Douce A.P. Thermodynamics of the Earth and Planets

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38 Energy in planetary processes

be state variables – they may not even be functions! This distinction is certainly correct

and has further mathematical implications (see, for example, Lewis & Randall, 1961,or

Glasstone, 1946) but these will not be pursued here, as they are not essential to our goals.

We now have a precise definition of heat capacity at constant volume. Another heat

capacity that is well defined is the heat capacity at constant pressure, C

, which is defined

in terms of heat transferred as:





. (1.53)

As with C

, we wish to recast this expression in terms of a state variable in order for it to be

of use in thermodynamic calculations. Internal energy is not a convenient choice, because

when heat is absorbed at constant pressure there is a change in volume and hence expansion

work takes place. Some of the heat does not become internal energy, but it is not a priori

obvious how much.

In order to place this discussion on firm ground it is necessary to give a precise mathemat-

ical definition of the First Law of Thermodynamics. Before we do that in the next section,

there is one additional issue that needs to be clarified. Our definitions of heat capacity and

internal energy make no mention of the size of the system. Clearly, the amount of heat that

is required to change the temperature of a system by a certain amount depends both on some

intrinsic material property of the system and on the size of the system. In thermodynamic

calculations we commonly wish to do so independently of system size. One way to do that

is to work with material properties and other thermodynamic variables on a molar (per mol)

basis. Variables such as molar heat capacity, molar volume and bulk modulus are intrinsic

characteristics of a substance or system: they are material properties that are independent of

the system’s size. A thermodynamic variable or property that is independent of system size

is called an intensive variable. Molar heat capacity, molar volume, molar internal energy

and bulk modulus are examples of intensive variables, and are thus material properties. The

total heat capacity of a system, its total internal energy, or its total volume, are examples of

extensive variables, and are the product of an intensive variable times the size of the system.

Pressure and temperature are intensive variables too, but they are not material properties.

Intensive variables will be represented with uppercase non-bold symbols (e.g. C

, E, V )

and their extensive counterparts with uppercase bold symbols (e.g. C

, E, V ). The inten-

sive variables P and T, of course, have no extensive counterparts. In some applications we

will find it convenient to consider thermodynamic properties per unit mass. These are called

specific properties, and will be represented with lowercase non-bold symbols, e.g. c

for

the specific heat capacity or e for specific internal energy. Molar and specific properties are

simply related to one another by the molecular weight of the substance. For example, for

constant pressure heat capacities:

ρV

, (1.54)

where w is molecular weight, V is molar volume and ρ is density. I note in passing that

Guggenheim (1967) has pointed out the sloppiness of the term “molecular weight”, given,

among other things, that what this quantity actually refers to is mass. He suggested the more

precise term proper mass. Unfortunately, it does not appear that this suggestion has gained

much acceptance.

39 1.10 The First Law of Thermodynamics

1.10 The First Law of Thermodynamics

The First Law of Thermodynamics is the mathematical statement of the law of conservation

of energy. If we restrict ourselves to systems and processes in which work and heat are the

only pathways for energy transfer, then the statement of the First Law is simply:

dE = dQ −dW , (1.55)

where dE is an (infinitesimal) change in the internal energy of the system, dQ is the heat

absorbed by the system and dW is the work performed by the system. This sign convention

must be clearly understood and adhered-to rigorously. It is not universally used, but it is

the one that is most intuitively appealing. It is consistent with the various sign conventions

that we have established in previous sections. For example, if a system performs expansion

work on its environment in the absence of any heat exchange (dQ =0), then dW > 0 and

its internal energy decreases. If the system loses heat without performing work (dW = 0),

then dQ < 0 and its internal energy also decreases.

The First Law as expressed by equation (1.55) is not specific about the type of work

represented by dW, but it excludes some types of energy. Processes in which there are

mass-energy conversions, such as radioactive decay and nuclear reactions, are not included

in (1.55). The First Law of Thermodynamics applies to such processes, but the equation

has to be modified in order to account for the mass-energy term. In systems that contain

non-negligible amounts of (macroscopic) kinetic energy this energy must also be added to

equation (1.51) as an extra term.

