Chapter 5
The Second Law

Thermodynamics is a fancy word that scientists use to make the study of heat sound important.
Tom Siegfried

Off[General::spell]

5.0 Introduction

The Industrial Revolution in the nineteenth century was driven by the invention of the steam engine. Previously, power was generated by natural sources like wind, water and draft animals. Steam engines, powered by burning fuel to produce steam from water, provided reliable, portable sources of power for manufacturing and transportation. The Second Law of Thermodynamics arose out of investigations for improving the efficiency of steam engines.

5.1 Heat Engines

Steam engines work on the principle that boiling water produces an expanding gas (steam) that produces work that can be converted into the mechanical work of locomotion.  Of course, the ultimate source of power of a steam engine is the chemical combustion of fuel to produce the heat to boil the water. Internal combustion engines, turbine engines, jet engines and rocket engines involve the same principles and the general term heat engine is used for such systems. A complete analysis of heat engines thus combines thermochemistry and physical energy conversion.

Recall that positive work increases the energy of a system. Recognizing steam or combustion products as gases, one sees that at constant pressure the work done by an expanding gas, which is equivalent to the negative of the work done on an expanding gas, produces mechanical work in the amount

= = - = -( -∫ P dV) > 0 for P > 0 and ΔV > 0

When the gas does work, the energy of the gas decreases. Energy is transferred to the engine by the First Law, and the gas cools. We can visualize the flow of heat into work as follows:

Note that the heat taken from the hotter resevoir (the "boiler" in a steam engine) is not totally converted into work, but some passes to a colder reservoir (the "condensor" in a steam engine). This necessity is one form of the Second Law of Thermodynamics, observed experimentally and studied theoretically in 1834 by Sadi Carnot (1796-1836). But this is only the begining of the story. Working engines are cyclic and return to some original state, so

= + = 0, or =

where all the contributions to the total quantities must be considered. A complete analysis requires consideration of all the changes taking place during the cycle, not just the expansion. Further, the conditions of the energy transfers affect the amounts of heat and work involved. For example, a general work term is computed from an integral over the path of the pressure-volume change; the constant pressure expression above being a special case. Carnot investigated the efficiency of conversion of heat into work by considering an ideal engine using an ideal gas as the working substance. A cycle of the engine consists of a path with four steps, an isothermal expansion at the higher temperature , followed by an adiabatic expansion (q = 0), followed by an isothermal compression at the lower temparature , followed by an adiabatic compression (q=0) returning to the original state.

In terms of incremental changes along the path, the First Law applied to an ideal gas gives

= dq + ⇒ dT = dq - PdV = dq - dV

Collecting terms in temperature and integrating (summing) over the cycle gives

∮dT = ∮dT - ∮ dV

where ∲  indicates an integral over the cyclic path (cyclic integral). Assuming , n and R are constant, we have

Since the logarithm of the recriprocal of an argument is the negative of the logarithm of the argument (why?), all the terms not involving heat cancel:

The thermodynamic efficiency of the ideal engine, ε, is the ratio of the work done by the engine to the heat taken in. Since we have seen that

= + = 0, or = = -(- ),

≡ = = 1 - = 1 - ⩽ 1

Note that a maximum efficiency is achieved only at a cold reservoir temperature of absolute zero. This is sometimes referred to as the Third Law of Thermodynamics.

If a human was considered a heat engine operating between body temperature (37 °C) and room temperature (25 °C), the efficiency would be only

= 1 - = .039

or 3.9%. Biochemical reactions convert chemical (carbohydrate) energy into (ATP) bond energy, producing an energy conversion efficiency ten times as great as straight combustion would produce.

Incidentally, a heat engine running in reverse, known as a heat pump has a theoretical efficiency (or coefficient of performance, COP)

≡ = ⩾ 1

Other engine cycles have been considered, such as the Otto cycle, which approximates the performance of a gasoline combustion engine, consisting of an adiabatic compression followed by an isochoric compression, followed by an adiabatic expansion and then an isochoric expansion. The efficiency can be shown to be

= 1 - ⩽ 1

where the ratio of heat capacities γ ≡ , and the ratio is known as the compression ratio.

5.2 Entropy

We have seen that for a Carnot engine the quantity is conserved over each cycle of the engine. This suggests some kind of state function, which became called entropy. Generalizing this quantity, we define the change in entropy as

ΔS ≡ ∫

such that.

