## Thermodynamics

Thermodynamics is the study of energy changes accompanying physical and chemical processes. The energy changes associated with chemical reactions are of considerable importance and will be briefly discussed to indicate the relationships of most interest in environmental engineering and science. First, it is necessary to review the relationship between heat and work.

### Heat and Work

Heat and work are related forms of energy. Heat energy can be converted into work,-and work can be converted into heat energy. Steam engines and frictional losses are examples of each conversion.

Heat is that form of energy which passes from one body to another solely as a result of a difference in temperature. On the molecular scale it is known that the temperature of a substance is related to the average translational energy of the molecules, and that flow of heat results from transfer of this molecular energy. The basic unit of heat is the calorie, the heat required to raise the temperature of one gram of water one degree Celsius. In engineering it is common to measure heat in British thermal units (Btu), which is the heat required to raise one pound of water one degree Fahrenheit. One Btu is equivalent to 252 calories (calj or 1054 joules (J).

The specific heat of a substance is the heat required to raise one gram of the material one degree Celsius, or where C = specific heat q — heat added, cal or J M = weight of material, g AT = rise in temperature of material, °C

For water the specific heat is 1.000 cal or 4.184 J per gram-degree Celsius only at 15°C, but varies from this value by less than 1 percent over the entire range from 0 to 100°C. Therefore, the assumption of a constant specific heat over a limited range in temperature is frequently a good one. Two values for specific heat are frequently reported: C„ is the specific heat at constant volume, and Cp is the specific heat at constant pressure. For liquids and solids there is little difference between these two values. For gases, however, Cp is greater than C„ because of the extra heat energy required to expand a gas against a constant pressure.

The heat of fusion is the heat required to melt a substance at its normal melting temperature. The heat of vaporization is the heat required to evaporate the substance at its normal boiling point. The values for water are 333 and 2258 J per gram, respectively. Environmental engineers and scientists encounter a wide variety of heating, evaporation, drying, and incineration problems involving a knowledge of specific heat, heat of fusion, and heat of vaporization. In addition, many industrial processes involve evaporation or drying in which cooling water is needed. In essence, the heat of vaporization is transferred to the coolant during condensation, and the warmed water may become a thermal pollution problem in terms of the receiving body of water. This may become a matter of concern when water use by industry and the public increases.

Work in chemical systems usually involves work of expansion. The system may either do work on its suiroundings or have work done on itself. This depends upon whether the volume of the system is expanding or contracting. Work (dw) is normally measured in terms of force times distance, which for a closed system is equivalent to pressure (P) times the change in volume (dV):

Work is measured in foot-pounds or in joules (J). Since heat and work are both forms of energy, they can be equated. Thus, 1 cal of heat is equivalent to 4.184 J. In other units, 1 Btu of heat is equivalent to 778 ft-lb.

### Energy

The first law of thermodynamics states that energy can be neither created nor destroyed. In chemical systems the energy involved is most easily handled in terms of three quantities: the work that is performed, the heat that flows, and the energy stored in the system. The chemical system may range from the contents of a laboratory beaker to a full-scale, activated-sludge plant. According to the conservation-of-energy law, any heat or work that flows into or out of the system must result in a change in the total energy stored in the system. In equation form,

where AE = change in internal energy of system q = heat flowing into system w = work done by system

It is important to note that if heat is absorbed by the system, q has a positive value. If the system gives off heat, q has a negative value. Also, if the system does work on the surroundings, w has a positive value, but if the surroundings do work on the system, w has a negative value.

In chemical systems, work performed is usually expansion work as given by Eq. (3.2). When the system expands in volume, it does work on its surroundings and w is positive. When it contracts in volume, work is done on the system and w is negative. If the volume of the system remains constant, then no expansion work can be done and w equals zero. For this case,

where <j„is the heat absorbed in a constant-volume system. Thus, in a constant-volume system, the change in internal energy is just equal to the heat absorbed. Although this is convenient for constant-volume systems, most chemical systems of interest to environmental engineers and scientists are open to the atmosphere and so operate under constant pressure, rather than constant volume. For such systems, the concept of enthalpy has been developed.