When solving an energy balance problem one starts by writing down the First Law of

Thermodynamics in a form that is appropriate to the nature of the problem at hand. For

example, if we restrict ourselves to chemical systems in which only expansion work is

possible, then equation (1.55) can be re-written as follows:

dE =dQ −PdV . (1.56)

Once we have chosen the appropriate expression for the First Law, the next step is to consider

whether the constraints of the problem allow any simplification. For example, for a constant

volume (isochoric) process dV = 0, so the First Law of Thermodynamics becomes:

=dQ

, (1.57)

where the subscript V reminds us of the constant volume constraint. One could then be

tempted to make the substitution dE

= C

dT , but before doing so the nature of the

system under study must be carefully considered. For a system that does not undergo a

phase transition (e.g. melting or vaporization), nor a chemical reaction, this substitution

is indeed valid, because in this case all of the heat that is exchanged is manifested as a

temperature change. This is called sensible heat. Heat associated with a phase transition or

with a chemical transformation is not reflected in a temperature change, and is sometimes

called latent heat. If latent heat is involved in the transformation then the substitution

dT is not valid.

The First Law applied to a constant pressure (isobaric) process:

=dQ

−PdV (1.58)

40 Energy in planetary processes

leads to the definition of a new state variable. Using internal energy as a thermodynamic

state variable for isobaric processes leads to cumbersome equations, because isobaric heat

transfer is accompanied by a change in volume and internal energy is generally a function of

volume. To see why this must be so, consider a system from the microscopic point of view. In

general, there are electrostatic attractions and repulsions among the system’s microscopic

components (e.g. molecules in a gas, ions in a crystal) and any work performed against

those forces during a change in volume becomes electrostatic potential energy, which must

be part of the system’s internal energy. Internal energy would be independent of volume

only if there were no interatomic forces in the system. This is true only in ideal gases

(Section 1.14). For all real substances E is a function of V. Given the simplicity of equation

(1.57), however, we would like to obtain a similarly compact equation that states the First

Law of Thermodynamics for isobaric processes. In order to do this, we design a new state

variable with the desired properties. This new state variable, called enthalpy, is universally

symbolized by H , and is defined as follows:

H ≡ E +P V . (1.59)

This trick – inventing new functions to make calculations easier – is a common one in

thermodynamics. It may not be a priori obvious why we would choose to define enthalpy

in this way. You can think of it initially as a useful mathematical device, but it will become

apparent that enthalpy has a clear physical meaning. Enthalpy has two important properties.

First, given that internal energy is a state variable, enthalpy is a state variable too, as pressure

and volume depend only on the equilibrium state of a system. Second, from the definition

of enthalpy it follows immediately that it has dimension of energy.

Applying the product rule for differentiation to (1.59) we see that an infinitesimal change

in enthalpy is given by:

dH = dE +PdV +V dP, (1.60)

which, for an isobaric process (dP = 0), simplifies to:

=dE

+PdV . (1.61)

Substituting equation (1.58)in(1.61):

=dQ

(1.62)

shows that the physical meaning of enthalpy is heat exchanged at constant pressure. The

enthalpy change includes a contribution from a change in internal energy and a contribution

from expansion work (equation (1.61)). Recalling the definition of heat capacity at constant

pressure (equation (1.53)), we see that, for a process in which only sensible heat is involved:



∂H

∂T



. (1.63)

1.11 Independent variables and material properties

The state of a thermodynamic system is fully determined once we fix the values of a small

number of variables that we can think of as the independent variables, or control variables.

41 1.12 Some applications of the First Law

In Chapter 6 we will prove that the state of a homogeneous system of fixed composition

at equilibrium is fully defined once we fix the values of two intensive variables. Thermo-

dynamics allows us remarkable freedom for choosing which intensive variables to use as

the independent variables. We can use this freedom to our advantage by choosing variable

combinations that are best suited to each specific problem. In chemical thermodynamics,

and especially in its applications to planetary sciences, pressure and temperature are the

two most “fundamental” independent variables, largely because they are the easiest ones

to measure, but also because understanding how a system changes as a function of these

variables, or how one of these variables changes as a function of the other (e.g. with depth

in a planetary body), tend to be some of our chief preoccupations. The molar volume of

a homogeneous systems is related to P and T through an equation of state (Section 1.4.3).

