Chapter02.pdf

Geochemistry

W. M. White

Chapter 2: Fundamental Concepts of Thermodynamics

Chapter 2: Energy, Entropy and Fundamental

Thermodynamic Concepts

2.1 The Thermodynamic Perspective

e defined geochemistry as the application of chemical knowledge and techniques to solve geo-

logical problems. It is appropriate, then, to begin our study of geochemistry with a review of

physical chemistry. Our initial focus will be on thermodynamics . Strictly defined, thermody-

namics is the study of energy and its transformations. Chemical reactions and changes of states of mat-

ter inevitably involve energy changes. By using thermodynamics to follow the energy, we will find that

we can predict the outcome of chemical reactions, and hence the state of matter in the Earth. In princi-

ple at least, we can use thermodynamics to predict at what temperature a rock will melt and the com-

position of that melt, and we can predict the sequence of minerals that will crystallize to form an igne-

ous rock from the melt. We can predict the new minerals that will form when that igneous rock under-

goes metamorphism, and we can predict the minerals and the composition of the solution that forms

when that metamorphic rocks weathers. Thus thermodynamics allows us to understand (in the sense

that we defined understanding in Chapter 1) a great variety of geologic processes.

Thermodynamics embodies a macroscopic viewpoint, i.e., it concerns itself with the properties of a sys-

tem, such as temperature, volume, heat capacity, and it does not concern itself with how these proper-

ties are reflected in the internal arrangement of atoms. The microscopic viewpoint, which is concerned

with transformations on the atomic and subatomic levels, is the realm of statistical mechanics and quan-

tum mechanics . In our treatment, we will focus mainly on the macroscopic (thermodynamic) viewpoint,

but we will occasionally consider the microscopic (statistical mechanical) viewpoint when our under-

standing can be enhanced by doing so.

In principle, thermodynamics is only usefully applied to systems at equilibrium . If an equilibrium system is

perturbed, thermodynamics can predict the new equilibrium state, but cannot predict how, how fast, or

indeed whether, the equilibrium state will be achieved. (The field of irreversible thermodynamics , which

we will not treat in this book, attempts to apply thermodynamics to non-equilibrium states. However,

we will see in Chapter 5 that thermodynamics, through the principle of detailed balancing and transition

state theory , can help us predict reaction rates.)

Kinetics is the study of rates and mechanisms of reaction. Whereas thermodynamics is concerned

with the ultimate equilibrium state and not concerned with the pathway to equilibrium, kinetics con-

cerns itself with the pathway to equilibrium. Very often, equilibrium in the Earth is not achieved, or

achieved only very slowly, which naturally limits the usefulness of thermodynamics. Kinetics helps us

to understand how equilibrium is achieved and why it is occasionally not achieved. Thus these two

fields are closely related, and together form the basis of much of geochemistry. We will treat kinetics in

Chapter 5.

2.2 Thermodynamic Systems and Equilibrium

We now need to define a few terms. We begin with the term system , which we have already used. A

thermodynamic system is simply that part of the universe we are considering. Everything else is re-

ferred to as the surrounding . A thermodynamic system is defined at the convenience of the observer in a

manner so that thermodynamics may be applied. While we are free to choose the boundaries of a sys-

tem, our choice must nevertheless be a careful one as the success or failure of thermodynamics in de-

scribing the system will depend on how we have defined its boundaries. Thermodynamics often al-

lows us this sort of freedom of definition. This can certainly be frustrating, particularly for someone

exposed to thermodynamics for the first time (and often even the second or third time). But this free-

dom allows us to apply thermodynamics successfully to a much broader range of problems than other-

wise.

September 9, 2009

Geochemistry

W. M. White

Chapter 2: Fundamental Concepts of Thermodynamics

A system may be related to its environment in

a number of ways. An isolated system can ex-

change neither energy (heat or work) nor matter

with its surroundings. A truly isolated system

does not exist in nature, so this is strictly a theo-

retical concept. An adiabatic system can ex-

change energy in the form of work, but not heat

or matter, with its surroundings, that is to say it

has thermally insulating boundaries. Though a

truly adiabatic system is probably also a fiction,

many geologic systems are sufficiently well insu-

lated that they may be considered adiabatic.

