PPT 107PHYSICAL CHEMISTRY Semester 2

CHAPTER 1 THERMODYNAMICS

CHAPTER 1 THERMODYNAMICS CONTENT: Physical Chemistry Thermodynamics Temperature The Mole Ideal Gases Differential Calculus Equations of State Integral Calculus

1.1Physical 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. bestudiedfromfromeitheramicroscopicora Chemical Systems The first half of this book uses mainly a macroscopic viewpoint; the second half uses mainly a microscopic 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 or subatomic levels which cannot be (even with a microscope!). seen directly The macroscopic world is the onewe can know by direct observations of physical properties such as mass, volume, etc.

4 Branches of Physical 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 equilibrium properties in processes. Quantum Chemistry 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. Statistical Mechanics 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 Kinetics is the study of rate processes such as diffusion, and electrochemical cell. portions of quantum chemistry, mechanics.

1.2Thermodynamics

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 compartmentfire.fire.Outsidethesystemareareitssurroundings.Thesystemandits 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, 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. 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 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.

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

For example, in figure above, the system of liquid water plus water vapor inthe no not sealed containerisclosed(since but mattercanenter or leave) isolated (since it can be warmed or cooled by the surrounding bathand by can the becompressedorexpanded 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). In Fig. 1.2, the system is separated from the bath by the 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 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. – Ifcondition (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 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 one depend system, remains whosevaluedoes not onthe sizeofthe providedthesystem Density of macroscopic. and pressure are examples of intensiveproperties.Wecan taketakeadrop of waterora 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 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.

1.3Temperature

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).

TheZeroth 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.

1.4The 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 (A r ):

Relative Molecular Mass, M r 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, CH 3 CH 2 OH, has a M r of 46 (Try it!).

Gram molecular mass Molecular mass expressed in grams is numerically equal to gram molecular mass substance. Molecular mass of O 2 = 32Gram of the

Calculation of Molecular Mass Molecular mass is equal to sum of the atomic masses of all atoms the substance. presentinonemoleculeof Example: – H 2 OMass of H atom = 18g – NaCl = 58.44g The statement that the molecular weight of H2O is means that a water molecule has on the average a mass that is /12 times the mass of a 12C atom.

WhyWhyunitless? Find out! Rememberthat 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 ).

10 23 Avogadro's number = 6.02 x Mole C atom = 6.02 xC atoms = 12g Mole Mg atom = 6.02 xMg 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 ) magnesium atoms weigh 24.3 grams X carbon atoms weigh a total of 12.0 grams X molecules of CO 2 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 24g24gcarbon? 24g of carbon = 24/12 =2 moles mole of atoms = 6.02 x Therefore 2 moles ofcarbon contains: = x atoms x 6.02 xatoms

Try This! How many atoms and moles of silicon are in sample of silicon that has a mass of 5.23g? a Answers==== mol Silicon; and xatoms

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 H 2 O, 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 g and 1 mol of oxygen atoms has a mass of g, so 1 mol of water has a mass of (2 x g) g = g. The molar mass of water is g/mol. 1 mol of oxygen atoms has a mass of g  Molar mass of O = 16 g/mol Molar mass of H 2 O = 2 mol of H + 1 mol of O = (2x1.008 g/mol of H) + (16 g/mol of O) = g/mol M = mass = m molen

1.5Ideal Gases

Ideal Gas Law 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 = J/mol K N = number of molecules k = Boltzmann constant = x J/K = x 10-5 eV/K k = R/NA NA = Avogadro's number = x 1023

TheIdealGasLaw or R = cm 3.atm mol. K PV = nRT P = Pressure (in kPa) V = Volume (in L) T = Temperature (in K) n = moles R = 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.

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 following simplification: inthe P1V1 = P2V2P1V1 = P2V2 or PV = constant (mathematical expression of Boyle's Law)

For theVolume-Temperature relationship: Charles's Law n1 = n2 and P1 = P2 therefore the n's and cancel in the original expression resulting following simplification: the P's inthe V 1 T 2 = V 2 T 1 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 the cancel in the above original expression: V's P 1 T 2 = P 2 T 1 or P / T = constant (mathematical expression of Gay Lussac's Law)

For theVolume-Mole relationship: Avagadro's Law P1 = P2 and T1 = T2 therefore the P's and T's cancel in the above original expression: V1n2 = V2n1V1n2 = 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 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 P 1 V 1 =P 2 V 2

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 (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 V 1 /V 2 =T 1 /T 2 P 1 V 1 /T 1 =P 2 V 2 /T 2

Pressure PRESSURE P (Pressure) = F (Force) andVolume Units VOLUME 1 L = 1 dm 3 = 1000 cm 3 A (Area) ATMOSPHERE 1 atm = 760 torr = x 10 5 Pa In SI: 1 Pa (Pascal) = 1 N/m 2 Chemists use: 2 1 torr = Pa or 2 = N/m or = kg/ms 1 bar =10 Pa = atm = 750 torr 5

Example 1.1: Density of an Ideal Gas Page 16 Find the density of F 2 gas at 20.0°C and 188 torr.

Exercise Find the molar mass of a gas whose 1.80 g/L at 25.0°C and 880 torr. density is (Answer: 38.0 g/mol.)

1.6Differential Calculus

Functions and Limits Dependent variableFunctionf limy=f(x) Independent variable x ax 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 as x approaches. 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: ofthefollowinggraphis0 as x approachesinfinity,becausethe y = f(x) Approaches 0 x Approaches ∞

Examples Lim (4x) x  3 Sample A: Find the limit of f(x) =4x,4x,asxapproaches3or 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 x – 3) x 1x 1 Follow the same steps, x 2 + 5x – 3 = (1)– 3 =3 x 2 + 5x – 3 as x approaches So, the limit of1 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 theslope Let (x 1,y 1 ) = (4, 9) and (x 2,y 2 ) = (2, 1) Slope, m = (y1 − y2) =(y1 − y2) = (9 − 1) Positive slope (x1 − x2)(x1 − x2) (4 − 2 ) = = 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 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 variableis 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 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 + 5y 2 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 Differentiation withrespecttotox "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. 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 + 5y 2. 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 of F(x,y) = y + 6 sin x + 5y 2 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 respecttotoy y2y2 + 5y 2. The the 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 of F(x,y) = y + 6 sin x + 5y2 with respect to y. The derivative of the y-parts with respect to y is y. 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:

1.7Equations of State

PT T dd 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:Liquids: VV (T, P)VV (T, P)V V dVdTdP TP Define a thermal expansion coefficient and an isothermal compressibility by V 1 VT P 1 V VP T 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 dV TP Vln V2Vln V2 TPP 2121 V1V1 Values of ҡ and β (β=α) can be found in many handbooks.

1.8Integral Calculus

Definition: A function F(x) isthe antiderivative of a function ƒ(x) if for all x in thedomainofƒ, F'(x) = ƒ(x)F'(x) = ƒ(x) ƒ(x)ƒ(x)dx=F(x)F(x)+C,C,whereCisaconstant. Example1: Evaluate Use formula (4): and get this:

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

EndofChapter1 Understanding, rather than mindless memorization, is the key to learning physical chemistry…