Given an EOS, we can always express one of the three variables, P – V – T, as a function

of the other two and substitute as needed in thermodynamic equations.

Equations of state are well known for some substances but not for many others.Amaterial,

however, always has an equation of state, even if we don’t know what this equation is. This

is equivalent to saying that a given substance at a given pressure and temperature has a single

and well-defined molar volume, or, equivalently, that molar volume is a state variable. But

because EOS for many substances are either unknown or mathematically unwieldy, it is

also possible, and often advantageous, to express P – V – T relations as a set of material

properties. In practice, there are three material properties that are relatively straightforward

to measure and that show up repeatedly in thermodynamic calculations. In Chapters 4 and

8 we will see that this last attribute is not a coincidence. Two of these properties describe

the mechanical properties of the material. They are the bulk modulus K, or its inverse,

the compressibility, β, and the coefficient of thermal expansion at constant pressure, α.

Recall that K (and β) can be defined at either isothermal or adiabatic conditions (Section

1.4.3). In order to complete the thermodynamic characterization of a material we need to

describe its thermal properties as well, and this additional information is contained in the

heat capacity, either C

or C

. We have already given the mathematical definition of K

(equation (1.21)). We repeat it here along with the definitions of molar heat capacity, C

coefficient of thermal expansion, α, and isothermal compressibility, β



∂H

∂T



(1.64)

=−V



∂P

∂V



; β

=−



∂V

∂P



⇒K

(1.65)

α =



∂V

∂T



. (1.66)

Material properties can be derived from the EOS (see Exercises 1.11 and 1.12) and they are

not independent of one another (Exercise 1.13).

1.12 Some applications of the First Law of Thermodynamics

1.12.1 Discontinuous phase transitions and latent heat

A phase transition is a change in the physical nature of a system that occurs in response

to a change in the values of some of the controlling intensive variables (e.g. temperature

42 Energy in planetary processes

or pressure) but without a change in chemical composition. We will see in Chapter 7 that

there are different types of phase transitions. The most familiar ones are discontinuous phase

transitions.As their name implies, these are phase transitions at which there is an observable

discontinuity in physical properties: melting, boiling and sublimation are familiar examples.

Latent heat is always absorbed or liberated during discontinuous phase transitions – in

fact, the existence of non-zero latent heat is the formal thermodynamic definition of a

discontinuous phase transition (Chapter 7).

Suppose that we want to know how much heat is required to convert, isobarically, 1 mol

of liquid H

O at a temperature T

to 1 mol of H

O gas at a temperature T

vap

where T

vap

is the temperature at which the liquid to gas phase transition for H

O takes place

for the pressure of interest. Because this is an isobaric transformation, the heat required to

effect this transformation equals the change in enthalpy between liquid H

OatT

and H

gas at T

(equation (1.62)). Calling this total enthalpy change H we can write it as follows:

H = H

liquid,T

⇒T

vap

+H

liquid to gas

+H

gas,T

vap

⇒T

. (1.67)

The first and last terms on the right-hand side of the equation are, respectively, the enthalpy

changes of the liquid, from the initial temperature to the vaporization temperature, and

of the gas, from the vaporization temperature to the final temperature. They involve only

sensible heat, so we make dH = C

dT (equation (1.63)) and integrate:

H

liquid,T

⇒T

vap



vap

P,liquid

dT (1.68)

and:

H

gas,T

vap

⇒T



vap

P,gas

dT . (1.69)

Note that if pressure were not constant, then the substitution dH =C

dT would not be valid.