Enthalpy

The enthalpy H of a system is defined as follows:

where E = internal energy of system P = pressure on system V = volume of system

Consider a constant-pressure system in which some chemical change has taken place, resulting in a change in internal energy. The heat absorbed at constant pressure is qp, and the work done by the system is given by an integration of Eq. (3.2). For constant temperature and pressure this integration gives w = P(V2 - V,), where yx is the initial volume of the system and V2 is then final volume after the change. The change in internal energy of such a system then becomes

AE = E2 — E\ = qp — w = qp — P(V2 - 7,) Rearranging, we obtain

The terms in parentheses are just equal to the final and initial enthalpy of the system, and thus

Thus, the quantity of heat absorbed by a system at constant temperature and pressure is equal to the change in system enthalpy. Chemical changes that are accompanied by the absorption of heat, making AH positive, are called endothermic reactions. Those accompanied by the evolution of heat, making AH negative, are called exothermic.

The total enthalpy (H) of a system would be difficult to measure. We are normally interested, however, only in the change in enthalpy and not in its absolute value. By developing a standard basis for comparison, it is possible to calculate the change in enthalpy or the heat of a given reaction from tabulated measurements from quite different reactions.

A convenient standard state for a substance may be taken as the stable state of the compound at 25°C and 1 atm pressure. For example, under these conditions, the standard state for oxygen is a gas, for mercury a liquid, and for sulfur it is rhombic crystals. By convention, the enthalpies of the chemical elements in this standard state are set equal to zero. The standard enthalpy of any compound is then the heat of the reaction by which it is formed from its elements; reactants and products all being in the standard state at 25°C and 1 atm.

For example,

H2(g) + |02(g) -> H20(Z) AH°f = -286,000 J CCi) + 02(g) CO2(g) AH°f = -394,000 J

The symbols in parentheses after each element or compound indicate the standard state of the elements or compounds. The superscript zero and subscript/on the enthalpy indicates a standard heat of formation with reactants and products at 1 atm and 25°C (298 K). Standard enthalpies for many compounds of interest are listed in Table 3.1. Many other values can be found in chemical handbooks such as those listed at the end of the chapter. An expanded listing is also given in App. A.

Standard enthalpy values can be used to determine the heat given off by a variety of reactions. In making such calculations it is very important to note the state

Table 3.11 Standard Free Energies and Enthalpies of Formation at 25"C

 Sub- ' stance ' St iti kj/inol . Ca2+ aq -553.6 CaC03 s -1128.76 CaF, s -1175.6 Ca(OH)2 s -896.76 CaS04 s -1322.0 ch4 S -50.79 CH3CH3 8 -32.89 CH3COOH aq -399.6 CH3COO~ aq -369,41 CH3CH2OH l -174,8 CSH!206 aq -917.22 Cls g 0 Clj aq 6.9 Cl- aq -131.3 CO, g -394.38 co2 aq -386.23 cor aq -527.80 p- aq -278.8 Fe2+ aq -78.87 Fe3+ aq -4.6 hco3" aq -586.85 h2co. aq -623

* aq = aqueous, s = solid, I = liquid Note: I kcal = 4.184 kJ

in which the products and reactants exist, as this can make a significant difference in the heat of reaction. The procedure used is to write a balanced equation for the reaction. The heat of the reaction is then equal to the sum of the standard enthalpies of the products minus the sum of the standard enthalpies of the reactants. It should be noted that standard enthalpies of formation are given in terms of kilo-joules per mole, and these values must be multiplied by the number of moles entering into the reaction.

Calculate both the net and gross heat of combustion of methane gas. :.;V.

