PPT 107 PHYSICAL CHEMISTRYSemester 2

Academic session 2012/2013

Dr Hayder Kh. Q. Ali

CHAPTER 1THERMODYNAMI

CS

CHAPTER 1THERMODYNAMI

CSCONTENT:

1.11.21.31.41.51.61.71.8

Physical ChemistryThermodynamicsTemperatureThe MoleIdeal GasesDifferential CalculusEquations of StateIntegral Calculus

1.1 Physical Chemistry

• What is Physical Chemistry?– Physical chemistry is the study of the underlying physical principles that govern the properties and behavior of chemical systems.

• What is Chemical Systems?– A chemical system can

macroscopic viewpoint.be studied from either a microscopic or a

ChemicalSystems

The first half of this book uses mainly a macroscopic viewpoint; the second half uses mainly amicroscopic viewpoint.

The macroscopic viewpoint studies large-scale properties of matter without explicit use of the molecule concept.

The microscopic viewpoint is based on the concept of molecules.

• "microscopic" implies detail at the atomic orsubatomic levels which cannot be(even with a microscope!).

seen directly

• The macroscopic world is the one we can knowby direct observations of physical propertiessuch as mass, volume, etc.

4 Branches ofPhysical

Chemistry

chemical reactions,

the flow of charge in an

Kinetics uses relevant

thermodynamics,

and statistical

Thermodynamics

Thermodynamics is a macroscopic

science that studies the interrelationships of the

various equilibrium properties

of a system and the changes in equilibriumproperties in processes.

QuantumChemistry

Molecules and the electrons and nuclei

that compose them do not obey classical

mechanics. Instead, their motions are

governed by the laws of quantum mechanics.

Application of quantum mechanics to atomic structure, molecular

bonding,and spectroscopy gives us quantum chemistry.

StatisticalMechanics

The molecular and macroscopic levels are

related to each other by the branch of science

called statistical mechanics. Statistical

mechanics gives insight into why the laws of

thermodynamics hold and allows calculation

of macroscopicthermodynamicproperties from

molecular properties.

Kinetics

Kinetics is the study of rate processes such as

diffusion, and

electrochemical cell.

portions of

quantum chemistry,

mechanics.

1.2 Thermodynamics

THERMODYNAMIC SYSTEM

An important concept in thermodynamics is the thermodynamic system. A thermodynamic system is one that interacts and exchanges energy with the area around it (transformation of energy). A system could be as simple as a block of metal or as complex as acompartment fire. Outside the system are its surroundings. The system and itssurroundings comprise the universe.

Systems:A region of the universe that we direct our attention to.

Surroundings:Everything outside a system is called surroundings.

Boundary:The boundary or wall separates a system from itssurroundings.

UNIVERSE

For example, to study the vaporpressure of water as a function oftemperature, we might put asealed container of water (withany air evacuated) in a constant-temperature bath and connect amanometer to the container tomeasure the pressure. Here, thesystem consists of the liquid waterand the water vapor in thecontainer, and the surroundingsare the constant-temperature bathand the mercury in themanometer.

A key property in

thermodynamics is temperature,

and thermodynamics is sometimes defined

as the study of the relation of temperature to the macroscopic

properties of matter.

For example we might consider a burning fuelpackage as the system and the compartment as thesurroundings. On a larger scale we might considerthe building containing the fire as the system and theexterior environment as the surroundings.

Energy transfer is studied in three typesof systems:

Open systems

Open systems can exchange both matter and energy with an outside system. They are portions of largersystems and in intimate contact with the larger system. Your body is an open system.

Closed systems

Closed systems exchange energy but not matter with an outside system. Though they are typicallyportions of larger systems, they are not in complete contact. The Earth is essentially a closed system; itobtains lots of energy from the Sun but the exchange of matter with the outside is almost zero.

Isolated systems

Isolated systems can exchange neither energy nor matter with an outside system. While they may beportions of larger systems, they do not communicate with the outside in any way. The physical universeis an isolated system; a closed thermos bottle is essentially an isolated system (though its insulation isnot perfect).