Heat capacities are not constant. They vary with temperature and, much less strongly, with

pressure. In order to integrate equations (1.68) and (1.69) it is necessary to express C

a function of temperature, and then integrate this function. This is discussed in Software

Box 1.1, and also in Chapter 5. The middle term on the right-hand side of equation (1.67),

H

liquid to gas

, is the energy associated with the phase transition, which for a liquid to gas

phase transition is called the enthalpy of vaporization, symbolized by H

vap

Software Box 1.1 Anintroductionto Maple: calculation of heatcapacityintegrals and enthalpy

of reaction as a function of temperature

Thermodynamic calculations are not difficult, but they can be tedious. In my opin-

ion, there is no point in doing any calculations or routine algebraic manipulations

by hand if they can be accomplished much faster and with much less possibility of

errors creeping in by relying on symbolic algebra software. Of the several products

available, I use Maple. Throughout this book I rely on a number of Maple proce-

dures that I have written myself. All of the code is available in Windows format from

www.cambridge.org/patino_douce, from where the files can be downloaded and run.

43 1.12 Some applications of the First Law

All you need to do is install Maple on your computer. The files can of course be opened

and edited. I am a self-taught programmer, however, so it is likely that many readers

will find better and more elegant ways of accomplishing the same tasks. If you prefer

different software, such as Mathematica or Matlab, and you are proficient in it, then it

will probably be very easy to translate the Maple code. If you have never used symbolic

algebra software before you may need to spend a few hours learning the basics of how

Maple works before you attempt to understand and run the procedures that accompany

this book. Maple can do many things. It can perform numeric calculations. It can per-

form algebraic manipulations. It can differentiate and integrate functions. It can solve

equations and systems of equations, linear and non-linear, algebraic and differential. It

can plot functions. It can read and write files. And it is a powerful programming lan-

guage, so that all of the things that Maple does can be part of a program, which is called

a Maple procedure.

A good introduction to the use of Maple in thermodynamic calculations is to apply

it to solve the integral of the heat capacity function, which is needed to calculate the

enthalpy change of a chemical reaction at any arbitrary temperature (equation (1.100)).

The heat capacity of all substances varies with temperature but there is generally no

strong physical basis to predict the form of the function C

(T). The approach that

is universally used is to measure heat capacity over a range of temperatures (see, for

example,Anderson, 2005) and fit the data empirically with a polynomial function. Differ-

ent functions are in use. The following, which is sometimes called the Shomate equation

(see Shomate, 1954; Shomate & Cohen, 1955), appears to work well for minerals and

fluids of geological interest:

T +a

−2

−1/2

, (S1.1.1)

where T is temperature in Kelvin and the a

s are empirical best-fit coefficients. Two

geologically oriented data bases that use this equation are those of Robie and Hemingway

(1995) and Holland and Powell (1998). Many, but not all, species listed in the NIST

Chemistry WebBook also use this equation. Throughout much of this book I use Holland

and Powell’s data base, so equation (S1.1.1) is the one that we will initially implement

in Maple. Holland and Powell truncate the heat capacity function after the fourth term,

but there is no harm in programming the full equation and setting a

=0 in Holland and

Powell’s data.

The problem that we wish to solve is to find the value of the following definite integral:



298

dT =



298



T +a

−2

+ a

−1/2

+ a



dT , (S1.1.2)

where T is the temperature of interest, and 298 stands for 298.15 K, the temperature at

which thermodynamic data are tabulated. The beauty of Maple is that we do not have

to deal with any cumbersome algebra and arithmetic. All we need to do is tell Maple

what is the function that we want to integrate and what the integration limits are, and

then ask it to integrate. Out pops a number.

The Maple procedure that performs the heat capacity integral is placed in a pack-

age, which is a file that contains a collection of Maple procedures that can be called

from other procedures. The name of this file is th_shomate.mw. The package contains

several procedures. The first one, named cp_sh, calculates the value of the heat

44 Energy in planetary processes

Software Box 1.1 Continued

capacity, i.e., of function (S1.1.1). The second procedure is named intcp_sh and

calculates the definite integral of the heat capacity equation, i.e., function (S1.1.2). The

fourth procedure, named HT_sh, calculates the enthalpy at the temperature of interest

by adding the heat capacity integral to the enthalpy at 298.15 K (e.g. equation (1.100)).