\ . The gross heat is the. heat released if the water vapor formed upon combustion is • .condensed to fonn liquid water. ; :

CH„0?).+ 202(g) ~> C02(g) + 2H,O(0 , ' • ; ' V •. -74:6 : .; 0 -393.51 2(-285.80)1 "/-x'-

Jhe. net heat is the heat released if the water remains as a vapor or gas. This is the value of usual interest. ■

CH4G?) + 20,(g) ~> C02(g) + 2H20fe) -74.6 0 -393 51 2(-241 8) AH0 for combustion = (-393.51) + 2(-24I.8) - (-74.6)

.";;•■''■'/ =.—802^51 kJ/rnol of methane -¡i-X^

Since thevalues of Al^ for the combustion are negative, heat isgiven off.by the combustion of methane gas. ' ■■ v '" ^-..T"- ■

: Calculate the approximate rise in solution temperature if-1 liter of 1 N H2SO., is mixed | EXAMPLE 3.2 with 1 Iiter.of.1NNaOH. ■: -.-.A;/ X^Pf

/ . • Sulfuric acid is'a strong-Acid and sodium hydroxide is a strong base and so they are: . completely ionized in solution, as is theNa2S04 which is formed when the given solutions are mixed. Therefore, the enthalpies of the aqueous solutions are equal.to the sum of the . enthalpies for the individual ions. \•.'^•v;/;.'. •.;;. '.'■>■■■:

= 2AH&,,., + Afif^o..-, = 2(—240 1) + (-909.3) = -1389.5 The neutralization reaction that occurs is ; . '

: - AH0 for neutralization = (-285.80) + ¿(-1389.5) - ¿(-909.3)- (-470.1) . ; = - 55.80 kJ/mol = T13.34 kcal/mol

. The final volume of the mixed solution is 2 liters, which would weigh about 2000 g. Since this solution is mainly water, the specific heat would be'l cal/g-°C, and, from Eq. (3.1),

:'''.-■ 13,340 _ , ¿-.Qf, ' ■ ■ •.■:■;. ,s rjsM:^

Thus, if the. initial temperatures of the solutions were 25°C, the temperature would rise to a value somewhat over 31°C. ■

j A knowledge of heats of reaction, as well as heats of fusion, vaporization, and

| specific heats, is used for incineration, combustion, wet-air oxidation, heating of di-| gesters, chemical handling, and thermal pollution studies, among others.

| Entropy

A large part of chemistry is concerned, in one way or another, with the state of equi-; librium and the tendency of systems to move spontaneously in the direction of the equilibrium state. The concept of entropy was developed from the search for a thermodynamic function that would serve as a general criterion of spontaneity for physical and chemical changes. The concept of entropy is based on the second law of thermodynamics, which in essence states that all systems tend to approach a state of equilibrium. The significance of the equilibrium state is realized from the fact that work can be obtained from a system only when the system is not already at equilibrium. If a system is at equilibrium, no process tends to occur spontaneously, and no chemical or physical changes are brought about.

The chemist's interest in entropy is related to the use of this concept to indicate something about the position of equilibrium in a chemical process. Entropy is defined by the following differential equation:

where S is the entropy of the system, and 2" is the absolute temperature. The quantity qm is the amount of heat that the system absorbs if a chemical change is brought about in an infinitely slow, reversible manner. As with enthalpy, it is the change in entropy in a system that is of usual interest, and this is evaluated as follows:

On the basis of the third law of thermodynamics, the entropy of a substance at 0 K is zero. Because of this, the absolute entropy of elements and compounds at some standard state can be determined by integration of Eq. (3.8), using the initial state to represent the equilibrium state of 0 K.

The significance of entropy is that when a spontaneous change occurs in a system, it will always be found that if the total entropy change for everything involved is calculated, a positive value is obtained. Thus, all spontaneous changes in an isolated system occur with an increase of entropy- If one wanted to determine whether a chemical or physical change from one state a to another state b could occur in a system, a calculation of entropy change would give the desired information. If AS for the whole system were positive, the change could occur spontaneously; if AS were negative, the change would tend to occur in the reverse direction, i.e., from b to a. However, if AS were zero, the system would be at equilibrium, and the change could not take place spontaneously in either direction.

On the molecular scale, entropy has a statistical basis. Systems tend to move from a highly ordered state to a more random state. The more highly probable or random a system becomes, the higher will be its entropy. For example, if two different ideal gases are placed together in a closed container, the gas molecules will not remain isolated, but will become randomly mixed. This spontaneous process occurs without a change in internal energy in the container, but with an increase in entropy.