Heat can be transferred between open systems and between closed systems, but not betweenisolated systems.

Example

Example

For example, in figure above, the systemof liquid water plus water vapor in the

nonot

sealed container is closed (sincebutmatter can enter or leave)

isolated (since it can be warmed orcooled by the surrounding bath and

bycanthebe compressed or expanded

mercury).

A thermodynamic system is either open or closed and is either isolated or non-isolated.Most commonly, we shall deal with closed systems

WALLS

A system may be separated from its surroundings by various kindsof walls.

1. A wall can be either rigid or nonrigid (movable).

In Fig. 1.2, the system isseparated from the bath bythe container walls

2. A wall may be permeable or impermeable.Impermeable means that it allows no matter to pass through it.

3. A wall may be adiabatic or nonadiabatic.An adiabatic wall is one that does not conduct heat at all,whereas a nonadiabatic wall does conduct heat.

EQUILIBRIUM

An isolated system is in equilibrium when its macroscopicproperties remain constant with time.

•

• A nonisolated system is in equilibrium when thefollowing two conditions hold:– The system’s macroscopic properties remain constant with

time;

removal of the system from contact with its surroundings causes no change in the properties of the system.

–

• If condition (a) holds but (b) does not hold, the system is in asteady state.

Types of Equilibrium:

1. Mechanical equilibrium• No unbalanced forces act on or within the system; hence the system undergoes

no acceleration, and there is no turbulence within the system.

2. Material equilibrium• No net chemical reactions are occurring in the system, nor is there any net transfer

of matter from one part of the system to another or between the system and its surroundings; the concentrations of the chemical species in the various parts of the system are constant in time.3. Thermal equilibrium between a system and its surroundings

• There must be no change in the properties of the system or surroundings when they are separated by a thermally conducting wall.

Likewise, we can insert a thermally conducting wall between two parts of a system to test whether the parts are in thermal equilibrium with each other. For thermodynamic equilibrium, all three kinds of equilibrium must be present.

THERMODYNAMIC PROPERTIES-used to characterize a system in equilibrium

extensive

Is one whose value is equal to the sum of its values for the parts of the system. Thus, if we divide a system into parts, the mass of the system is the sum of the masses of the parts; mass is an extensive property. So is volume.

intensive

Is onedependsystem,remains

whose value does noton the size of theprovided the system

Densityof macroscopic.and pressure are examples ofintensive properties. We cantake a drop of water or aswimming pool full of water, andboth systemswill have the same density.

Phase A phase is a region of space (a thermodynamic system), throughout which allphysical properties of a material are essentially uniform. Examples of physicalproperties include density, index of refraction, and chemical composition

• Extensive Parameters:– Parameters which values for the composite system are

the sum of the values for each of the subsystems. These parameters are non-local in the sense that they refer to the entire system.Examples are: Volume, internal energy, mass, length.

–

• Intensive Parameters:– These parameters are identical for each subsystem into

which we might subdivide our system.– Examples are: Pressure, temperature, and density.

• Homogenous system :

– A system is homogenous when it has

composition throughout.

– e.g. mixture of gases or true solution

some chemical

of solid in liquid.

• Heterogenous system :

– Two or more different phases which are homogenous

but separated by a boundary.

– e.g. Ice in water.

1.3 Temperature

• To determine whether or not thermal equilibriumexists between systems.

• By definition, two systems in thermal equilibriumwith each other have the same temperature; twosystems not in thermal equilibrium have differenttemperatures.

• Symbolized by θ (theta).

The Zeroth Law

Two systems that are each found to be inthermal equilibrium with a third systemwill be found to be in thermalequilibrium with each other.

It is so calledbecause only after the first, second, and thirdlaws of thermodynamics had been formulatedwas it realized that the zeroth law is needed forthe development of thermodynamics.Moreover, a statement of the zeroth lawlogically precedes the other three. Thezeroth law allows us to assert the existence oftemperature as a state function.