= 0

This is an alternate statement of the Second Law of Thermodynamics.

5.2.1 Adiabatic Entropy

Entropy changes depend on path and may be calculated for a variety of processes. For an adiabatic process

= ∫ = 0

where the subscript reminds us this is an adiabatic process.

5.2.2 Isothermal Entropy

For an isothermal process

= ∫ =

For example, a phase change (transition) takes place at constant temperature and

= ⇒ =

5.2.3 Heating Entropy

A heating (or cooling) process taking place between temperature and has

dq = C dT ⇒ = ∫ = ∫ ≃ C

5.2.4 Mixing Entropy

When substances are throughly mixed, entropy increases. This may be illustrated for gases by noting that when samples of different gases A, B, ... are mixed by removing partitions between the gases at constant temperature, each gas expands to fill to total final volume.

= + + ... =  

By Dalton's Law of Partial Volumes,

where is the mol fraction of species i. Thus

=  - > 0

since and R are positive quantities and is a fraction less than one.

5.2.5 Third Law Entropy

Experimental values of entropies may be obtained by heating a substance and measuring the heat capacity (as a function of temperature) and the enthalpies of any transitions. Entropies at very low temperature are obtained by fitting the heat capacity to a cubic in temperature. For example, a simple substance which is a gas at room temperature has a standard entropy

= + + + + + +

where the subscripts s, l and g refer to the solid, liquid and gas phases and m and b refer to melting and boiling, respectively. Other phase transitions insert terms at the transition temperatures.

The term refers to any residual entropy at absolute zero due to lack of complete order in the crystalline phase. For a symmetrical molecule like , = 0, but for an unsymmetrical molecule like CO, = nR ln(2) due to the two random orientations the molecules may assume in the solid phase. Molecules having m equivalent orientations may have a residual entropy of up to nR ln(m) if the various orientations are completely mixed.

5.2.6 Reaction Entropy

Since entropy is a state function, the entropy of chemical reactions may be computed from the contributions of each species. For the abstract reaction R = P, or P - R = 0,

=   > 0

where is the number of mols of species i in the balanced chemical reaction (stoichiometric coefficient).

Given the entropy of reaction at a given temperature, the entropy of reaction can be calculated at other temperatures using the heating entropy formula

- = ≃

where the approximation assumes the heat capacities of the various species do not depend on temperature. For careful work, the temperature dependence of heat capacity must be taken into account.

5.3 Thermodynamic Potentials

In 1875 Josiah Willard Gibbs published a series of articles on thermodynamics that generalized and summarized most theoretical and practical aspects of the subject. His starting point is the realization that thermodynamic quantities like energy and entropy are analogous to mechanical potential energy. Processes such as falling objects in the gravitational field have analogues in thermodynamic process as well.

According to Gibbs, there is just one thermodynamic quantity of interest, the thermodynamic potential, from which all other thermodynamic information may be derived. This is similar to the claim that the wave function of quantum mechanics is the key description of atoms and molecules, from which all properties of interest can be obtained. In thermodynamics, however, there are a number of choices of independent variables upon which thermodynamic potentials depend. This results in a variety of thermodynamic potentials depending on the set of variables that are "natural" to their potentials. The basic assumption of thermodynamics in this scheme is the existence of a thermodynamic potential which depends on a particular set of variables. Historically, energy was recognized first as a fundamental quantity of matter. According to Gibbs, this potential has for its natural variables entropy, volume and amount of matter, measured as number of molecules:

E(S,V,)

where the arrow indicates a set of molecules with numbers {, ... } for the various molecular species in the system of interest. This relation may be termed the Thermodynamic Equation of State for the system, analogous to a mechanical equation of state that interrelates mechanical variables such as pressure, volume, temperature and mols. For the Ideal Gas, if we choose n, T and V as the independent variables, for example, the mechanical equation of state is P(V,T,n) = nRT/V. The task of thermodynamics is to provide the means of obtaining thermodynamic quantities and relations of interest in a given set of independent variables from the thermodyanmic equation of state.

