PPT 107 PHYSICAL CHEMISTRY Semester 2

Academic session 2011/2012 NURUL AIN HARMIZA ABDULLAH

CHAPTER 1 THERMODYNAMICS

CHAPTER 1 THERMODYNAMICS

CONTENT: 1.1 Physical Chemistry 1.2 Thermodynamics 1.3 Temperature 1.4 The Mole 1.5 Ideal Gases 1.6 Differential Calculus 1.7 Equations of State 1.8 Integral 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 be studied from either a microscopic or a

macroscopic viewpoint.

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

The microscopic viewpoint is based on the concept of molecules.

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

Chemical Systems

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

• The macroscopic world is the one we can know by direct observations of physical properties such as mass, volume, etc.

Thermodynamics

4 Branches of Physical

Chemistry

Quantum Chemistry

Statistical Mechanics

Kinetics

Thermodynamics is a macroscopic

science that studies the interrelationships of the

various equilibrium properties

of a system and the changes in equilibrium

properties in processes.

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.

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 macroscopic thermodynamic properties from

molecular properties.

Kinetics is the study of rate processes such as

chemical reactions, diffusion, and

the flow of charge in an electrochemical cell. Kinetics uses relevant

portions of thermodynamics,

quantum chemistry, and statistical

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 a compartment fire. Outside the system are its surroundings. The system and its surroundings 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 its

surroundings. UNIVERSE

For example we might consider a burning fuel package as the system and the compartment as the surroundings. On a larger scale we might consider the building containing the fire as the system and the exterior environment as the surroundings.

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, to study the vapor pressure of water as a function of temperature, we might put a sealed container of water (with any air evacuated) in a constant-temperature bath and connect a manometer to the container to measure the pressure. Here, the system consists of the liquid water and the water vapor in the container, and the surroundings are the constant-temperature bath and the mercury in the manometer.

Energy transfer is studied in three types of systems:

Open systems

Open systems can exchange both matter and energy with an outside system. They are portions of larger systems 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 typically portions of larger systems, they are not in complete contact. The Earth is essentially a closed system; it obtains 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 be portions of larger systems, they do not communicate with the outside in any way. The physical universe is an isolated system; a closed thermos bottle is essentially an isolated system (though its insulation is not perfect).

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

Example

Example

For example, in figure above, the system of liquid water plus water vapor in the sealed container is closed (since no matter can enter or leave) but not isolated (since it can be warmed or cooled by the surrounding bath and can be compressed or expanded by the 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 kinds of walls.

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

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.

In Fig. 1.2, the system is separated from the bath by the container walls

EQUILIBRIUM

• An isolated system is in equilibrium when its macroscopic properties remain constant with time.

• A nonisolated system is in equilibrium when the following 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 a steady 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 intensive

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.

Is one whose value does not depend on the size of the system, provided the system remains of macroscopic. Density and pressure are examples of intensive properties. We can take a drop of water or a swimming pool full of water, and both systems will have the same density.

Phase A phase is a region of space (a thermodynamic system), throughout which all physical properties of a material are essentially uniform. Examples of physical properties 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 some chemical composition throughout.

– e.g. mixture of gases or true solution 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 equilibrium exists between systems.

• By definition, two systems in thermal equilibrium with each other have the same temperature; two systems not in thermal equilibrium have different temperatures.

• Symbolized by θ (theta).

The Zeroth Law

Two systems that are each found to be in thermal equilibrium with a third system will be found to be in thermal equilibrium with each other.

It is so called because only after the first, second, and third laws of thermodynamics had been formulated was it realized that the zeroth law is needed for the development of thermodynamics. Moreover, a statement of the zeroth law logically precedes the other three. The zeroth law allows us to assert the existence of temperature as a state function.

1.4 The Mole

Relative Atomic Mass, Ar

• The ratio of the average mass of an atom of an element to 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 an instrument called a mass spectrometer.

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

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 of the substance.

• Molecular mass of O2 = 32Gram

Calculation of Molecular Mass

• Molecular mass is equal to sum of the atomic masses of all atoms present in one molecule of the substance.

• Example: – H2OMass of H atom = 18g

– NaCl = 58.44g

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

Why unitless? Find out!

Remember that relative atomic

mass/relative molecular mass is a ratio

and 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).

• 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 x 1023 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.

Avogadro's number = 6.02 x 1023

1 Mole C atom = 6.02 x 1023 C atoms = 12g

1 Mole Mg atom = 6.02 x 1023 Mg atoms = 24.3g

Example

• How many atoms are there in 24g carbon?

24g of carbon = 24/12 moles = 2 moles

1 mole of atoms = 6.02 x 1023

Therefore 2 moles of carbon contains:

2 x 6.02 x 1023 atoms = 1.204 x 1024 atoms

Try This!

• How many atoms and moles of silicon are in a sample of silicon that has a mass of 5.23g?

• Answers = 0.186 mol Silicon; and

• = 1.12 x 1023 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.