### Free Energy

In the original concepts of thermodynamics it was incorrectly assumed that energy given out by a reaction was a measure of the driving force. It is now seen that in an isolated system, where energy cannot be gained or lost, the entropy change is the

driving force. In more general systems, such as those used in environmental engineering practice, both energy and entropy factors must be considered in order to determine what processes will occur spontaneously. For this purpose the concept of free energy has been developed. Free energy (G) is defined as

where H = enthalpy, J

T = absolute temperature, K (K = °C + 273.15) S — entropy, J/K

At constant temperature and pressure, the change in free energy for a given reaction is

AH = Hi - Hx = (E2 + P2V2) - (E, + P,Vi) = AE + PAV

Combining with Eq. (3.3) for E, we obtain AH = q ~ w + P AF.

From Eq. (3.8) at constant, T, we see that T AS is equal to qm. If the system change is brought about very slowly so that energy losses are at a minimum, then q becomes and w becomes wmiK? the maximum quantity of work that could be obtained from the change.

Therefore,

AG = qKV - wmax + P A V - qK, and -AG = wm„ - P AV

P AV gives the portion of the work that must be "wasted" in expanding the system against the confining pressure. Therefore, -AG gives the difference between the maximum work and the wasted work; in other words, it gives the useful work available from the system change:

In principle, any process that tends to proceed spontaneously can be made to do useful work. Since the free-energy change measures the useful work that might be obtained from a constant-pressure process, it is a measure of the spontaneity of the process.

Consider a change from a to b in a constant-pressure system. If AG for such a change is negative, then the process can proceed; if AG is positive, the process can proceed, but in the reverse direction, i.e., from b to a; if AG is zero, the system is in equilibrium, and the process cannot proceed in either direction. This is a particularly significant relationship, and for the chemist it is one of the most important in thermodynamics.

In order for the concept of free energy to be useful, a reference point for determining free-energy changes must be available. As in the case of enthalpy, a zero value is assigned to free energies of the stable form of the elements at 25°C and 1 atm pressure. In addition, the hydrogen ion at unit activity (approximately one normal solution) is assigned a standard free energy of zero.

The standard free energy of a compound (AG?) is the free energy of formation of that compound from its elements, considering reactants and products all to be m the standard state at 25°C and 1 atm (that is, at unit activity). A list of standard free energies of formation of various compounds is given in Table 3.1 and App. A.

Free-energy changes accompanying a chemical reaction can now be calculated, and the direction in which the reaction will proceed can be determined m a qualitative manner from the sign of the free-energy change. It is apparent from a knowledge of equilibrium reactions that a reaction will proceed in a given direction only until the system reaches a state of equilibrium. It can be shown that free energies can also be used to determine the equilibrium state to which the reaction carries the system as well as the reaction direction. Without going into the details, it should also be apparent that the direction of a reaction is dependent upon the concentration of reactants and products, and so this must be considered in free-energy calculations. Consider the following reaction:

aA + cC + dD The free energy of this reaction, considering the concentration of the various reactants and products, is given by the following equation:

where AG = reaction free-energy change, AG® = standard free-energy change,

R = universal gas constant = 8.314 J/K-mol - 1.99 caVK-mol T = absolute temperature, K The terms in the braces are the activities of the various reactants and products. The convention outlined in Sec. 2.12 for expressing activities must be followed. The term IC}c{D}'i/{ A WB}* is called the reaction quotient Q.

The development of Eq. (3.12) is beyond the scope of this book but is given m most books on physical chemistry. This equation is important, as it allows the prediction of reaction direction for any activity (or concentration, approximately) at products and reactants of interest- If the free-energy change (AG) is negative, the reaction will proceed to the right as written. If AG is positive, the reaction will proceed in the reverse direction. If a reaction is allowed to proceed to a state of equilibrium, it will reach a position for which no further driving force is operative. At this point AG will be zero, and thus

The subscript "equilibrium" is to indicate that this equation is true only when the system is at equilibrium. It should now be noted that at equilibrium die ratio of product concentrations to reactant concentrations is equal to the equilibrium constant:

Thus, Eq. (3.13) can be written in the following form:

This equation is one of the most important results of thermodynamics. It shows the relationship between the equilibrium constant of a reaction and its thermochemical properties. By use of this equation, the equilibrium constant for a reaction can be calculated from thermochemical properties of the reactants and products which may have been determined from entirely different reactions.