Energy, entropy, volume and number of molecules are all extensive variables, that is, they are proportional to the amount of matter in the system. There is a corresponding set of intensive variables which are independent of the amount. They may be defined in terms of derivatives of the the extensive variables as follows. Taylor's Theorem states that the differential of a given function of some number of variables f() is given in terms of a sum of partial derivative quantities as:

df() = d≡ d  = · d  

where the tensor notation for partial derivative using a single partial symbol is used in the second form. Applying this to the energy state function, we have

dE(S,V, =   d = dS +   dV +   d

Since in general there are mechanical pressure-volume work terms and chemical work terms for chemical systems,

dE(S,V, =   q + w = TdS -  PdV +   d

where is called the chemical potential of the ith chemical species. Comparing coefficients of each variable (S, V and ), we obtain the thermodynamic definitions of temperature, pressure and chemical potential:

T ≡   P ≡ - ,   ≡  

The new (intensive) variables are said to to conjugate to original (extensive) variables from which they are derived via derivatives. For the energy representation, the conjugate pairs are (S,T), (V,P) and (.

Historically, a number of other chemical potentials had been explored by thermodynamicists. Gibbs recognized these as resulting from transformation of the independent (natural) variables by way of a procedure discovered by Legendre, called a Legendre transformation.  Given a function of independent variable x, its derivative ("rise over run")

= ≡  y

where g is the intercept on the f axis of a line tangent to f(x) at value x, and a new variable y is defined to be the derivative of the old function in terms of the old variables. In effect, we have replaced the equation for a straight line in terms of pairs of independent and dependent values, by an equivalent expression in terms of independent variable values and intercepts of slopes evaluated at the indepedent variable values. Rearanging and generalizing to several independent variables give the transformation

= - •

The prescription for creating a function of new variable, defined as derivatives of a given function with respect to its variables, from old variables is easy to state. Create the new function (g) of new variables () by subtracting the product of the new and old variables from the old function (f) of its old variables (), where the new independent variables are derivativies of the old function with respect to its old variables.

Applying the Legendre transformation to energy E(S,V,to replace, say the old variable volume (V) with its conjugate variable pressure P generates a new thermodynamic potential as a function of S, P and . This thermodynamic function had been identified by Clausius, who called it the enthalpy H:

≡ - V( = + V

(Note that the product VP is equivalent to the product written in the customary order PV.)

The differential of the enthalpy given in terms of its natural variables equated to the differential of the enthalpy from its definition gives

dH(S,P, =   d = dS +   dP +   d = dE + PdV + VdP = TdS + VdP + d

since the term -PdV from dE (see its expression above) cancels the term PdV from dH. Equating coefficients of like differential terms gives the following intensive variables in enthalpy space:

T ≡   V ≡ ,   ≡  

It is interesting that the historical quantity enthalpy, related to convenient measurement processes, has a strict mathematical relationship to energy. In this sense, the only difference between energy and enthalpy as thermodynamic potentials is the independent variables on which they depend.

Another thermodynamic potential can be obtained as the Legendre transformation of energy which replaces the old variable entropy (S) with its conjugate variable temperature T. This generates a new thermodynamic potential as a function of T, V and , which had been identified by Helmholtz and named the Helmholtz potential A:

≡ - S() = - TS

Considering the Taylor expansion of Helmholtz potential gives

dA(T,V, =   d = dT +   dV +   d = dE -TdS - SdT = -SdT -PdV + d

since the term TdS from dE cancels the term -TdS from dA. In this space we have the intensive variables

S ≡-   P ≡ - ,   ≡  

Finally, Gibbs proposed a thermodynamic potential, obtained as the Legendre transformation of energy which replaces the old variables volume and entropy (V, S) with their conjugate variables pressure and temperature (P, T). This generates a new thermodynamic potential as a function of T, P and , named the Gibbs potential G (sometimes called the Gibbs "free" energy):

≡   - ) - ) = - TS + PV = H - TS = A + PV

Considering the Taylor expansion of Gibbs potential gives

dG(T,P, =   d = dT + dP + d = dE -TdS - SdT +PdV +VdP  = -SdT + VdP + d

since the term TdS from dE cancels the term -TdS from dG. In this space we have the intensive variables

S ≡ -   V ≡ ,    ≡

5.4 Partial Derivative Relations

For convenience or necessity, it may be useful to express thermodynamic quantites in terms of some given set of independent variables. Mathematical identities between partial derivatives provide a mechanism for expressing thermodynamic quantities in terms of a given set of variables. The fundamental identities for partial derivatives were given by Leonhard Euler. Given some function of three independent variables, ), an inverse relation between any two variables exists, called Euler's inverse relation:

=         

From the Taylor expansion of the differential of one variable in terms of the others, we have

= +   = 0 ⇒ = - )(

Using the inverse relation gives Euler's cyclic relation

))( = -1     

Finally, for well-behaved functions, the order of differentiation is immaterial:

where is Euler's reciprocity relation.