1.4 The Mole

Relative Atomic Mass, Ar

The ratio of the average mass of an atom of an elementto the mass of some chosen standard.

The Relative Atomic Mass of a chemical element gives us an idea of how heavy it feels (the force it makes when gravity pulls on it).The relative masses of atoms are measured using aninstrument called a mass spectrometer.

Look at the periodic table, the number at the bottom of the symbol is the Relative Atomic Mass (Ar ):

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•

•

•

Relative Molecular Mass, Mr

• Most atoms exist in molecules.• To work out the Relative Molecular Mass, simply add up the

Relative Atomic Masses of each atom in the molecule: • A relative molecular mass can be calculated easily by adding

together the relative atomic masses of the constituent atoms. For example, ethanol, CH3CH2OH, has a Mr of 46 (Try it!).

Gram molecular mass

Molecular mass expressed in grams is numerically equal to gram molecular mass substance.Molecular mass of O2 = 32Gram

•of the

•

Calculation of Molecular Mass

Molecular mass is equal to sum of the atomic•masses of all atomsthe substance.

present in one molecule of

• Example:– H2OMass of H atom = 18g– NaCl = 58.44g

The statement that the molecular weight of H2O is 18.015 means that a water moleculehas on the average a mass that is 18.015/12 times the mass of a 12C atom.

Why unitless? Find out!

Remember

that relative atomicmass/relative molecular mass is a

ratioand has no units while gram

molecular mass and gram atomic mass are expressed in grams.

Mole Concept and Avogadro’s Number

• It is convenient to consider the number of atoms needed to make 12g of carbon and for this number to be given a name - one mole of carbon atoms.

• Avogadro's number and the mole are very important to the understanding of atomic structure.

• The Mole is like a dozen. You can have a dozen guitars, a dozen roosters, or a dozen rocks. If you have 12 of anything then you would have what we call a dozen. The concept of the mole is just like the concept of a dozen.

• You can have a mole of anything. The number associated with a mole is Avogadro's number. Avogadro's number is

602,000,000,000,000,000,000,000 (6.02 x 1023).

1023Avogadro's number = 6.02 x

10231 Mole C atom = 6.02 x C atoms = 12g

10231 Mole Mg atom = 6.02 x Mg atoms = 24.3g

• A mole of marbles would spread over the surface of the earth, and produce a layer about 50 miles thick. A mole of sand, spread over the United States, would produce a layer 3 inches deep. A mole of dollars could not be spent at the rate of a billion dollars a day over a trillion years. This shows you just how big a mole is.

• Probably the only thing you will ever have a mole of is atoms or

molecules. One mole of magnesium atoms (6.02 X 1023 ) magnesium atoms

weigh 24.3 grams. 6.02 X 1023 carbon atoms weigh a total of 12.0 grams.

6.02 X1023 molecules of CO2 gas only weigh a total of 44.0 grams.

• The actual number of atoms that is needed to give the relative atomic mass expressed in grams is called Avogadro's number.

Example

How many atoms are there in• 24g carbon?

24g of carbon = 24/12 = 2 moles

10231 mole of atoms = 6.02 x

Therefore 2 moles of carbon contains:

= 1.204 x 1024 atoms10232 x 6.02 x atoms

Try This!

How many atoms and moles of silicon are insample of silicon that has a mass of 5.23g?

• a

••

Answers =

=0.186 mol Silicon; and

10231.12 x atoms

Molar Mass, M

The mole is just a number; it can be used for atoms, molecules, ions,electrons, or anything else we wish to refer to.

Because we know the formula of water is H2O, for example, then we can say one mole of water molecules contains one mole of oxygen atoms and two moles of hydrogen atoms.One mole of hydrogen atoms has a mass of 1.008 g and 1 mol of oxygen atoms has a mass of 16.00 g, so 1 mol of water has a mass of (2 x 1.008 g) + 16.00 g = 18.02 g. The molar mass of water is 18.02 g/mol.