Closed systems may exchange energy, in the form

of both heat and work with their surrounding

but cannot exchange matter. An open system

may exchange both matter and energy across it

boundaries. The various possible relationships

of a system to its environment are illustrated in

Figure 2.1.

Depending on how they behave over time, sys-

tems are said to be either in transient or time-

invariant states. Transient states are those that

change with time. Time-independent states may

be either static or dynamic . A dynamic time-independent state, or steady-state , is one whose thermody-

namic and chemical characteristics do not change with time despite internal changes or exchanges of

mass and energy with its surroundings. As we will see, the ocean is a good example of a steady-state

system. Despite a constant influx of water and salts from rivers and loss of salts and water to sedi-

ments and the atmosphere, it composition does not change with time (at least on geologically short

time scales). Thus a steady-state system may also be an open system. We could define a static system

is one in which nothing is happening. For example, an igneous rock or a flask of seawater (or some

other solution) is static in the macroscopic perspective. From the statistical mechanical viewpoint,

however, there is a constant reshuffling of atoms and electrons, but with no net changes. Thus static

states are generally also dynamic states when viewed on a sufficiently fine scale.

Let’s now consider one of the most important concepts in physical chemistry, that of equilibrium . One

of the characteristics of the equilibrium state is that it is static from a macroscopic perspective, that is, it

does not change measurably with time. Thus the equilibrium state is always time-invariant. While a

reaction A → B may appear to have reached static equilibrium on a macroscopic scale this reaction may

still proceed on a microscopic scale but with the rate of reaction A → B being the same as that of B → A.

Indeed, a kinetic definition of equilibrium is that the forward and reverse rates of reaction are equal.

The equilibrium state is entirely independent of the manner or pathway in which equilibrium is

achieved. Indeed, once equilibrium is achieved, no information about previous states of the system can

be recovered from its thermodynamic properties. Thus a flask of CO 2 produced by combustion of

graphite cannot be distinguished from CO 2 produced by combustion of diamond. In achieving a new

equilibrium state, all records of past states are destroyed.

Time-invariance is a necessary but not sufficient condition for equilibrium. Many systems exist in

metastable states. Diamond at the surface of the Earth is not in an equilibrium state, despite its time-

invariance on geologic time scales. Carbon exists in this metastable state because of kinetic barriers that

inhibit transformation to graphite, the equilibrium state of pure carbon at the Earth’s surface. Over-

coming these kinetic barriers generally requires energy. If diamond is heated sufficiently, it will trans-

form to graphite, or in the presence of sufficient oxygen, to CO 2 .

Surface free

to do work

Isolated System

Adiabatic System

Closed System

Open System

Figure 2.1. Systems in relationship to their sur-

roundings. The ball represents mass exchange, the

arrow represents energy exchange.

September 9, 2009

Geochemistry

W. M. White

Chapter 2: Fundamental Concepts of Thermodynamics

The concept of equilibrium versus metas-

table or unstable (transient) states is illus-

trated in Figure 2.2 by a ball on a hill. The

equilibrium state is when the ball is in the

valley at the bottom of the hill, because its

gravitational potential energy is minimized

in this position. When the ball is on a slope,

it is in an unstable, or transient, state and

will tend to roll down the hill. However, it

may also become trapped in small depres-

sions on the side of the hill, which represent

metastable states. The small hill bordering

the depression represents a kinetic barrier.

This kinetic barrier can only be overcome

when the ball acquires enough energy to roll

up and over it. Lacking that energy, it will

exist in the metastable state indefinitely.