To illustrate the usefulness of Euler's relations, consider a non-reactive thermodynamic system consisting of a single substance ). If an equation of state is known for substnace, it will relate three state variables, reducing the number of independent state variables to two. A thermodynamic quantity ) will be a function of only two independent variables. This means there will be only three independent second derivatives of the quantity, , , and ( = by ). This permits the choice of a convenient set of three derivative quantities in which all other thermodynamic quantities can be expressed.

We can choose convenient measureable independent variables, such as temperature, volume and pressure and convenient derivative quantities, such as heat capacity, thermal expansion coefficient and isothermal compressibility. Other thermodynamic quantities may then be expressed in terms of these accessible quantities.

≡ T

≡ T

≡

≡

≡

The subscripts remind us of the set of independent variables we have chosen.

For an ideal gas, we have the equations of state PV = nRT and E = nRT. The derivative quantites for the ideal gas are

Typical values for solids and gases are

In 1871 Clerk Maxwell wrote a treatise on thermodynamics in which he presented the consequences of  immaterial order of differentiation of thermodynamic potentials (Euler recriprocity), called Maxwell's Equations.

G(T,P) ⇒ ⇒ =   =     

A(T,V) ⇒ ⇒ = - =     

H(S,P) ⇒ ⇒ = - =     

E(S,V) ⇒ ⇒ = - -    

where ideal gas values are presented as a special case.

5.5 Partial Derivative Applications

5.5.1 Energy

The natural independent variables for the energy of a substance are entropy and volume. But suppose we are interested in the energy change along an isobaric followed by an isothermal path. The derivative relationships provide the answer. Given E(T,P),
since

dE = TdS - PdV ⇒ = T - P = - αPV

dE = TdS - PdV ⇒ = T - P =  -αTV + κPV

Therefore, by Taylor's expansion

dE = +

and

ΔE(T,P) = ∫ ( - αPV)dT + ∫(-αTV + κPV)dP

all measureable quantities as functions of measureable variables T and P.

Similarly, if volume and temperature are chosen as independent variables, we can show that

dE = TdS - PdV ⇒ = T - P = - P

dE = TdS - PdV ⇒ = T=  

( = 0). So

dE =   +

and

ΔE(V,T) = ∫(- P)dV + ∫ dT

= - P shows how energy depends on volume and relates how the internal pressure (lhs) depends on the external pressure on the rhs and a term that can be interpreted as attractive pressure due to intermolecular interactions. For an ideal gas, the right hand side evaluates to zero, indicating no intermolecular interactions. This demonstates that the energy of an ideal gas depends only on temperature, = E(T) = nRT (the latter equality derived from the Equipartition Theorem). For a van der Waals gas, = , a measure of the intermolecular attractions.

5.5.2 Entropy

It is not possible to measure entropy directly, but changes in entropy can be determined for special paths. Similar to the derivations for energy, we can determine the entropy change along a temperature, pressure path as follows

dS = + = dT - αVdP

For the ideal gas, this gives

ΔS(T,P) = dT - dP = - nR

5.5.3 Heat Capacity

Derivative relations may be used to derive a useful general relationship between heat capacity at constant pressure and heat capacity at constant volume.

5.6 Thermodynamic Equilibrium

5.6.1 General Equilibrium

When an object falls as far as it can in the gravitational field and stops, it is said to have reached equilibrium. This is the state of minimum potential energy. Consider a child on a swing that has come to rest. The potential is a relative minimum, constrained by the ropes or chain of the swing. Thermodynamic potentials have an analogous property. The minimum or equilibrium thermodynamic energy E occurs at constant entropy S (isentropic), constant volume V (isochoric), non-reactive (constant amount) conditions. Similarly, minimum enthalpy H occurs at constant S, P and , minimum Helmholtz potential A at constant T, V and , and minimum Gibbs potential G at constant T, P and . One can visualize thermodynamic potential surfaces shaped like bowls with a system like a ball rolling in the bowl coming to equilibrium at the bottom of the bowl.

