NENG 301 Lecture 2 – The structure of

thermodynamics (DeHoff, Chap. 2)

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“All hail the mighty Glow Cloud”

Learning objectives for Chapter 2

At the end of this chapter you will be able to:

– Understand the general statements of the laws of thermodynamics

– Understand the basic terminology of thermodynamics as presented and used in this chapter and be able to give examples of each: system, surroundings, state of a system, state function

– Understand the classification of thermodynamic systems into different categories

– Know the meaning of the terms heat and work from a thermodynamic perspective

– Understand the difference between exact and inexact differentials

– Understand the classification of thermodynamic relationships: the laws of thermodynamics; definitions; coefficient relationships; Maxwell relations; and conditions for equilibrium

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General statements of the laws of thermodynamics as applied to the universe

• There exists a property of the universe, called its energy (U), which cannot change no matter what processes occur in the universe

• There exists a property of the universe, called its entropy (S), which can only change in one direction no matter what processes occur in the universe

• A universal absolute temperature scale exists and has a minimum value, defined to be absolute zero, and the entropy of all substances is the same at that temperature

Don’t worry for now: we will have a lot to say about these in Chapter 3…

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Again: important (if obvious) concepts

• System: any region of the universe, large or small, that is being considered in our analysis

• Boundary: the interface between the system and its surroundings

• Surroundings: regions outside the boundaries of the system but can alter the system by interacting with it

• Properties: Physical characteristics that define the condition of the system and its surroundings

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This is a system….

• The subset of the universe in focus in a particular application of thermodynamics is usually called the system

• At any given instant of observation the condition of the system is described by an appropriate set of properties

• Limitations on changes in these properties are set by the nature of its boundary

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This is a system that goes through a process

• The subset of the universe in focus in a particular application of thermodynamics is usually called the system

• At any given instant of observation the condition of the system is described by an appropriate set of properties

• Limitations on changes in these properties are set by the nature of its boundary

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Classification of systems

• Thermodynamic systems can be classified into several categories:

1. unary (one chemical component) versus multicomponent (two or more chemical components in multicomponent systems, the chemical composition may vary

2. homogeneous (single phase) versus heterogeneous (two or more phases, eg. ice/water)

3. closed (no exchange of matter by the system across its boundary with the surroundings) versus open (exchange of matter by the system with the surroundings) [note: also isolated]

4. non-reacting versus reacting (specifically chemical reactions)

5. “simple” versus “complex”

o simple: only energy exchanges involve thermal, mechanical or chemical changes

o complex: gravitational, electrical, magnetic or surface factors

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A typical thermodynamic system

• Cross-section through a MOSFET (metal oxide semiconductor field effect transistor) thin film device shows it to be a multicomponent, multiphase system in which chemical reactions and the influence of an electric field are important

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More definitions • State functions (or state variables): a system is said

to be in a certain state when all of its properties have these specific values – depend on the current condition of the system and not on how the system got there

– temperature

– pressure

– volume

– chemical composition

– (internal) energy

– entropy

– others….

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Simple Exercise

• If Z is a state function that depends on other state functions X and Y,

Z = Z (X,Y), show:

(

Still more definitions • Process: processes are described by quantities that

only have meaning for changing systems

• Consider a transition from state A to state B: represented by a curved path in the X-Y plane

• The change in Z (or DZ) is independent of the path

• There are two very important types of processes:

– work done on the system as it changes

– heat absorbed by the system as it changes

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The study of energy transformations and the relationships among physical properties of substances which are affected by these transformations.

-- K. Wark, Thermodynamics, 5th Edition (1988)

What is Thermodynamics?

Thermodynamics is mainly concerned with transformations of heat into mechanical work and the opposite transformations of mechanical work into heat.

-- E. Fermi, Thermodynamics (1937)

Work

• The easiest way to discuss work is through classical mechanics

• Consider the application of a force F: if the point of application of the force moves, then the force does work

• The increment of work done by the displacement is:

F

F=P A

P A P

ext

ext ext

w dx

w dx dV

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Work

Type of work intensity factor capacity factor formula

mechanical force change in distance f Δx

gravitational gravitational potential (a function of height)

mass mgh

electrical potential difference quantity of charge QΔV

surface surface energy/tension change in area g DA

• Work, like energy, can take various forms: mechanical, electrical, gravitational, etc.

