**2. Energy, work and heat**

### **2.1 The first law of thermodynamics**

Generally, when a system passes through a process it exchanges energy U with its environment. The energy change in the system ΔU may result from performing work w on the system or letting the system perform work, and from exchanging heat q between the system and the environment

$$
\Delta \mathcal{L} I = q + w \tag{2}
$$

The heat and the work supplied to a system are withdrawn from the environment, such that, according to the first law of thermodynamics

$$
\Delta l I\_{system} + \Delta l I\_{environment} = 0 \tag{3}
$$

The First Law of thermodynamics states that the energy content of the universe (or any other isolated system) is constant. In other words, energy can neither be created nor annihilated. It implies the impossibility of designing a perpetuum mobile, a machine that performs work without the input of energy from the environment. The First Law also implies that for a system passing from initial state 1 to final state 2 the energy change

between the system and surroundings. In practice, if a reactor is used to carry out a chemical reaction, the walls of the reactor that are in contact with the thermo stated liquid medium around the reactor may be assumed to be the surroundings of the experimental system. For particles such as colloids, the medium in which they are immersed may act as the surroundings, provided nothing beyond this medium influences the particle. An isolated system is defined as a system to or from which there is no transport of matter and energy. When a system is isolated, it cannot be affected by its surroundings. The universe is assumed to be an isolated system. Nevertheless, changes may occur within the system that are detectable using measuring instruments such as thermometers, pressure gauges etc. However, such changes cannot continue indefinitely, and the system must eventually reach a final static condition of internal equilibrium. If a system is not isolated, its boundaries may permit exchange of matter or energy or both with its surroundings. A closed system is one for which only energy transfer is permitted, but no transfer of mass takes place across the boundaries, and the total mass of the system is constant. As an example, a gas confined in an impermeable cylinder under an impermeable piston is a closed system. For a closed system, this interacts with its surroundings; a final static condition may be reached such that the system is not only internally at equilibrium but also in external equilibrium with its surroundings. A system is in equilibrium if no further spontaneous changes take place at constant surroundings. Out of equilibrium, a system is under a certain stress, it is not relaxed, and it tends to equilibrate. However, in equilibrium, the system is fully relaxed. If a system is in equilibrium with its surroundings, its macroscopic properties are fixed, and the system can be defined as a given thermodynamic state. It should be noted that a thermodynamic state is completely different from a molecular state because only after the precise spatial distributions and velocities of all molecules present in a system are known can we define a molecular state of this system. An extremely large number of molecular states correspond to one thermodynamic state, and the application of statistical thermodynamics can form the link between them (Lyklema, J. 2005), (Dabrowski A., 2001).

Generally, when a system passes through a process it exchanges energy U with its environment. The energy change in the system ΔU may result from performing work w on the system or letting the system perform work, and from exchanging heat q between the

The heat and the work supplied to a system are withdrawn from the environment, such

0 *U U system environment* (3)

The First Law of thermodynamics states that the energy content of the universe (or any other isolated system) is constant. In other words, energy can neither be created nor annihilated. It implies the impossibility of designing a perpetuum mobile, a machine that performs work without the input of energy from the environment. The First Law also implies that for a system passing from initial state 1 to final state 2 the energy change

*Uqw* (2)

**2. Energy, work and heat** 

system and the environment

**2.1 The first law of thermodynamics** 

that, according to the first law of thermodynamics

1 2 *U* does not depend on the path taken to go from 1 to 2. A direct consequence of that conclusion is that U is a function of state: when the macroscopic state of a system is fully specified with respect to composition, temperature, pressure, and so on (the so-called state variables), its energy is fixed. This is not the case for the exchanged heat and work. These quantities do depend on the path of the process. For an infinitesimal small change of the energy of the system

$$
\Delta \mathcal{L} I = \hat{c}\hat{q} + \hat{c}w \tag{4}
$$

For w and, hence, *w* , various types of work may be considered, such as mechanical work resulting from compression or expansion

of the system, electrical work, interfacial work associated with expanding or reducing the interfacial area between two phases, and chemical work due to the exchange of matter between system and environment. All types of work are expressed as *XdY* , where X and Y are state variables. X is an intensive property (independent of the extension of the system) and Y the corresponding extensive property (it scales with the extension of the system). Examples of such combinations of intensive and extensive properties are pressure p and volume V, interfacial tension γ and interfacial area A, electric potential Ψ and electric charge Q, the chemical potential µi of component i, and the number of moles ni of component i. As a rule, X varies with Y but for an infinitesimal small change of Y, X is approximately constant. Hence, we may write

$$d\mathcal{U} = \hat{c}q - p dV + \chi dA + \varphi dQ + \sum\_{i} \mu\_{i} dn\_{i} \tag{5}$$