Comparison of Q with K indicates whether the reaction is proceeding to the right or the left, or if the reaction is at equilibrium. ïf Q is less than K, the reaction will proceed from left to right; if Q is greater than K, the reaction will proceed from right to left; if Q equals K, the reaction is at equilibrium.

. ' The equatipn and free energies for the first ionization are

. ' The equatipn and free energies for the first ionization are

AG0 for reaction = (-586.85) + (0) - (-623) = 36 15 kJ

In AT, = -AG°/RT = -36,150/8.314(298) = -14.59 Therefore, = 4.61 X 10"7

EXAMPLE 3.3

v .'The equation and free energies of interest are . ■ v.-.

-386.23 - 394.38 AG0 for reaction = (-394.38) - (-386.23) = -8.15 kJ

v .'The equation and free energies of interest are . ■ v.-.

-386.23 - 394.38 AG0 for reaction = (-394.38) - (-386.23) = -8.15 kJ

Therefore, JK* = 26.8 atm-L/mol j. So-,'., „ ,:,/ V, , ■ '. = 26.8atm-L/mol . ■

Thus, the Henry's law constant discussed in Sec. 2.9 can be determined from free-energy

EXAMPLE 3.4

isfi'-SÍ"'v«'«• !>-•■* '*.■ -í-"t""".•71T-"¿ii' : Í-Viv<r v¿"tft^C •Jirr^ffi-.í-'i'iív.'ji^ij'lir-i^v^y^^viv^.5'-

EXAMPLE 3.6 | If a water sample has a Ca2P concentration of 20 mg/L and a F~ concentration of 0.1 mg/L, will CaF2(i) dissolve or precipitate, or is the water m equilibrium With, respect to CaF2(i)? Ignore activity corrections ^ ,

' [Ca2^ = (20 mg/L)/(40,000 mg/mol) = 5 0 X 10""1 M ' - ' , [F-] =(01 mg/L)/(19,000 rag/mol) = 5 3 X 10 M , "

Q= (5 0 x 10""4 M)(5 3 X 10-6 M)2 - 1.4 * 10^4

Since Q is less than Ksp (1 4 X 10"14 < 5 14 x 10~12), the reacUon is proceeding from left to nght and CaFi(i) will dissolve further' * i

Another way to approach the problem is to realize fromEqs (3.12) anc th;

Tn this case, Q is less than K, making AG negative Thus, the reaction proceeds from left

### Temperature Dependence of Equilibrium Constant

From a consideration of the relationships between free energy and the equilibrium constant and between free energy and enthalpy, a thermodynamic basis for predicting the change in equilibrium constant with temperature can be obtained. In differential form, this relationship is d\nK = AH0 (3.15)

dT RT2

This equation indicates that for exothermic reactions the equilibrium constant decreases with increasing temperature, while for endothermic reactions it increases.

chapter 3 Basic Concepts from Physical Chemistry 63

Over the rather limited temperature range of environmental interest, AH° is normally constant enough that the integrated form can be used:

Thus, if the equilibrium constant for a reaction is known for one temperature, the value for another temperature can be calculated from standard enthalpy values for the products and reactants of the reaction.