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1 mol of oxygen atoms has a mass of 16.00 g Molar mass of O = 16 g/mol

Molar mass of H2O = 2 mol of H + 1 mol of O= (2x1.008 g/mol of H) + (16 g/mol of O)= 18.02 g/mol M = mass = m

mole n

1.5 Ideal Gases

Ideal Gas LawAn ideal gas is defined as one in which all collisions between atoms or molecules are perfectly elastic and in which there are no intermolecular attractive forces. One can visualize it as a collection of perfectly hard spheres which collide but which otherwise do not interact with each other. In such a gas, all the internal energy is in the form of kinetic energy and any change in internal energy is accompanied by a change in temperature. An ideal gas can be characterized by three state variables: absolute pressure (P), volume (V), and absolute temperature (T). The relationship between them may be deduced from kinetic theory and is called the

Where:n = number of molesR = universal gas constant = 8.3145 J/mol KN = number of moleculesk = Boltzmann constant = 1.38066 x 10-23 J/K = 8.617385 x 10-5 eV/Kk = R/NANA = Avogadro's number = 6.0221 x 1023

The Ideal Gas Law

orR = 82.06 cm 3.atmmol . K

PV = nRTP = Pressure (in kPa) V = Volume (in L)T = Temperature (in K) n = moles

R = 8.3145 kPa • Lmol • K

R is constant. If we are given three of P, V, n, or

T, we can solve for the unknown value.

For the Volume-Pressure relationship:Boyle’s Law

• n1 = n2 and T1 = T2 therefore the n's and T'scancel in the above expression resultingfollowing simplification:

in the

• P1V1 = P2V2 or PV = constant(mathematical expression of Boyle's Law)

For the Volume-Temperature relationship:Charles's Law

• n1 = n2 and P1 = P2 therefore the n's andcancel in the original expression resultingfollowing simplification:

the P'sin the

• V1T2 = V2T1 or V / T = constant(mathematical expression of Charles's Law)

For the Pressure-Temperature Relationship:

Gay-Lussac's Law

• n1 = n2 and V1 = V2 therefore the n's and thecancel in the above original expression:

V's

• P1T2 = P2T1 or P / T = constant(mathematical expression of Gay Lussac's Law)

For the Volume-Mole relationship:Avagadro's Law

• P1 = P2 and T1 = T2 therefore the P's and T'scancel in the above original expression:

• V1n2 = V2n1 or V / n = constant(mathematical expression of Avagadro's Law)

Boyle’s Law

At constant temperature, the volume of a given quantity of gas is inverselyproportional to its pressure : V 1/PSo at constant temperature, if the volume of a gas is doubled, its pressure is halved.ORAt constant temperature for a given quantity of gas, the product of its volume and itspressure is a constant : PV = constant, PV = k

•

• At constant temperature for a given quantity of gas : PiVi = PfVfwhere Pi is the initial (original) pressure, Vi is its initial (original) volume, Pf is its finalpressure, Vf is its final volume

Pi and Pf must be in the same units of measurement (eg, both in atmospheres), Viand Vf must be in the same units of measurement (eg, both in litres).

All gases approximate Boyle's Law at high temperatures and low pressures. A hypothetical gas which obeys Boyle's Law at all temperatures and pressures is called an Ideal Gas. A Real Gas is one which approaches Boyle's Law behaviour as the temperature is raised or the pressure lowered.

•

•

Boyle’s Law

P1V1=P2V2

Charles Law

At constant pressure, the volume of a given quantity of gas is directly proportional to theabsolute temperature : V T (in Kelvin)So at constant pressure, if the temperature (K) is doubled, the volume of gas is also doubled.ORAt constant pressure for a given quantity of gas, the ratio of its volume and the absolutetemperature is a constant : V/T = constant, V/T = kAt constant pressure for a given quantity of gas : Vi/Ti = Vf/Tfwhere Vi is the initial (original) volume, Ti is its initial (original) temperature (in Kelvin), Vf is itsfinal volme, Tf is its final tempeature (in Kelvin)Vi and Vf must be in the same units of measurement (eg, both in litres), Ti and Tf must be inKelvin NOT celsius.temperature in kelvin = temperature in celsius + 273 (approximately)