*i*

The terms of type XdY in Eq. above represent mechanical (volume), interfacial, electric, and chemical works, respectively. *i* implies summation over all components in the system. It

is obvious that for homogeneous systems the γdA term is not relevant.

#### **2.2 The second law of thermodynamics: entropy**

According to the First Law of thermodynamics the energy content of the universe is constant. It follows that any change in the energy of a system is accompanied by an equal, but opposite, change in the energy of the environment. At first sight, this law of energy conservation seems to present good news: if the total amount of energy is kept constant why then should we be frugal in using it? The bad news is that all processes always go in a certain direction, a direction in which the energy that is available for performing work continuously decreases.

Entropy, S, is the central notion in the Second Law. The entropy of a system is a measure of the number of ways the energy can be stored in that system. In view of the foregoing, any spontaneous process goes along with an entropy increase in the universe that is, ΔS > 0. If as a result of a process the entropy of a system decreases, the entropy of the environment must increase in order to satisfy the requirement ΔS > 0 (Levine, I.N., 1990).

Based on statistical mechanics, the entropy of a system, at constant U and V can be expressed by Boltzmann's law

$$S\_{u,v} = k\_s \ln w \tag{6}$$

To avoid impractical conditions when expressing intensive variables as differential quotients as, for example auxiliary functions are introduced. These are the enthalpy H,

Since U is a function of state, and p, V, T, and S are state variables, H, A, and G are also

*ii i dH TdS Vdp dA dQ dn* 

*ii i dA SdT pdV dA dQ dn* 

*ii i dG SdT Vdp dA dQ dn* 

Expressing γ, Ψ, or µi as a differential quotient requires constancy of S and V, S and p, T and V, and T and p, when using the differentials dU, dH, dA, and dG, respectively. In most cases the conditions of constant T and V or constant T and p are most practical. It is noted that for heating or cooling a system at constant p, the heat exchange between the system and its environment is equal to the enthalpy exchange. Hence, for the heat capacity, at constant p,

> ()( ) *<sup>p</sup> <sup>p</sup> <sup>p</sup> <sup>q</sup> dH <sup>C</sup> dT dT*

In general, for a function of state f that is completely determined by variables x and y, df = Adx + Bdy. Cross-differentiation in df gives ( / ) ( / ) *Ay Bx x y* , known as a Maxwell relation. Similarly, cross-differentiation in dU, dH, dA, and dG yields a wide variety of Maxwell relations between differential quotients. For instance, by cross-differentiation in dG

,,, ,, , , , () () *pAQni s T pQni s*

Molar properties, indicated by a lowercase symbol, are defined as an extensive property Y per mole of material: y = Y/n. Since they are expressed per mole, molar quantities are

For a single component system Y is a function of T; p; . . . ; n. Many extensive quantities vary linearly with n, but for some (e.g., the entropy) the variation with n is not proportional. In the latter case y is still a function of n. In a two-, three- or multi-component system (i.e., a mixture), the contribution of each component to the functions of state, say, the energy of the

*T A* 

*S*

*H U pV* (12)

*A U TS* (13)

(15)

(16)

(17)

*G U pV TS H TS A pV* (14)

 

> 

 

(18)

(19)

defined as

the Helmholtz energy

and the Gibbs energy

functions of state. The corresponding differentials are

we find, (Lyklema, L. 1991), (Pitzer, K.S. and Brewer L. 1961).