Each of these procedures receives from the calling procedure an array that contains

the values of the thermodynamic properties, and a separate variable that contains the

temperature in K. The remaining procedures work in the same way but calculate other

thermodynamic functions. They will be introduced in later chapters. The procedures in

th_shomate.mw are placed in a table (one of the many Maple data structures – we

won’t get into that here), and this table is saved into a package, which is an executable

file that other Maple procedures can call. The last line in th_shomate.mw takes care

of this, and here is a very important point: packages must be saved in a directory, or

library, that must be known by the calling procedure. The name of this directory can

be anything you want. I have chosen c:/thcalc, so you should create a directory

(or “folder”) with this name before attempting to run any of the Maple procedures

from the website. When you download th_shomate.mw you can put it in this or any

directory you wish. Then open the file and execute it (you can do this in the “Edit”

pull down menu). Execution of the file places the th_shomate executable package

in c:/thcalc, and the package is ready to be used by other procedures.

The file th_template_1.mw contains a number of commands and procedures that

are used by many of the thermodynamic calculation worksheets that we will discuss in

this and subsequent chapters. It is important to understand what each of these commands

does. I describe them in the order in which they appear in the Maple worksheet.

libname := ... tells Maple the location of the library where the packages are

stored. It must match the name used when the package was created (see above).

with (th_shomate) tells Maple to load the th_shomate package.

with (spread) tells Maple to load a standard package that enables spreadsheet

functions.

RefStateData := CreateSpreadsheet() creates a spreadsheet named

RefStateData, that is used to store standard state thermodynamic properties at

the reference conditions, 298.15 K and 1 bar. Each row in the spreadsheet contains

data for one chemical species. For now we will only use the first nine columns, but

more data columns will be occupied in later chapters. The first column contains

correlative numbers that will serve to identify the species (more on this below).

The second column stores the name of the species. Each additional column after the

second one corresponds to a thermodynamic property, generally at 298K and 1 bar.

Successive columns store the values of 

1,298

, S

1,298



1,298

(the functions S

and G will be defined in later chapters) and the a

coefficients of the C

function.

Enthalpy and Gibbs free energy must be entered in kJoules, all other quantities

in Joules. The data can be entered directly in Maple, but I find it easier to store

the data (e.g. from Holland & Powell, 1998, or some other data base) in a regu-

lar spreadsheet such as QuattroPro and then copy whatever is needed to Maple.

Once you have all of the data that you will need for the calculations in the Maple

spreadsheet it is best to save it in Tab Delimited format by right-clicking anywhere

in the spreadsheet and then Export Data. It can then be imported directly into

Maple by performing the inverse operation. The data for this example are stored

in tab-delimited format in a file named spgrt.

45 1.12 Some applications of the First Law

The next block of statements is a procedure called load that loads a one-dimensional

array with the thermodynamic data for a species, identified by the number in the

first column of the spreadsheet. This is used by other procedures, for example the

following.

deltareax, that calculates 

1,298

, 

1,298

, 

1,298

and the 

coefficients

that are used to integrate the 

equation (see equation (1.100)). These are the

differences in thermodynamic properties at the 298.15 K reference temperature

(entropy and Gibbs free energy will be covered in later chapters).

Finally, procedure delH calculates the enthalpy of reaction at the temperature of

interest and 1 bar, 

1,T

(see equation (1.100)) simply by calling HT_sh in the

th_shomate package with the thermodynamic properties specific to the reaction

of interest.