In Figure 2.2, the ball is at equilibrium when its (gravitational) potential energy lowest (i.e., at the bot-

tom of the hill). This is a good definition of equilibrium in this system, but as we will soon see, is not

adequate in all cases. A more general statement would be to say that the equilibrium state is the one

toward which a system will change in the absence of constraints . So in this case, if we plane down the bump

(remove a constraint), the ball rolls to the bottom of the hill. At the end of this Chapter, we will be able

to produce a thermodynamic definition of equilibrium based on the Gibbs Free Energy. We will find

that, for a given pressure and temperature, the chemical equilibrium state occurs when the Gibbs Free

Energy of the system is lowest.

Natural processes proceeding at a finite rate are irreversible : under a given set of conditions; i.e., they

will only proceed in one direction. Here we encounter a problem in the application of thermody-

namics: if a reaction is proceeding, then the system is out of equilibrium and thermodynamic analysis

cannot be applied. This is one of the first of many paradoxes in thermodynamics. This limitation might

at first seem fatal, but we get around it by imagining a comparable reversible reaction. Reversibility and

local equilibrium are concepts that allow us to 'cheat' and apply thermodynamics to non-equilibrium

situations. A “reversible” process is an idealized one where the reaction proceeds in sufficiently small steps that

it is in equilibrium at any given time (thus allowing the application of thermodynamics).

Local equilibrium embodies the concept that in a closed or open system, which may not be at equi-

librium on the whole, small volumes of the system may nonetheless be at equilibrium. There are many

such examples. In the example of mineral crystallizing from magma, only the rim of the crystal may be

in equilibrium with the melt. Information about the system may nevertheless be derived from the rela-

tionship of this rim to the surrounding magma. Local equilibrium is in a sense the spatial equivalent to

the temporal concept of reversibility and allows the application of thermodynamics to real systems,

which are invariably non-equilibrium at large scales. Both local equilibrium and reversibility are exam-

ples of simplifying assumptions that allow us to treat complex situations. In making such assumptions,

some accuracy in the answer may be lost. Knowing when and how to simplify a problem is an impor-

tant scientific skill.

2.2.1 Fundamental Thermodynamic Variables

In the next two chapters we will be using a number of variables, or properties, to describe thermody-

namic systems. Some of these will be quite familiar to you, others less so. Volume, pressure, energy,

heat, work, entropy, and temperature are most fundamental variables in thermodynamics. As all other

thermodynamic variables are derived from them, it is worth our while to consider a few of these prop-

erties.

Unstable

Transient

Metastable

(Time-Invariant)

Equilibrium

(Time-Invariant)

Figure 2.2. States of a system.

September 9, 2009

Geochemistry

W. M. White

Chapter 2: Fundamental Concepts of Thermodynamics

Energy is the capacity to produce change. It is a fundamental property of any system, and it should

be familiar from physics. By choosing a suitable reference frame, we can define an absolute energy

scale. However, it is changes in energy that are generally of interest to us rather than absolute

amounts. Work and heat are two of many forms of energy. Heat, or thermal energy, results from ran-

dom motions of molecules or atoms in a substance and is closely related to kinetic energy. Work is

done by moving a mass, M, through some distance, x = X, against a force F :

∫

w = −

2.1

where w is work and force is defined mass times acceleration:

F = –M dv

2.2

(the minus signs are there because of the convention that work done on a system is positive , work done by a

system is negative ). This is, of course, Newton’s first law. In chemical thermodynamics, pressure–

volume work is usually of more interest. Pressure is defined as force per unit area:

P = F

2.3

Since volume is area times distance, we can substitute equation 2.3 and dV = Adx into 2.1 and obtain:

A A

x 1

V 1

∫

w = −

dx = −

PdV

2.4

x 0

V 0

Thus work is also done as a result of a volume change in the presence of pressure.

Potential energy is energy possessed by a body by virtue of its position in a force field, such as the

gravitational field of the Earth, or an electric field. Chemical energy will be of most interest to us in this

book. Chemical energy is a form a potential energy stored in chemical bonds of a substance. Chemical

energy arises from the electromagnetic forces acting on atoms and electrons. Internal energy, which

we denote with the symbol U, is the sum of the potential energy arising from these forces as well as the

kinetic energy of the atoms and molecules (i.e., thermal energy) in a substance. It is internal energy that

will be of most interest to us.