**2.5 Molar properties and partial molar properties** 

intensive.

where w is the number of states accessible to the system and kB is Boltzmann's constant. For a given state w is fixed and, hence, so is S. It follows that S is a function of state. It furthermore follows that S is an extensive property: for a system comprising two subsystems (a and b) w= wa + wb and therefore, because of, S = Sa+ Sb. The entropy change in a system undergoing a process 1 2 is thermodynamically formulated in terms of the heat *q* taken up by that system and the temperature T at which the heat uptake occurs(sraelachvili, J. 1991):

$$
\Delta S \geq \left\| \frac{\partial q}{T} \right\|\tag{7}
$$

Because the temperature may change during the heat transfer is written in differential form (Pitzer, K.S. and Brewer L. 1961).

### **2.3 Reversible processes**

In contrast to the entropy, heat is not a function of state. For the heat change it matters whether a process 1 2 is carried out reversibly or irreversibly. For a reversible process, that is, a process in which the system is always fully relaxed

$$
\Delta S = \int\_1^2 \frac{\partial q\_{rev}}{T} \tag{8}
$$

Infinitesimal small changes imply infinitesimal small deviations from equilibrium and, therefore, reversibility. The term *q* in (5) may then be replaced by TdS, which gives

$$d\,d\mathcal{U} = Tds - p\,dV + \mathcal{\mathcal{Y}}\,dA + \mathcal{\mathcal{Y}}\,d\mathcal{Q} + \sum\_{i} \mu\_{i}dn\_{i} \tag{9}$$

where all terms of the right-hand side are now of the form XdY. Equation (9) allows the intensive variables X to be expressed as differential quotients, such as, for instance,

$$\gamma = \left(\frac{\partial \mathcal{U}}{\partial A}\right)\_{S,V,Q,n\_{i,s}} \tag{10}$$

where the subscripts indicate the properties that are kept constant. In other words, the interfacial tension equals the energy increment of the system resulting from the reversible extension of the interface by one unit area under the conditions of constant entropy, volume, electric charge, and composition. The required conditions make this definition very impractical, if not in operational. If the interface is extended it is very difficult to keep variables such as entropy and volume constant.

The other intensive variables may be expressed similarly as the change in energy per unit extensive property, under the appropriate conditions (Tempkin M. I. and Pyzhev V., 1940).

#### **2.4 Maxwell relations**

At equilibrium, implying that the intensive variables are constant throughout the system, (9) may be integrated, which yields

$$dL = TdS - pV + \gamma A + \mu Q + \Sigma\_i \,\mu\_i n\_i \tag{11}$$

where w is the number of states accessible to the system and kB is Boltzmann's constant. For a given state w is fixed and, hence, so is S. It follows that S is a function of state. It furthermore follows that S is an extensive property: for a system comprising two subsystems (a and b) w= wa + wb and therefore, because of, S = Sa+ Sb. The entropy change in a system undergoing a process 1 2 is thermodynamically formulated in terms of the heat *q* taken up by that system and the temperature T at which the heat uptake

2

(7)

(8)

(10)

(9)

(11)

*i i i*

 

1 *<sup>q</sup> <sup>S</sup> T*

Because the temperature may change during the heat transfer is written in differential form

In contrast to the entropy, heat is not a function of state. For the heat change it matters whether a process 1 2 is carried out reversibly or irreversibly. For a reversible process,

2

1 *rev <sup>q</sup> <sup>S</sup> T*

Infinitesimal small changes imply infinitesimal small deviations from equilibrium and,

*dU Tds pdV dA dQ dn* 

where all terms of the right-hand side are now of the form XdY. Equation (9) allows the

,,, , ( )*SVQni s U A*

where the subscripts indicate the properties that are kept constant. In other words, the interfacial tension equals the energy increment of the system resulting from the reversible extension of the interface by one unit area under the conditions of constant entropy, volume, electric charge, and composition. The required conditions make this definition very impractical, if not in operational. If the interface is extended it is very difficult to keep

The other intensive variables may be expressed similarly as the change in energy per unit extensive property, under the appropriate conditions (Tempkin M. I. and Pyzhev V., 1940).