All gases approximate Charles' Law at high temperatures and low pressures. A hypothetical gas which obeys Charles' Law at all temperatures and pressures is called an Ideal Gas. A Real Gas is one which approaches Charles' Law as the temperature is raised or the pressure lowered.As a Real Gas is cooled at constant pressure from a point well above its condensation point, its volume begins to increase linearly. As the temperature approaches the gases condensation point, the line begins to curve (usually downward) so there is a marked deviation from Ideal Gas behaviour close to the condensation point. Once the gas condenses to a liquid it is no longer a gas and so does not obey Charles' Law at all.Absolute zero (0K, -273oC approximately) is the temperature at which the volume of a gas would become zero if it did not condense and if it behaved ideally down to that temperature.

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Charles Law

V1/V2=T1/T2

P1V1/T1=P2V2/T2

Pressure

PRESSUREP (Pressure) = F (Force)

and Volume Units

VOLUME1 L = 1 dm3 = 1000 cm3

A (Area)

ATMOSPHERE1 atm = 760 torr = 1.01325 x 10 5

Pa

In SI:1 Pa (Pascal) = 1 N/m

2

Chemists use:

21 torr = 133.322 Pa or

2= 133.322 N/m or= 13 3.322 kg/ms

1 bar =10 Pa = 0.986923 atm = 750 torr5

Example 1.1: Density of an Ideal GasPage 16

• Find the density of F2 gas at 20.0°C and 188 torr.

Exercise

Find the molar mass of a gas whose1.80 g/L at 25.0°C and 880 torr.

• density is

• (Answer: 38.0 g/mol.)

1.6 Differential Calculus

Functions and Limits

Dependent variable Function f

lim y= f (x)

Independent variable

x a

Limit of the function f(x)as x approaches the value of a

To say that the variable y is a function of the variable x means that for any given value of x there is specified a value of y; we write y=f(x).

What is a limit?

A limit is a certain value to which a function approaches. Finding a limit means finding what value y is as x approaches a certain number. You would typical say that the limit of a certain function is <a number> as x approaches <some x coordinate>. For example, imagine a curve such that as x approaches infinity, that curve may come closer and closer to y=0 while never actually getting there. So, how do we algebraically find that limit? One way to find the limit is by the SUBSTITUTION METHOD.

•

• For example, the limit graph approaches 0:

of the following graph is 0 as x approaches infinity, because the

y = f(x)

Approaches 0

x

Approaches ∞

Examples

Lim (4x)x 3

Sample A: Find the limit of f(x) = 4x, as x approaches 3 or

Steps:1) Replace x for 3.2) Simplify.

f(x) = 4x becomes f(3) = 4(3) = 12.

So, the limit of f(x) = 4x as x approaches 3 is 12; or

Examples

Sample B: Find the limit:

lim (x 2+ 5x – 3)x 1

Follow the same steps,

x 2+ 5x – 3

= 12+ 5(1) – 3= 3

x 2+ 5x –

3 as x approachesSo, the limit of 1 is 3.

Slope•••

What is slope?If you have ever walked up or down a hill, then you have already experienced a real life example of slope

By definition, the slope is the measure of the steepness of a line.

Example: How to find

Examples: How to find the slope when points are given

the slope

Let (x1,y1) = (4, 9) and (x2,y2) = (2, 1)

Slope, m = (y1 − y2) = (9 − 1)Positiveslope

(x1 − x2) (4 − 2 )

= 82

4=

If we write the equation of the straight line in the form y=mx+b, it follows from this definition that the line’s slope equals m.The intercept of the line on the y axis equals b, since y=b when x=0.

The Derivative

Derivatives

The derivative tells us the rate of change of one quantity compared to another at aparticular instant or point (so we call it "instantaneous rate of change").

