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Thermodynamics Lecture Notes for CHEM 3615

Importance of Units: Physical Quantity = Numerical Value + Unit No unit or wrong unit meaningless result In this class, we will attempt to use S.I. Units (S.I. = Système International) 1. S.I. Units are widely used and recommended in scientific journals 2. S.I. units form a coherent set. For example, let us assume that “u” is a function of the variables Z and X (so

we write u(Z,X)). If u(Z,X) is given by:

u(Z,X)=AZ+ B2

CX2 + D3 lnDE

exp

ED

where A, B, C, D and E are parameters, then the following must hold:

AZ must have the same unit as B2 and CX2 must have the same unit as D3.

ln( ) and exp( ) are dimensionless quantities, thus they are unit-less. Their

argument (D/E for ln and E/D for exp) must also be dimensionless. Therefore

E must have the same unit as D.

The unit of u(Z,X) must be the same as that of the ratio B2 / D3.

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If S.I. units are used for A, B, C, D, E, X and Z then the result u(Z, X) will be

correctly expressed in S.I. units.

When writing any equation A = B the units of A and B MUST be the same.

Therefore a simple consideration of the units on the R.H.S. and L.H.S. of an

equation can help you find potential error in the formulation of an equation.

The Five Primary S.I. Units

Quantity Symbol Unit Symbol for Unit

Length L meter m

Mass m kilogram kg

Time t second s

Temperature T Kelvin K

Electrical Current I Ampere A

Secondary S.I. Units

Quantity Symbol Unit Symbol for Unit

Volume V = L3 m3

Acceleration a = d2x/dt2 m.s-2

Force f = m a Newton 1 N = 1 kg.m.s-2

Work, Heat, Energy w = f ∆L Joule 1 J = 1 N.m

Power P = w / ∆t Watt 1 W = 1 J.s-1

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Pressure P = f / A Pascal 1 Pa = 1 N.m-2

Electric Charge Q = I t Coulomb 1 C = 1 A.s

Electric Potential ∆V = E / Q Volt 1 V = 1 J.C-1

Electric Resistance R = ∆V / I Ohm 1 Ω = 1 V.A-1

Electric Capacitance C = Q / ∆V Farad 1 F = 1 C.V-1

Unit Conversion:

Although I will not ask you to memorize conversion factors between different

unit systems, you must know the following:

1 micrometer (1 µm) = 10-6 m

1 nanometer (1 nm) = 10-9 m

1 kilogram (1 kg) = 103 g

1 deciliter (1 dL) = 10-1 L

1 milimeter (1 mm) = 10-3 m

1 centimeter (1 cm) = 10-2 m

Frequently Used Conversion Factors:

For Pressure:

1 atm = 101,325 Pa 1 bar = 105 Pa 1 torr = 133.222 Pa

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For Force:

1 dyne = 10-5 N

For Energy, Heat, Work:

1 calorie (cal) = 4.184 J 1 erg = 10-7 J

Thermodynamic Language: Definitions

System = Part of the Universe one is trying to understand and model

Surroundings (of a system): Part of the Universe that is “interacting” (i.e.

exchanging energy or matter) with the System

Universe = System + Surroundings.

Thermodynamics (dynamics of heat) is concerned with the transformation of

energy of one kind into others (for example mechanical or electrical work into

heat).This branch of science is based on the adoption of four fundamental

laws (known as the Laws of Thermodynamics), which have been derived

“empirically” from experimental observation but which CANNOT be proven

from first principles. The strength of Thermodynamics lies in the simple fact

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that two centuries of studies have not shown the least evidence that any of

these laws are incorrect.

Thermodynamics allows the prediction of whether a given process (evolution

of a system) can occur spontaneously, reversibly or not at all. It also allows

one to obtain specific relationships between the physical properties of a

system or the change in these properties (heat capacity, density, melting

temperature, energy, expansion coefficient, etc...) and the change in the

experimental conditions (perturbation) imposed on the system (temperature,

pressure).

The most important aspects of any thermodynamic calculation are 1)

the precise definition of the system to be studied and 2) the determination of

the surroundings for that system. The surroundings are generally defined to

include only the part of the universe which exchanges energy and matter with

the system. Considering “larger” surroundings obviously enables a more

accurate prediction of the system’s behavior during a given process, but also

leads to more difficult calculations.

To visualize what is meant by systems, surroundings and processes

think of the following cases: Human Body, Heart, Living Cell, Reaction Flask,

Chemical Reactor, Combustion Engine, Car, Blacksburg, Earth, the Universe,

a Metal Block, a Sponge, etc..

