that the complete

cessation of all molecular motion contrary to quantum mechanics stipulation.

Since the only energy the gas possesses is translational (because the collision is elastic

Epotential = 0) the degrees of freedom (x, y. z) each translational degree of freedom has ½

RT (per mole). This is a special case of a more general theorem known as Principle of

Equipartition of Energy. Evans Adei CHEM 155

MOLECULAR SPEEDS

The ability of the kinetic theory to account for Boyle’s Law suggests that it is a

valid model of perfect gas behaviour.

We thus take a major step.

If PV = 1/3 nMc2

is to be precisely the equation of state of a perfect gas then the RHS must be equal to

nRT

PV = 1/3 nMc2 = nRT

c = (3RT/M)1/2

The root mean square speed of the molecules of a gas is proportional to the

square root of the temperature and inversely proportional to the square root of

the molar mass.

i.e. c α (T/M) 1/2

The higher the temperature, the faster the molecules travel and, at a given

temperature, heavy molecules travel more slowly than light molecules.

Evans Adei CHEM 155

Maxwell Distribution of Molecular Speeds

The chaotic continuous motion of molecules in a gas

leads to numerous collisions with consequent redistributnof molecular speeds of individual molecules that span a wide range.

The velocity of a molecule changes after a span of even less than 10-9 seconds, thus making it difficult to known the speed of individual molecules.

However, there must be a fraction of the total number of molecules that have a particular velocity at any time.

This fraction was obtained theoretically by Maxwell and Boltzman by utilizing probability considerations.

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Maxwell Distribution of Molecular Speeds

According to Maxwell, the fraction f of molecules that has a speed in a narrow range s

to s + Δs is

Maxwell equation is a particular case of a general expression which finds application in

many aspects of physical chemistry. The expression is known as Boltzman’s

distribution (probability distribution). The probability distribution for molecular states in

a dilute gas is:

This has a number of important properties

States of higher energy are less probable than state of lower energy.

At higher temps, the different in population between states of high energy and

states of low energy decreases, until, as T approaches infinity, all states

approach equal probability.

As T approaches zero on the Kelvin scale, only the states of lowest energy are

populated. Evans Adei CHEM 155

Maxwell Distribution of Molecular Speeds

On dividing both sides of equation (1) by ds, we have

The expression gives the probability P of finding the molecules with the velocity

s as equation (1) and (2) are used forms of the equation for the Maxwell’s law of

distribution of velocities.

Evans Adei CHEM 155