At equilibrium, implying that the intensive variables are constant throughout the system, (9)

*U TdS pV A Q n i ii* 

 

therefore, reversibility. The term *q* in (5) may then be replaced by TdS, which gives

intensive variables X to be expressed as differential quotients, such as, for instance,

.

occurs(sraelachvili, J. 1991):

(Pitzer, K.S. and Brewer L. 1961).

that is, a process in which the system is always fully relaxed

variables such as entropy and volume constant.

**2.4 Maxwell relations** 

may be integrated, which yields

**2.3 Reversible processes** 

To avoid impractical conditions when expressing intensive variables as differential quotients as, for example auxiliary functions are introduced. These are the enthalpy H, defined as

$$H \equiv \mathcal{U} + pV \tag{12}$$

the Helmholtz energy

$$A \equiv \mathcal{U}I - TS \tag{13}$$

and the Gibbs energy

$$\mathbf{G} \equiv \mathbf{L}\mathbf{I} + p\mathbf{V} - \mathbf{T}\mathbf{S} = \mathbf{H} - \mathbf{T}\mathbf{S} = \mathbf{A} + p\mathbf{V} \tag{14}$$

Since U is a function of state, and p, V, T, and S are state variables, H, A, and G are also functions of state. The corresponding differentials are

$$dH = TdS + Vdp + \mathcal{y}dA + \Psi dQ + \sum\_{i} \mu\_{i} d\mathfrak{n}\_{i} \tag{15}$$

$$dA = -SdT - pdV + \gamma dA + \Psi dQ + \sum\_{i} \mu\_{i} d\mathfrak{n}\_{i} \tag{16}$$

$$dG = -SdT + Vdp + \gamma dA + \Psi \, dQ + \sum\_{i} \mu\_{i} dn\_{i} \tag{17}$$

Expressing γ, Ψ, or µi as a differential quotient requires constancy of S and V, S and p, T and V, and T and p, when using the differentials dU, dH, dA, and dG, respectively. In most cases the conditions of constant T and V or constant T and p are most practical. It is noted that for heating or cooling a system at constant p, the heat exchange between the system and its environment is equal to the enthalpy exchange. Hence, for the heat capacity, at constant p,

$$C\_r = \left(\frac{\partial \eta}{dT}\right)\_p = \left(\frac{dH}{dT}\right)\_r \tag{18}$$

In general, for a function of state f that is completely determined by variables x and y, df = Adx + Bdy. Cross-differentiation in df gives ( / ) ( / ) *Ay Bx x y* , known as a Maxwell relation. Similarly, cross-differentiation in dU, dH, dA, and dG yields a wide variety of Maxwell relations between differential quotients. For instance, by cross-differentiation in dG we find, (Lyklema, L. 1991), (Pitzer, K.S. and Brewer L. 1961).

$$(\frac{\partial \mathcal{Y}}{\partial T})\_{\mathfrak{r},\mathbb{A}\mathbb{Q},\mathfrak{n}\_{\mathfrak{i},\mathbb{S}}} = -(\frac{\partial \mathcal{S}}{\partial A})\_{\mathfrak{r},\mathfrak{p},\mathbb{Q},\mathfrak{n}\_{\mathfrak{i},\mathbb{S}}}\tag{19}$$

#### **2.5 Molar properties and partial molar properties**

Molar properties, indicated by a lowercase symbol, are defined as an extensive property Y per mole of material: y = Y/n. Since they are expressed per mole, molar quantities are intensive.

For a single component system Y is a function of T; p; . . . ; n. Many extensive quantities vary linearly with n, but for some (e.g., the entropy) the variation with n is not proportional. In the latter case y is still a function of n. In a two-, three- or multi-component system (i.e., a mixture), the contribution of each component to the functions of state, say, the energy of the

*i*

0

*<sup>i</sup>* is an integration constant that is independent of pi; 0 ( 1) *i ii*

0

0

where ci is the concentration of i in the solution. In more general terms, for an ideal mixture

where Xi is the mole fraction of i in the mixture, defined as *Xn n i i ii* / . Note that the

mixture. This value deviates from the real value of µi for pure i, because in the case of pure i

and pi, respectively, and their values are therefore independent of the configurations of *i* in the mixture. They do depend on the interactions between i and the other components and therefore on the types of substances in the mixture. Because Xi, ci, and pi are expressed in