5.6.1 Chemical Equilibrium

Which thermodynamic potential is appropriate for a given process depends on the variables that can be measured or controlled. Gibbs pointed out that chemical reactions taking place in beakers open to and controled by the atmosphere isobaric and also are essentially isothermic for solutions having enough water to absorb the heat of the chemical reaction. Thus, the Gibbs potential is the appropriate potential for chemists to use, and equilibrium occurs at the minimum of G() for constant T and P.

For a general chemical reaction R = P, or P - R = 0, since G is a state function,

≡ - = =

where is the number of mols of the ith species (stoichiometric coefficients in the balanced reaction). For an ideal gas,

= = =

Integrating,

- =

where the superscript refers to a standard state of one atmosphere, = 1.

Applying this to each species in the chemical reaction, we have

- =   = RT   = RT ≡ RT ln(Q)

Generalizing from partial pressures to concentrations measured in any convenient units, Q is called the reaction quotient of the reaction. At equilibrium, Q becomes a constant, called the equilibrium constant, K.

Also at equilibrium, , and

= -RT ln(K)

Since at constant temperature,

= - T

the equilibrium "constant" has the temperature dependence

K(T) = =

This relationship we will call the van't Hoff equation.

Summarizing,

= RT ln()

from which we see that at the standard state, Q → 1 ⇒ = -RT ln(K), and at equilibrium = 0 ⇒ Q → K.

5.6.2 Temperature Dependence of Chemical Equilibrium

Since G ≡ H - TS,

S ≡ - = = - + = -=

giving the Gibbs-Helmholtz equation

=

Applying the Gibbs-Helmholtz relation to each specied in a reaction,

= =

This says that a plot of ln(K) vs. 1/T is a straight line having slope equal to . If is not constant, the line has curvature and the slope at any temperature equals .

Integrating both sides over temperature for constant gives

= - )

Rearranging gives a convenient formula for computing the value of the equilibrium constant at given the value at .

Van't Hoff Parameters by Linear Regression
Van't Hoff equation: K(T) = exp(ΔS/R - ΔH/RT)

Parameters

<<Miscellaneous`Units`
<<Miscellaneous`PhysicalConstants`
R = MolarGasConstant
Rvalue = R Kelvin Mole/Joule

     8.3144 Joule
     ------------
     Kelvin Mole

     8.3144

Data:   {T, K(T)} pairs

CelsiusMmData = {{20,17.54}, {30,31.82}, {40,55.32}, {50,92.51}, {60,149.38}, {70,233.7}}

     {{20, 17.54}, {30, 31.82}, {40, 55.32}, {50, 92.51}, {60, 149.38}, {70, 233.7}}

KelvinMmData = Map[{ConvertTemperature[#[[1]], Celsius, Kelvin],#[[2]]}&, CelsiusMmData]

     {{293.15, 17.54}, {303.15, 31.82}, {313.15, 55.32}, {323.15, 92.51}, {333.15, 149.38}, 
      
       {343.15, 233.7}}

Two point fit of ln[K(T2)/K(T1)] = ΔH*ΛT/(R*T1*T2)

Print["Ti = ", T1 = KelvinMmData[[1]][[1]] Kelvin]
Print["Tf = ", T2 = KelvinMmData[[Length[KelvinMmData]]][[1]] Kelvin]
Print["K(Ti) = ", K1 = KelvinMmData[[1]][[2]]]
Print["K(Tf) = ", K2 = KelvinMmData[[Length[KelvinMmData]]][[2]]]

Print["ΔH = R*Ti*Tf*Ln[K(Tf)/K(Ti)]/(Tf-Ti) = ", R*T1*T2*Log[K2/K1]/(T2-T1)]

Ti = 293.15 Kelvin
Tf = 343.15 Kelvin
K(Ti) = 17.54
K(Tf) = 233.7
                                       43317.1 Joule
ΔH = R*Ti*Tf*Ln[K(Tf)/K(Ti)]/(Tf-Ti) = -------------
                                           Mole