•

• Let y f (x). Let the independent variable change its value from x to x+ h; this will change y from f (x) to f (x +h). The average rate of change of y with x over this interval equals the change in y divided by the change in x and is

• The instantaneous rate of change of y with x is the limit of this average rate of change taken as the change in x goes to zero. The instantaneous rate of change is called the derivative of the function f and is symbolized by f :

• Wherever a quantity is always changing in value, we can use calculus (differentiation and integration) to model its behaviour.

• The derivative of a function with respect to the variable is defined as

• but may also be calculated more symmetrically as

• the second derivative may be defined as

• and calculated more symmetrically as

The Partial DerivativePartial derivatives are defined as derivatives of a function of multiple variables when allbut the variable of interest are held fixed during the differentiation.Example - Function of 2 variables

Here is a function of 2 variables, x and y:F(x,y) = y + 6 sin x + 5y2

• The derivative is carried out in the same way as ordinary differentiation with thisconstraint. For example, given the polynomial in variables x and y,

• the partial derivative with respect to x is written

• and the partial derivative with respect to y is written

Partial Differentiation with respect to x

• "Partial derivative with respect to x" means"regard all other letters as constants, and justdifferentiate the x parts".In our example (and likewise for every 2-variable function), this means that (in effect)we should turn around our graph and look atit from the far end of the y-axis. We are

•

looking at the x-z plane only.

We see a sine curve at the bottom and this comes from the 6 sin x part of our function F(x,y) = y + 6 sin x +5y2. The y parts are regarded as constants.(The sine curve at the top of the graph is just where the software is cutting off the surface - it could have been made it straight.)

•

•

Now for the partial derivative ofF(x,y) = y + 6 sin x + 5y2

with respect to x:

The derivative of the 6 sin x part is 6 cos x. The derivative of the y-parts is zero since they are regarded as constants.Notice that we use the symbol "∂" to denote "partial differentiation", rather than "d" which we use for normal differentiation.

Partial Differentiation with

"Partial derivative with respect to y" means "regard all other letters as constants, just differentiate the y parts".As we did above, we turn around our graph and look at it from the far end of the x-axis. So we see(and consider things from) the y-z plane only.We see a parabola. This comes from

respect to y

•

•

•y2 + 5y2. Thethe and y terms in F(x,y) = y + 6 sin x

"6 sin x" part is now regarded as a constant.

Now for the partial derivative ofF(x,y) = y + 6 sin x + 5y2with respect to y.

The derivative of the y-parts with respect to y is 1 + 10y. The derivative of the 6 sin x part is zero since it is regarded as a constant when we are differentiating with respect to y.

If now both x and y undergo infinitesimal changes, the infinitesimal change in z is the sum of the infinitesimal changes due to dx and dy:

In this equation, dz is called the total differential of z(x, y). This equation is often used in thermodynamics. An analogous equation holds for the total differential of a function of more than two variables. For example, if z=z(r, s, t), then:

1.7 Equations of State

P T

T

dd

Please Read Topic 1.7 (Equations ofState) in page 22 to 25.

Equations of State:An equation of state is a relation between P, V, and T (for a pure material). For a mixture, it must also involve the composition of the mixture (usually in mole fractions).

•

• Liquids:

V V (T , P)

V VdV dT dP

T P

Define a thermal expansion coefficient and an isothermal compressibility by•

V1

V TP

1 V

V PT

••

These can be assumed constant for liquids, as long as we are not near the critical point.The equation of state can then be written as

dVT P

V

ln V2

•

T P P2 1 2 1V1

• Values of ҡ and β (β=α) can be found in many handbooks.

1.8 Integral Calculus

Definition:A function F(x) is the antiderivative of a function ƒ(x) if for all x in the domain of ƒ,

F'(x) = ƒ(x)

ƒ(x) dx = F(x) + C, where C is a constant.

• Example 1: Evaluate

• Use formula (4):

• and get this:

Logarithms

Integration of 1/x gives the natural logarithm lnx.

•

End of Chapter 1

Understanding, rather than mindlessmemorization, is the key to learning

physical chemistry…