The RTln Xi term in Eq. (30) or, for that matter, the RTlnci and RTlnpi terms in (28) and (29) do not contain any variable pertaining to the types of substances in the mixture. Hence,

*<sup>i</sup>* is the one obtained for mi by extrapolating to Xi=1 assuming ideality of the

*<sup>i</sup>* differ (Keller J.U., 2005)

0

*i*

(31)

<sup>0</sup> *s s RX ii i* ln (32)

*<sup>i</sup> s* ,which is independent of the

*R X*

ln *i i*

configurations of i in the mixture but dependent on the interactions of i with the other components, and a part Rln Xi, which takes into account the possible configurations of *i*. It follows that the RTlnXi (or RTlnci or RTlnpi) term in the expressions for µi stems from the

 

depending on the units in which pi is expressed. Similarly, without giving the derivation here, it is mentioned that for the chemical potential of component i in an ideal solution

 

 

 

,... ,

*T T* 

*<sup>p</sup> ni s*

the ''mixture'' is as far as possible from ideal. As said µi and 0

The partial molar entropy of i is composed of a part 0

*v*

*i RT*

in which R is the universal gas constant. Combining (26) and (27) gives, after integration, an

*<sup>p</sup>* (27)

*ii i RT p* ln (28)

*ii i RT c* ln (29)

*i i RT X* ln *<sup>i</sup>* (30)

 

*p* , its value

*<sup>i</sup>* are defined per unit Xi, ci,

For an ideal gas

where 0 

value of 0 

expression for µi(pi) in an ideal gas

different units, the values for µi and 0

RTlnXi term follows from

which, because of (23), gives

configurationally possibilities as well.

these terms are generic. Interpretation of the

system cannot be assigned unambiguously. This is because the energy of the system is not simply the sum of the energies of the constituting components but includes the interaction energies between the components as well. It is impossible to specify which part of the total interaction energy belongs to component i. For that reason partial molar quantities yi are introduced. They are defined as the change in the extensive quantity Y pertaining to the whole system due to the addition of one mole of ni under otherwise constant conditions. Because by adding component i the composition of the mixture and, hence, the interactions between the components are affected, yi is defined as the differential quotient (Prausnitz, J.M., and et al. 1999).

$$\mathbf{y}\_{i} = (\frac{\partial Y}{\partial \mathbf{n}\_{i}})\_{\mathbf{r}\_{\cdot}, \mathbf{s}\_{\cdot}, \dots, \mathbf{s}\_{j} \neq \mathbf{1}} \tag{20}$$

The partial molar quantities are operational; that is, they can be measured. Now , ,... *Tp ni s*, *Y* can be obtained as *i ii n y* A partial molar quantity often encountered is the partial molar Gibbs energy (Aveyard, R. and Haydon, D.A., 1973),

$$\mathbf{g}\_{i} \equiv \left(\frac{\partial \mathbf{G}}{\partial \mathbf{n}\_{i}}\right)\_{\mathbf{r}\_{,\mathbf{p}\_{r},\ldots,\mathbf{p}\_{j}\neq i}}\tag{21}$$

According to (17)

$$\mathbf{g}\_{i} \equiv \left(\frac{\partial \mathbf{G}}{\partial \mathbf{n}\_{i}}\right)\_{\mathbf{r}\_{\cdot \cdot \eta \ldots \eta \, j \neq i}} = \mu\_{i} \tag{22}$$

that is, at constant *Tp n* , ,..., *<sup>j</sup><sup>i</sup>* ,the chemical potential of component *i* in a mixture equals its partial molar Gibbs energy.