Least squares linear fit y = b + mx for ln[K] = ΔS/R -ΔH/RT

LinearizedData = Map[{1/(#[[1]]),N[Log[#[[2]]]]}&, KelvinMmData]

     {{0.00341122, 2.86448}, {0.0032987, 3.4601}, {0.00319336, 4.01313}, {0.00309454, 4.52732}, 
      
       {0.00300165, 5.00649}, {0.00291418, 5.45404}}

Print["Regression fit: ln(K) = ", lnK = Fit[LinearizedData, {1,x}, x], ", where x = 1/T"]

Regression fit: ln(K) = 20.6449 - 5210.18 x, where x = 1/T

Extract Parameters

b = Coefficient[lnK, x, 0];
m = Coefficient[lnK, x, 1];
Print["ΔH = ", -R*m Kelvin]
Print["ΔS = ",  R*b]

     43319.5 Joule
ΔH = -------------
         Mole
     171.65 Joule
ΔS = ------------
     Kelvin Mole

Plot Data and Fit

lp = ListPlot[LinearizedData, Prolog -> PointSize[.03], DisplayFunction -> Identity];
fp = Plot[lnK, {x, LinearizedData[[1]][[1]], LinearizedData[[Length[LinearizedData]]][[1]]}, DisplayFunction -> Identity];
Show[lp, fp, PlotLabel -> "     Van'Hoff Plot", AxesLabel -> {"1/T", "ln(K)"},
             DisplayFunction -> $DisplayFunction];

Least squares fit to the Van't Hoff equation

Needs["Statistics`NonlinearFit`"]

Print["van't Hoff fit: K = ", 
VH = NonlinearFit[KelvinMmData, Exp[ΔS/Rvalue - ΔH/(Rvalue*t)], t, {ΔS,ΔH}, ShowProgress->True]]

Iteration:1  ChiSquared:88567.9  Parameters:{1., 1.}
                                    7
Iteration:2  ChiSquared:2.62029 × 10   Parameters:{393.701, 110726.}
                                    6
Iteration:3  ChiSquared:3.33751 × 10   Parameters:{389.672, 112033.}
Iteration:4  ChiSquared:380880.  Parameters:{383.836, 112505.}
Iteration:5  ChiSquared:38625.  Parameters:{345.635, 101390.}
Iteration:6  ChiSquared:1505.37  Parameters:{198.379, 52107.4}
Iteration:7  ChiSquared:5.58351  Parameters:{170.516, 42927.1}
Iteration:8  ChiSquared:0.639649  Parameters:{170.384, 42903.}
Iteration:9  ChiSquared:0.639582  Parameters:{170.383, 42902.9}
Iteration:10  ChiSquared:0.639582  Parameters:{170.383, 42902.9}
Iteration:11  ChiSquared:0.639582  Parameters:{170.383, 42902.9}
                     20.4925 - 5160.07/t
van't Hoff fit: K = E

Extract Parameters

Print["ΔH = ", -VH[[2]][[2]]*t Kelvin*R]
Print["ΔS = ", VH[[2]][[1]]*R]

     42902.9 Joule
ΔH = -------------
         Mole
     170.383 Joule
ΔS = -------------
      Kelvin Mole

Plot Data and Fit

y[x_] := Log[VH]/.t->1/x

lp = ListPlot[LinearizedData, Prolog -> PointSize[.03], DisplayFunction -> Identity];
fp = Plot[y[x], {x, 1/KelvinMmData[[1]][[1]], 1/KelvinMmData[[Length[KelvinMmData]]][[1]]},           DisplayFunction -> Identity];
Show[lp, fp, PlotLabel -> "     Van'Hoff Plot", AxesLabel -> {"1/T", "ln(K)"},
             DisplayFunction -> $DisplayFunction];

5.6.3 Thermodynamic Tables

From the previous discussion, we can see that thermodynamic values of can be obtained from measurement of equilibrium concentrations (K), and that knowledge of K as a function of temperature gives . Combining these values produces values of = . In this way, values of , , can be obtained for various substances, which, together with Third Law entropies and heat capacities C provide entries in thermodyanmic tables for substances. From the tables, standard thermodynamic potentials for any reaction involving entries in the tables can be computed. From van't Hoff''s equation, values can be computed at arbitrary temperatures.

Exercises

  1. Show that for an isentropic process (ΔS = 0), an ideal gas obeys the law = .

  2.