All that remains now is to tell Maple what is the reaction that we want to calculate, and

where to find the data. We do this by creating a table with two columns. The first column

contains the stoichiometric coefficient of each chemical species in the reaction, positive

if it is a product, negative for a reactant. The second column contains the row number

that identifies the species in the spreadsheet that contains the data (i.e. the number in

the first column of the spreadsheet). The example given in th_template_1.mw is

for the reaction:

MgAl

+2Mg

⇒Mg

SiO

+Mg

. (1.70)

Suppose we enter the properties for spinel in row 1, enstatite in row 2, forsterite in row

3 and pyrope in row 4. We name the table that identifies this reaction spgrt, and the

four entries in the table are: [1,3], [1,4], [-1,1], [-2,1]. All we need to do

now to calculate the enthalpy change for this reaction at any temperature we wish is to

run the procedure delH, providing it with the name of the reaction and the temperature

(see th_template_1.mw).

An isobaric discontinuous phase transition is associated with expansion work, as density

changes across discontinuous phase transition. Because pressure is constant, the magnitude

of this expansion work, W

transition

, is given by:

transition

=P V

transition

. (1.70)

The molar volume of liquid H

O at 1 bar and 373 K is 1.88 J bar

−1

mol

−1

(convert this

number to density in kg m

−3

, so as to get a feeling for the relationship between density and

molar volume). The molar volume of H

O vapor at the same pressure and temperature is

3.05 × 10

J bar

−1

mol

−1

. The expansion work associated with the phase transition is thus

∼ 3kJmol

−1

. From equation (1.61) we have:

E

vap

=H

vap

−W

transition

. (1.71)

The enthalpy of vaporization of H

O is 40.7 kJ mol

−1

. Thus, when H

O boils its internal

energy increases by ∼37.7 kJ mol

−1

. This is about one order of magnitude greater than the

expansion work. The increase in internal energy reflects the fact that molecules in the gas

state carry significantly more translational kinetic energy than in the liquid state (Section

1.14). The increase in internal energy (i.e. molecular kinetic energy) is a microscopic

46 Energy in planetary processes

contribution to the enthalpy of vaporization, that is distinct from the macroscopic expansion

work that is performed against the pressure exerted by the environment.

Let us define a variable T

pot

= H

vap

with dimension of temperature. We can

call this variable a potential temperature, because it represents the temperature difference

that would be caused by full conversion of latent heat to sensible heat. The heat capacities

of liquid water and water vapor at temperatures in the neighborhood of the boiling point

are C

P,liquid

≈ 75.9 J K

−1

mol

−1

and C

P,gas

≈ 37.4 J K

−1

mol

−1

. Thus, for vaporization

of water T

pot

is of order 100–1000 K. This means that the first and last terms on the

right-hand side of equation (1.67) are negligible compared to the thermal effect of the

phase transition. Water vapor in the terrestrial atmosphere stores a large amount of thermal

energy, and is a major factor in driving the planet’s weather patterns (more on this in

Chapter 4). Enthalpies of crystallization are generally of smaller magnitude than enthalpies

of condensation, with T

pot

of order 10–100 K (Chapter 10). These values are nevertheless

large enough to have important effects on the energetics of planetary systems undergoing

melting or crystallization. For instance, enthalpy of crystallization of metallic Fe is almost

certainly the immediate source of heat that drives convection in the Earth’s core, and may

also drive convection in large silicate magma chambers (as an aside, I see no reason why a

partially molten metallic planetary core cannot be called a magma chamber, except perhaps

for tradition). Conversely, the large energy requirement of melting plays an important role

in the origin of planetary magmas (Chapter 10).

1.12.2 Adiabatic expansion of gases

The destructive power of pyroclastic eruptions, or of any conventional (i.e., chemical)

explosion, results from fast expansion of a gas driven by conversion of internal energy of

the gas to expansion work. Although in reality the very fast rate of expansion means that

the process is not an equilibrium one, we can at least get an idea of the magnitude of the

energy involved by treating the problem as a reversible adiabatic expansion. The First Law

of Thermodynamics for an adiabatic process (dQ = 0) is:

dE =−PdV . (1.72)

We equate the energy liberated by a pyroclastic eruption, W

pyr oclastic,

to the expansion

work of the volcanic gas. Because volume is not constant during this process we cannot

make the substitution: dE = C

dT, as in general internal energy is a function of volume as

well as temperature. We need to consider the total change in internal energy as a function of

the partial derivatives of E relative to the variables that we choose as independent variables