We will discuss all these fundamental variables in more detail in the next few sections.

2.2.1.1 Properties of State

Properties or variables of a system that depend only on the present state of the system, and not on the

manner in which that state was achieved are called variables of state or state functions . Extensive proper-

ties depend on total size of the system. Mass, volume, and energy are all extensive properties. Exten-

sive properties are additive, the value for the whole being the sum of values for the parts. Intensive

properties are independent of the size of a system, for example temperature, pressure, and viscosity.

They are not additive, e.g., the temperature of a system is not the sum of the temperature of its parts. In

general, an extensive property can be converted to an intensive one by dividing it by some other exten-

sive property. For example, density is the mass per volume and is an intensive property. It is generally

more convenient to work with intensive rather than extensive properties. For a single component sys-

tem not undergoing reaction, specification of 3 variables (2 intensive, 1 extensive) is generally sufficient

to determine the rest, and specification of any 2 intensive variables is generally sufficient to determine

the remaining intensive variables.

A final definition is that of a pure substance . A pure substance is one that cannot be separated into

fractions of different properties by the same processes as those considered. For example, in most proc-

esses, the compound H 2 O can be considered a pure substance. However, if electrolysis were involved,

this would not be the case.

September 9, 2009

Geochemistry

W. M. White

Chapter 2: Fundamental Concepts of Thermodynamics

2.3 Equations of State

Equations of state describe the relationship that exists among the state variables of a system. We

will begin by considering the ideal gas law and then very briefly consider two more complex equations

of state for gases.

2.3.1 Ideal Gas Law

The simplest and most fundamental of the equations of state is the ideal gas law † . It states that pres-

sure, volume, temperature, and the number of moles of a gas are related as:

PV = NRT 2.5

where P is pressure, V is volume, N is the number of moles, T is thermodynamic, or absolute tempera-

ture (which we will explain shortly), and R is the ideal gas constant* (an empirically determined con-

stant equal to 8.314 J/mol-K, 1.987 cal/mol-K or 82.06 cc-atm/deg-mol). This equation describes the re-

lation between two extensive (mass dependent) parameters, volume and the number of moles, and two

intensive (mass independent) parameters, temperature and pressure. We earlier stated that if we de-

fined two intensive and one extensive system parameter, we could determine the remaining parame-

ters. We can see from equation 2.5 that this is indeed the case for an ideal gas. For example, if we know

N, P, and T, we can use equation 2.5 to determine V.

The ideal gas law, and any equation of state, can be rewritten with intensive properties only. Di-

viding V by N we obtain the molar volume , – . Substituting – for V and rearranging, the ideal gas equa-

tion becomes:

V = RT

2.6

The ideal gas equation tells us how the volume of a given amount of gas will vary with pressure and

temperature. To see how molar volume will vary with temperature alone, we can differentiate equa-

tion 2.6 with respect to temperature holding pressure constant and obtain:

(

)

∂ V

∂ T

= ∂ NRT / P







 P

2.7

∂ T

∂ V

∂ T

= NR







 P

which reduces to:

2.8

It would be more useful to know to fractional volume change rather than the absolute volume change

with temperature, because the result in that case does not depend on the size of the system. To convert

to the fractional volume change, we simply divide the equation by V:

∂ V

∂ T

= NR







 P

2.9

Comparing equation 2.9 with 2.5, we see that the right hand side of the equation is simply 1/T, thus



 



  P

∂ V

∂ T

= 1

2.10

The left hand side of this equation, the fractional change in volume with change in temperature, is

known as the coefficient of thermal expansion, α :

† Frenchman Joseph Gay-Lussac (1778-1850) established this law based on earlier work of Englishman Robert Boyle

and Frenchman Edme Mariotte.

* We will often refer to it merely as the gas constant.

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