• All have in common the fact that they are the product of two factors, an intensity term and a capacity term. – the simplest form of mechanical work arises when an object

moves a certain distance against an opposing force. – example: electrical work is done when a body having a certain

charge moves through a potential difference

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Heat

• Heat and work are both measured in energy units, so they must both represent energy

• Energy can take many forms: mechanical, chemical, electrical, radiation (light), and thermal, or heat

• Heat is a form of energy, but it differs from all the others in one crucial way: complete conversion of heat into other forms of energy is impossible

• Thermal energy can be transferred from one body (i.e., one system) to another (we often refer to this as a "flow" of heat)

• Heat can only flow spontaneously from a system at a higher temperature to one at a lower temperature

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Heat and Work • Consider a gas in a cylinder that is subjected it to two processes:

― compress from Pi, Vi to Pf, Vf

― heat from Ti to Tf

Mass

Piston

Pi, Vi

Mass

Pf, Vf

Piston

Tf

mechanical stops

Ti

Exact and inexact differentials • An “ordinary” (exact) differential, when integrated, yields a

finite difference given by the limits of integration:

• In other words: an exact differential integrates to a finite difference, independent of the path of the integration

• In contrast, an inexact differential integrates to a total quantity which depends on the path of integration taken:

• The cyclic integration of an exact differential is exactly zero for all cycles, while the cyclic integral of an inexact differential is usually non-zero:

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12 1

x

xdx x x x D

2

1Q Q

0dy (all cycles) 0Q exact inexact

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Exact and inexact differentials

• Quantities of heat Q and work W are path dependent and hence depend on the path taken by a given process

• The terms DQ and DW in this case are meaningless

– If DW meant anything it would mean W2 - W1

– The system in either the initial state or the final state does not have any work W1 or W2, nor does it have any heat Q1 or Q2

• Work and heat appear during a change in state; they are not properties of the state, but instead are properties of the path

• Properties of the state of the system (T, P, V, U) have differentials which are exact, while differentials of properties of the path (Q and W) are inexact

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Peculiarities of heat, work and energy

• Heat and work both appear at the boundary of a system

• Work can be completely converted into heat (by friction, for example), but heat can only be partially converted to work – conversion of heat into work is accomplished by a heat engine

• Heat and work are best thought of as processes by which energy is exchanged

• Energy is measured in terms of its ability to perform work or to transfer heat

• The basic unit of energy is the joule – one joule is the amount of work done when a force of 1 newton acts over a distance of 1 m; thus 1 J = 1 N-m

• The newton is the amount of force required to accelerate a 1-kg mass by 1 m/sec2, so the dimensions of the joule are kg m2 s–2

• The other two units in wide use: the calorie and the BTU (British thermal unit) are defined in terms of the heating effect on water

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2.2. It is not an overstatement to say that without state functions thermodynamics would be useless. Discuss this assertion.

2.4. Why is heat a process variable?

Sample homework problems

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If there were no state functions (like T, P, V, composition), i.e., properties that depend only upon the current condition of the system, and not on how it arrived at that condition) then the behavior of all aspects of matter would depend explicitly upon the history of the system. There would be no variables that, by themselves, explicitly describe the current condition of any system. Thus, even the history experienced by the system could not be described in terms of some sequence of change of its properties.

Heat is fundamentally a flow of energy. Heat is transferred between two systems, or between parts of the same system; this rearrangement of the distribution of energy is necessarily accompanied by changes in at least some of the properties of the systems involved. Such a change is by definition a process.

More definitions

• equations of state or state functions: relationships between dependent variables of state and independent variables of state

• intensive property: a property that is independent of the quantity of matter in a system (temperature, pressure, concentration, etc.)

• extensive property: a property that is dependent of the quantity of matter in a system (volume, heat capacity)

• Note that we can derive intensive properties from extensive properties (example: mole fraction)

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2.3. Determine which of the following properties of a thermodynamic system are extensive properties and which are intensive. a. The mass density. b. The molar density. c. The number of gram atoms of aluminum in a chunk of alumina. d. The potential energy of the system in a gravitational field. e. The molar concentration of NaCl in a salt solution. f. The heat absorbed by a the gas in a cylinder when it is compressed

gram atoms: the quantity of an element whose weight in grams is numerically equal to the atomic weight of the element. 0.450 gram of Fe contains how many atoms?

Classification of relationships • Just wait and see: you will become familiar with a large

number of thermodynamic relationships!