(Box 1.3). Choosing temperature and volume as the independent variables:

dE =



∂E

∂T



dT +



∂E

∂V



dV =C

dT +



∂E

∂V



dV. (1.73)

The energy liberated by a pyroclastic eruption is then given by:

pyroclastic

=PdV =−





dT +





∂E

∂V





, (1.74)

where T

, V

are the eruption temperature and the molar volume of the gas at that tem-

perature, and T

, V

are atmospheric temperature and the molar volume of the gas at that

temperature. In order to solve the integrals we need explicit expressions for C

= C

(T )

and for E =E(V ). The problem is simplified enormously if we assume that volcanic gases

47 1.12 Some applications of the First Law

behave as ideal gases, because for an ideal gas C

is constant and internal energy is a

function of temperature only, i.e. (∂E/∂V)

=0. We derive these properties of ideal gases

in Section 1.14.1. For an ideal gas, then, but only for an ideal gas:

pyroclastic

=−C

(

−T

)

. (1.75)

Because eruption temperature is greater than atmospheric temperature, W

pyroclastic

> 0.

According to our sign convention a positive value means that the system (expanding volcanic

gas) performs work on its environment (hapless bystanders). Typical eruption temperatures

for silicic magmas are ∼850

◦

C, and we can take a typical atmospheric temperature to

be 15

◦

C. The major component of volcanic gases is H

O, which consists of polyatomic

molecules. In Section 1.14 we shall see that for such gases C

= 3R. Substituting these

numerical values we get an energy release per mol of erupted volcanic gas of:

pyroclastic

≈2.08 ×10

J mol

−1

. (1.76)

As an example, the 1980 eruption of Mt. St. Helen’s extruded ∼ 1km

of tephra. Assuming

a pre-eruptive H

O content of 3 wt% and a magma density of 2700 kg m

−3

we see that some

4.5 × 10

mols of H

O were erupted. The estimated energy yield of the Mt. St. Helen’s

eruption is ∼9.36 × 10

The yield of nuclear weapons is measured in kilotons (kt), where 1 kt = 4.184 × 10

The 1980 Mt. St. Helen’s eruption was thus equivalent (less the high energy electromagnetic

radiation and radioactive fallout) to a 22.4 megaton nuclear device – or about fifteen hundred

times larger than the bomb that destroyed Hiroshima. This yield is also larger than that of

the largest thermonuclear weapon ever detonated by the USA (the ∼18 megaton Bravo

test), but less than half the size of the largest man made thermonuclear explosion: the 50

megaton Soviet “Tsar Bomba” (I don’t cease to be amazed by the fact that the pilots who

dropped this bomb, and the crews flying observer aircraft nearby, managed to survive).

You may have noticed that we calculated the magnitude of the mechanical energy liber-

ated by a pyroclastic eruption without actually integrating the expansion work term. Two

conditions made this possible: (i) our assumption that the expansion is adiabatic, and (ii) our

knowledge (or reasonable assumption) of the initial and final temperatures of the process.

Lacking this last piece of information, we would have had to derive the magnitude of the

expansion work by some other means, independent of temperature.

1.12.3 Frictional heating in faults and shear zones

When displacement occurs along a brittle fault or a ductile shear zone the frictional force

that opposes motion performs work. Frictional forces are dissipative, which means that the

work that they perform is converted to heat. What is the maximum temperature increase at

a fault as a result of this energy dissipation? Let us assume that frictional heat is distributed

uniformly throughout a finite volume of width z, bracketing the fault or shear zone (Fig. 1.8).

The actual magnitude of z depends on the rate of dissipation of mechanical energy (i.e. the

strain rate) – more on this in Chapter 3. We consider the volume of rock enclosed within

width z to be an adiabatic system. The change in internal energy of the heated rock volume

is then given by (compare with equation (1.72)):

dE =−PdV −

(

−dW

)

. (1.77)

In this equation we distinguish between expansion work arising from the change in tem-

perature of the rocks, and work performed by the frictional force. The work performed by