M = mass = m mole n

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

1.5 Ideal Gases

An 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 moles R = universal gas constant = 8.3145 J/mol K N = number of molecules k = Boltzmann constant = 1.38066 x 10-23 J/K = 8.617385 x 10-5 eV/K k = R/NA NA = Avogadro's number = 6.0221 x 1023

Ideal Gas Law

The Ideal Gas Law

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

R = 8.3145 kPa • L

mol • K

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

T, we can solve for the unknown value.

R = 82.06 cm .atm 3

mol . K or

For the Volume-Pressure relationship: Boyle’s Law

• n1 = n2 and T1 = T2 therefore the n's and T's cancel in the above expression resulting in the following simplification:

• 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 and the P's cancel in the original expression resulting in the following simplification:

• V1T1 = V2T2 or VT = 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 the V's cancel in the above original expression:

• P1T1 = P2T2 or PT = 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's cancel in the above original expression:

• V1n1 = V2n2 or Vn = constant

(mathematical expression of Avagadro's Law)

Boyle’s Law

• At constant temperature, the volume of a given quantity of gas is inversely proportional to its pressure : V 1/P So at constant temperature, if the volume of a gas is doubled, its pressure is halved. OR At constant temperature for a given quantity of gas, the product of its volume and its pressure is a constant : PV = constant, PV = k

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

• Pi and Pf must be in the same units of measurement (eg, both in atmospheres), Vi and 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 the absolute temperature : V T (in Kelvin) So at constant pressure, if the temperature (K) is doubled, the volume of gas is also doubled. OR At constant pressure for a given quantity of gas, the ratio of its volume and the absolute temperature is a constant : V/T = constant, V/T = k

• At constant pressure for a given quantity of gas : Vi/Ti = Vf/Tf where Vi is the initial (original) volume, Ti is its initial (original) temperature (in Kelvin), Vf is its final 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 in Kelvin 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.

Charles Law

V1/V2=T1/T2

P1V1/T1=P2V2/T2

Pressure and Volume Units

PRESSURE VOLUME P (Pressure) = F (Force)

A (Area)

In SI: 1 Pa (Pascal) = 1 N/m Chemists use: 1 torr = 133.322 Pa or = 133.322 N/m or = 133.322 kg/ms 1 bar =10 Pa = 0.986923 atm = 750 torr

2

2

2

ATMOSPHERE 1 atm = 760 torr = 1.01325 x 10 Pa 5

5

1 L = 1 dm = 1000 cm 3 3

Example 1.1: Density of an Ideal Gas

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

Page 16

Exercise

• Find the molar mass of a gas whose density is 1.80 g/L at 25.0°C and 880 torr.

• (Answer: 38.0 g/mol.)

1.6 Differential Calculus

Functions and Limits

lim y= f (x)

Function f Dependent variable

Independent variable

x a

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

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

y = f(x)

x

Approaches 0

Approaches ∞

Examples

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

Lim (4x) x 3

Examples

Sample B: Find the limit:

Follow the same steps,

lim (x + 5x – 3) x 1

2

x + 5x – 3 2

= 1 + 5(1) – 3 2

= 3

So, the limit of as x approaches 1 is 3. x + 5x – 3 2

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 the slope

Examples: How to find the slope when points are given

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

(x1 − x2) (4 − 2 )

= 8

2

= 4

Slope, m = Positive slope

The Derivative

Derivatives

• The derivative tells us the rate of change of one quantity compared to another at a particular 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 Derivative

• The derivative is carried out in the same way as ordinary differentiation with this constraint. 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 derivatives are defined as derivatives of a function of multiple variables when all but 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

Partial Differentiation with respect to x

• 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.)

• "Partial derivative with respect to x" means "regard all other letters as constants, and just differentiate 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 at it from the far end of the y-axis. We are looking at the x-z plane only.

Now for the partial derivative of F(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 respect to y

• "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 the y2 and y terms in F(x,y) = y + 6 sin x + 5y2. The "6 sin x" part is now regarded as a constant.

Now for the partial derivative of F(x,y) = y + 6 sin x + 5y2 with 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.

1.7 Equations of State

Please Read Topic 1.7 (Equations of State) 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:

Define a thermal expansion coefficient and an isothermal compressibility by •

• 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

•

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

V V T P

dVV

TdT

V

PdP

P T

( , )

1

1

V

V

T

V

V

P

P

T

1212

1

2ln PPTTV

V

dPdTV

dV

Example 1.4: Expansion due to a temperature increase

• Estimate the percentage increase in volume produced by a 10°C temperature increase in a liquid with the typical a value 0.001 K1, approximately independent of temperature.

• Equation (1.43) gives dVP aV dTP. Since we require only an approximate answer and since the changes in T and V are small (a is small), we can approximate the ratio dVP/dTP by the ratio VP/TP of finite changes to get VP/V a TP (0.001 K1) (10 K) 0.01 1%.

1.8 Integral Calculus

• Example 1: Evaluate

• Use formula (4):

• and get this:

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.

Logarithms

• Integration of 1/x gives the natural logarithm ln x.

Understanding, rather than mindless memorization, is the key to learning

physical chemistry…

End of Chapter 1