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As one defines system and surroundings for a given problem, it

becomes extremely important to understand the nature of the “Boundaries”

between the System and the Surroundings. These boundaries can be artificial

(as in the case of making weather forecasts for Blacksburg) or can be very

real, as in most examples listed above.

Different qualifiers are used to describe a system and the type of

exchanges that can take place between a system and its surroundings.

Open System: refers to the fact that both matter and energy can be

exchanged between the system and its surroundings. Closed System: refers to

the fact that matter cannot flow either in or out of the system (a closed system

is therefore characterized by a fixed mass. (Note the number of moles and the

composition are not necessarily constant during a process in a closed system,

as one may envision the process to be a chemical reaction where the total

number of moles and the concentration is changing). Isolated System: refers

to a system such that neither mass nor energy can flow in or out of the system.

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Homogeneous vs Heterogeneous Systems:

A system is homogeneous if any property of the system is uniform over

the system (has the same value regardless of where it is measured in the

system). Such a system is called a single-phase system. A system is

heterogeneous if the measured property varies with the location where it is

evaluated. Such a system is called a multiple-phase system.

Heterogeneity refers to a length-scale of the material which is much larger

than the size of the atoms, ions or molecules it is made of. A phase is defined

as a macroscopic assembly (number of molecules of the order of Avogadro’s

number) of atoms, ions, molecules. Obviously, the distinction between

homogeneous and heterogeneous becomes very vague if the “phase” size

corresponds to a small collection of atoms, ions or molecules.

Intensive vs. Extensive Properties of a System:

Systems can be characterized by a number of properties (Temperature,

Pressure, Volume, Composition, Density, Heat Capacity, Energy, Mass,

Number of Moles, Expansion Coefficient, etc...). When the magnitude of a

property is proportional to the amount of material considered, it is called an

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extensive property. On the other hand, when the magnitude of the property is

independent of the amount of material considered, it is called an intensive

property.

Intensive Property Extensive Property

Temperature (T) Mass (m)

Pressure (P) Number of Moles (n)

Density (ρ) Volume (V)

Mole Fraction (x) Energy (U)

Dielectric Constant (ε) Enthalpy (H)

Molar Mass (M) Entropy (S)

Expansion Coefficient (α)

A molar quantity, Xm, can be defined for any extensive property, X,

by dividing the property, X, by the total number of moles of molecules present

in the system. Molar quantities are therefore intensive properties.

A number of rules apply when we consider whether physical quantities

are intensive or extensive.

1. Thermodynamic variables or properties are either extensive or intensive.

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2. Two quantities can only be added to, equated to or subtracted from one-

another if they are both extensive or both intensive.

3. The ratio of two extensive quantities is an intensive quantity.

4. The product of an intensive quantity by an extensive quantity is an

extensive quantity.

The amount of material is defined for simplicity using the “mole” unit

(1 mol = 6.022 1023), using Avogadro’s number (the number of Carbon atoms

in 12 grams of 12C).

Concept of Equilibrium (Intro):

A system is said to be at equilibrium if none of its macroscopic (large

scale compared to atomic scale) properties are changing with time. A system

at equilibrium is said to be in a stable state, where all its properties have well

defined average values.

The macroscopic properties of a system which define the state of that

system are called the state variables or the thermodynamic coordinates (P,

V, T, ρ, ...). “Thermodynamics” will allow us to define and calculate state

functions, which define additional properties of that system in a given

equilibrium state (energy, entropy, free energy, etc..). These State Functions

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are usually expressed mathematically in terms of State Variables and

Materials Properties.

Example: Relating the change in energy of a system (∆U) to the change in

temperature (∆T), when the system volume is kept constant (during the whole

process). If, instead of considering the whole process, one focuses first on a

part of the process, where we change the temperature by a small increment dT.

The energy of the system changes by dU such that: dU = Cv dT

Cv is the material heat capacity measured at constant volume. If this

quantity can be considered to be constant during the whole process (we know

that although Cv depends on T, it increases only slowly with T), then we can

integrate the previous equation between initial and final states.

∆U = Cv ∆T

where ∆U = Ufinal - Uinitial (U is a state function)

∆T = Tfinal - Tinitial (T is a state variable)

Note that when dealing with state properties and state variables, we use the

symbol “d” to characterize an increment in the property and “∆” to

characterize the overall change in state property or state variable for the whole

process. Also note the self-consistent aspect of thermodynamic equations with

respect to the concept of intensive/extensive properties.

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Concept of Equation of State for a Homogeneous System:

When a system is at equilibrium, all its properties are entirely defined

by the values of the state variables (P, T, concentration, etc..) and not by the

history of the system. This suggests that the system can be described by some

law that relates the various state variables to each other. Such relationship

between State Variables is called an Equation of State.