• In order to sort through the coming confusion it will be useful to classify these relationships

1. Laws of thermodynamics – these form the physical basis for all subsequent relationships

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There exists a property of the universe, called its energy (U), which cannot change no matter what processes occur in the universe

There exists a property of the universe, called its entropy (S), which can only change in one direction no matter what processes occur in the universe

A universal absolute temperature scale exists and has a minimum value, defined to be absolute zero, and the entropy of all substances is the same at that temperature

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2.6. Describe what the notion of equilibrium means to you. List as many attributes as you can think of that would be exhibited by a system that has come to equilibrium. Why do you think these characteristics of a system in equilibrium are important in thermodynamics?

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Answer to 2.6.

Attributes of equilibrium: 1. A state of rest: state of the system does not change with time. 2. A stable state: if the state is displaced from the equilibrium state, it will return to it. 3. A state of internal uniformity; (in the absence of external fields) gradients of intensive properties vanish.

The equilibrium state is the final state of every process. The primary goal of thermodynamics is the prediction of the properties of the final equilibrium state for any given initial condition of any system. "How far the system is" from the equilibrium state is a measure of the driving force for processes changing the system toward equilibrium, and controls the rate of approach to the final state of rest.

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The Zeroth Law of Thermodynamics

Equilibrium is characterized by a function of thermodynamic state variables. This function specifies the equation of state.

Classification of relationships

2. Definitions – new parameters, quantities and variables based on prior ones

• Energy: U = U(S,V) dU = T dS P dV

• Enthalpy: H = U + PV dH = T dS + V dP

• Helmholtz free energy: F = U – TS dF = S dT P dV

• Gibbs free energy: G = U + PV – TS = H - TS

dG = S dT + V dP

2. Coefficient relationships – describe how the value of state variable changes during an infinitesimal step in a process:

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Y X

dZ MdX NdY

Z ZdX dY

X Y

X, Y, Z are all state variables

Y X

dZ MdX NdY

Z ZdX dY

X Y

Coefficient relationships between state functions • If Z is a state function that depends

on other state functions X and Y, then we can related a change in Z with respect to X and Y:

• If Z = f(X,Y) represents a surface in (X,Y,Z) space, then dZ is the sum of the components in the X and Y directions

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Y X

Z ZdZ MdX NdY dX dY

X Y

Coefficient relationships between state functions – apply to an ideal gas

• Consider a mole of an ideal gas: PV = RT or V = f(P,T)

• Express V as a function of T and P:

• Determine the partial derivatives:

• So the final relationship for dV is:

1( )

RTV R T

P P

2;

T P

V RT V R

P P T P

2

RT RdV dP dT

P P

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Classification of relationships (con’t) 4. Maxwell relationships – provide descriptions of partial

derivatives involving state functions

X Y

M NdZ MdX NdY

Y X

• Where did this come from? Go back to:

Y X

Z ZM N

X Y

and

• Now take the derivatives:

X Y Y XX Y

M Z N Z

Y Y X X X Y

and

• Since the order of differentiation doesn’t matter:

X Y X YX Y

M Z Z N

Y Y X X Y X

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2.5. Write the total differential of the function

a. Identify the coefficients of the three differentials in this expression as appropriate partial derivatives.

b. Show that three Maxwell relations hold among these coefficients.

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Classification of relationships (con’t)

• As stated previously: Maxwell relationships provide descriptions of partial derivatives involving state functions

• Some are entirely unimportant, while others are extremely important, for example:

5. Conditions for equilibrium – sets of equations that describe the relations between state functions that must exist when a system is at equilibrium; these are the relationships that are used to calculate equilibrium maps

– We’ll have a lot of these!

T P

S VdG SdT VdP

P T

the isothermal pressure dependence of entropy is given by the easily-measured thermal expansion

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Simple Exercise

• If Z is a state function that depends on other state functions X and Y,

Z = Z (X,Y), show:

(

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Group Exercise

Group I: B,C,W (7) Group II: D, E, F, Y (7)

Group III: G,J, T (7) Group IV: K, M, U (7)

Group V: N, S (7) Group VI: P, R (6)

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Writing the differential dX directly from X = X(Y,Z):

Comparing the coefficients of dY in the two expressions:

dX =

Chapter 2 – So what have we learned?

• We have three laws of thermodynamics: – energy is conserved

– entropy is created

– temperature has a zero

• We can classify thermodynamic systems into categories: – number of components

– number of phases

– nature of the system boundary

– chemical reactivity

– complexity with respect to non-mechanical forces

• Thermodynamic relations can be classified as well: – the laws of thermodynamics

– definitions

– coefficient relationships

– Maxwell relations

– conditions for equilibrium 38