' Calculate, the,,iomzat >n ;on\$tam*tf, it 10°C for carbonic acid, assuming ^ — 4:61 X'-^

1 À""7 ot >?<Î0(^ be rflÎr-iiîflifiiHI .irj -'F.Yamnlp::^ "A'.v'Thp Wmatfnn'Vanift /> n tko 1 n i 0 ô' 'Xf fA^tvi^iïXV» iWr-'

' Calculate, the,,iomzat >n ;on\$tam*tf, it 10°C for carbonic acid, assuming ^ — 4:61 X'-^

1 À""7 ot >?<Î0(^ be rflÎr-iiîflifiiHI .irj -'F.Yamnlp::^ "A'.v'Thp Wmatfnn'Vanift /> n tko 1 n i 0 ô' 'Xf fA^tvi^iïXV» iWr-'

H2C03(ô3) = H \aq) + HC03"(û?) Arf , -699 0 -692.00

H2C03(ô3) = H \aq) + HC03"(û?) Arf , -699 0 -692.00

K2} 8.314 [(298X283)j , Therefore, Kl0 = eT0150(4.61 X 10"7) = 3.97 X 10"7. ' ,

### 3.3 I VAPOR PRESSURE OF LIQUIDS

According to the kinetic-molecular theory, liquids as well as gases are in constant agitation, and molecules are constantly flying from the surface of the liquid into the atmosphere above. In open systems most of these particles never return, and the liquid is said to be undergoing evaporation. In a closed system, however, particles return to the liquid phase in proportion to their concentration in the gaseous phase. Eventually the rate of return equals the rate of flight, and a condition of equilibrium is established. The vapor is then said to be saturated. The pressure exerted by the vapor under these conditions is known as the vapor pressure. The vapor pressure of all liquids increases with temperature. The vapor-pressure values of water and a few organic liquids are given in Table 3.2. It will be noted that vapor pressures do not rise in a regular manner. For rough approximations, the vapor pressure may be considered to increase about 1.5 times for each 10°C rise in temperature.

The Rankine formula, or one of its modifications, is commonly used to calculate vapor pressures, logp = ^ +.Slog 7 + C (3.17)

where p ~ vapor pressure

EXAMPLE 3.7

 liipiiil ifpr ¡111111® % Alcofa»! . . 1 ■ ' 0 0.006 0.016 0.060 0.035 10 0.012 0.031 0.099 0.060 20 0.023 0.058 0.158 0.098 30 0.042 0.104 0.244 0.156 40 0.073 0.178 0.364 0.238 50 0.122 0.292 0.528 0.354 60 0.197 0.464 0.745 0.511 70 0.308 0.714 1.036 0.720 80 0.467 1.068 1.397 0.968 90 0.692 1.562 1.851 1.337 100 1.000 2.228 2.416 1.768

Vapor-pressure data for many of the compounds of environmental interest can be found in the "International Critical Tables of Numerical Data, Physics, Chemistry and Technology" or standard handbooks of chemistry. Appropriate formulas and constants are usually given to allow calculation of vapor-pressure values for any temperature.

When the vapor pressure of a liquid becomes equal to the pressure of the atmosphere above it, the liquid is said to have reached its boiling point. Violent agitation of the liquid occurs under these conditions as a result of the transformation of liquid to gas at the source of heat, migration of the bubbles of vapor through the liquid, and their escape from the liquid surface.

Liquids with appreciable vapor pressure may be caused to boil over a wide range of temperatures by decreasing or increasing the pressure. Water boils at room temperature if the pressure above it is reduced to about 0.028 atm. On the other hand, the boiling point of water in a steam boiler operating at 13.6 atm is 194.4°C. Application of this principle can be employed in the wet oxidation process for combustion of organic sludges and other organic wastes. In this process part of the organic matter is chemically oxidized in an aqueous phase by dissolved oxygen in a specially designed reactor in which the water temperature is elevated to between 250 and 300°C. To maintain a temperature in this range without boiling the water requires pressures between 40 and 85 atm, respectively. The oxidizability of the waste organics, as well as the oxidation rate, increases markedly with temperature. However, the maximum temperature that can be used in such a reactor is set by the critical temperature above which water can no longer exist in a liquid phase, regardless of pressure. This temperature is 374°C. The critical pressure that suffices to keep water in liquid form just below this critical temperature is 218 atm.

## Renewable Energy 101

Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable. The usage of renewable energy sources is very important when considering the sustainability of the existing energy usage of the world. While there is currently an abundance of non-renewable energy sources, such as nuclear fuels, these energy sources are depleting. In addition to being a non-renewable supply, the non-renewable energy sources release emissions into the air, which has an adverse effect on the environment.

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