For example the Ideal or Perfect Gas Equation Of State (E.O.S.) is

given by:

PV = nRT

where: R = 8.3145 J.K-1.mol-1 (S.I.), if P, V, T are expressed in S.I. units.

Example of Calculation:

Assuming n = 1.00 mol, T = 100.0 K and V = 10 m3, calculate P

P = 1.00 mol x 8.3145 J.K-1.mol-1x100.0 K / 10 m3 = 83.145....Pa

The result should be given as: P = 83 Pa, since the quantity V in the above

equation has only 2 significant digits. The result cannot have more significant

digits than any of the quantities involved in its calculation. A more rigorous

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evaluation of the number of significant digits is achieved through

differentiation of the equation:

P =nRTV

dPP

= d ln P( ) = d ln nRTV

= d ln n( ) + ln R( ) + ln T( ) − ln V( )( )

dPP

=dnn

+dRR

+dTT

−dVV

∆PP

=∆nn

+∆RR

+∆TT

+∆VV

∆P = P∆nn

+∆RR

+∆TT

+∆VV

∆n, ∆R, ∆T, ∆V, are the uncertainties on n, R, T and V. If n is given to be 1.00

mol, it implies that its value could be between 0.995 and 1.005, so the

uncertainty is ∆n = 0.005. Do the same thing for other quantities and calculate

∆P.

Gibbs Rule of Phase (First Visit)

For a single component, single phase system (i.e. pure gas, pure liquid,

or pure solid), only two intensive variables are necessary and sufficient to

describe fully all other intensive properties of the system (i.e. to fully describe

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the state of the system). For the ideal gas, knowledge of P and T, allows

calculation of Vm, ρ, etc..

Vm = RT / P (molar volume)

ρ = Μ / Vm = M R T / P (density)

We will see later that for more complicated systems (heterogeneous with

different species and under external fields) one may need to know the

magnitude of a larger number of state variables to fully define the state of the

system (the rule allowing to calculate the minimum number of state variables

necessary to fully define the state of the system is the Gibbs Phase Rule).

Zeroth Law of Thermodynamics (Law of Thermal Equilibrium)

If one brings two closed systems with fixed volume in thermal contact,

eventually these systems will reach thermal equilibrium (no net heat flow

between them will be detected) and both systems will be at the same

temperature.

The Zeroth Law of Thermodynamics states that if a system A is in

thermal equilibrium with a system B and system B is in thermal equilibrium

with system C, then system A is in thermal equilibrium with system C and

systems A and C are characterized by the same temperature.

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Note that this statement is derived from observations, cannot be proven

from first principles and has never been found to be wrong.

More Definitions:

Diathermic boundary: heat can flow through that boundary

Adiabatic boundary: heat cannot flow through that boundary

Following the experimental work by Boyle (PV = constant at constant T),

Charles and Gay-Lussac (V/T = constant at constant P) and the definition of

the mole by Avogadro leads to the definition of an ABSOLUTE

Temperature Scale (the Kelvin Temperature Scale).

P.V = constant. n.T

where the constant is denoted R and called the Gas constant

T (K) = θ (°C) + 273.15

The Ideal or Perfect Gas Equation of State is an approximation

which is only valid when the gas pressure is sufficiently low and the

temperature sufficiently high. It is however a very good approximation in

most areas of thermodynamics, provided that you are not interested in the

condensation of gases or critical phenomena (see later).

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Dalton’s Law for Gas Mixtures

This law is concerned with the application of the Perfect Gas law to a

mixture of non-reacting perfect gases. This will be useful when dealing with

chemical reactions or when dealing with specific gas mixtures (atmosphere,

natural gas, etc...).

Dalton’s Law states that the pressure exerted by a mixture of perfect gases is

the sum of the pressures exerted by the individual gases, assuming that each

gas is occupying the same volume as the mixture at the same temperature. The

pressure exerted by a component “i” of the gas mixture is called the Partial

Pressure, Pi .

A, B, C, , ,“i”, , , , , X species in the mixture

PA, PB, PC, , , Pi, , , , PX partial pressures

Pi =ni RT

VP = PA + PB + .... + Pi + ... + PX = Pi

i∑

P =RTV

nii

∑ =nRT

V

xi =nini

i∑

=nin

=PiPi

i∑

=PiP

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The Dalton law, simply put, is a definition for the Partial Pressure. It is

obeyed when one can exchange one type of molecules for another, i.e. when

the molecules do not care whether they are surrounded by molecules of one

type or another. This will be the case when molecules do not interact with one

another (i.e. when the gas is perfect (low P and high T)).