By cross-differentiation in (17) the temperature- and pressure-dependence of µi can be derived as

$$\left(\frac{\partial \mu\_i}{\partial T}\right)\_{r,\dots,n\_{i,\text{s}}} = -\left(\frac{\partial S}{\partial n\_i}\right)\_{r,\dots,n\_{j\text{s}}\neq i} = -s\_i\tag{23}$$

With

$$\mu\_i = \mathcal{g}\_i \equiv \mathcal{H}\_i - \mathcal{T}\mathcal{s}\_i \tag{24}$$

it can be deduced that

$$\left(\frac{\partial(\mu\_i/T)}{\partial T}\right)\_{\nu\_r = i, s} = -\frac{H}{T^2} \tag{25}$$

The pressure-dependence of mi is also obtained from (17):

$$\left(\frac{\partial \mu\_i}{\partial p}\right)\_{\mathbb{T}\_{-}, \eta\_{i,s}} = \left(\frac{\partial V}{\partial \mathfrak{n}\_i}\right)\_{\mathbb{T}\_{-}, \mathfrak{p}\_{-}, \eta\_j \neq i} = \upsilon\_i \tag{26}$$

For an ideal gas

206 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

system cannot be assigned unambiguously. This is because the energy of the system is not simply the sum of the energies of the constituting components but includes the interaction energies between the components as well. It is impossible to specify which part of the total interaction energy belongs to component i. For that reason partial molar quantities yi are introduced. They are defined as the change in the extensive quantity Y pertaining to the whole system due to the addition of one mole of ni under otherwise constant conditions. Because by adding component i the composition of the mixture and, hence, the interactions between the components are affected, yi is defined as the differential quotient (Prausnitz,

> , ,...., <sup>1</sup> ( ) *<sup>i</sup> Tp n i*

The partial molar quantities are operational; that is, they can be measured. Now , ,... *Tp ni s*, *Y* can be obtained as *i ii n y* A partial molar quantity often encountered is the partial molar Gibbs

*G*

 

*G*

*n*

*n*

, ,...,

*n*

, ,..., *i i <sup>i</sup> Tp nj i*

that is, at constant *Tp n* , ,..., *<sup>j</sup><sup>i</sup>* ,the chemical potential of component *i* in a mixture equals its

By cross-differentiation in (17) the temperature- and pressure-dependence of µi can be

,..., , , ,...,

*T n*

( /) *<sup>i</sup>*

*p n*

,..., , ,... ,

*T ni s <sup>i</sup> Tp nj i V*

 

*i*

*p ni s <sup>i</sup> Tp nj i*

*ii i i*

,... ,

*p ni s T H T T*

*S*

*<sup>i</sup> Tp nj i*

2

*i*

*v*

*i*

*g H Ts* (24)

(25)

*s*

*Y*

*y*

*i*

*g*

*g*

*i*

The pressure-dependence of mi is also obtained from (17):

*j*

(20)

(21)

(22)

(23)

(26)

J.M., and et al. 1999).

According to (17)

derived as

With

partial molar Gibbs energy.

it can be deduced that

energy (Aveyard, R. and Haydon, D.A., 1973),

$$w\_i = \frac{RT}{p\_i} \tag{27}$$

in which R is the universal gas constant. Combining (26) and (27) gives, after integration, an expression for µi(pi) in an ideal gas

$$
\mu\_i = \mu\_i^\upsilon + RT \ln p\_i \tag{28}
$$

where 0 *<sup>i</sup>* is an integration constant that is independent of pi; 0 ( 1) *i ii p* , its value depending on the units in which pi is expressed. Similarly, without giving the derivation here, it is mentioned that for the chemical potential of component i in an ideal solution

$$
\mu\_i = \mu\_i^0 + RT\ln c\_i \tag{29}
$$

where ci is the concentration of i in the solution. In more general terms, for an ideal mixture

$$
\mu\_i = \mu\_i^0 + RT\ln X\_i \tag{30}
$$

where Xi is the mole fraction of i in the mixture, defined as *Xn n i i ii* / . Note that the value of 0 *<sup>i</sup>* is the one obtained for mi by extrapolating to Xi=1 assuming ideality of the mixture. This value deviates from the real value of µi for pure i, because in the case of pure i the ''mixture'' is as far as possible from ideal. As said µi and 0 *<sup>i</sup>* are defined per unit Xi, ci, and pi, respectively, and their values are therefore independent of the configurations of *i* in the mixture. They do depend on the interactions between i and the other components and therefore on the types of substances in the mixture. Because Xi, ci, and pi are expressed in different units, the values for µi and 0 *<sup>i</sup>* differ (Keller J.U., 2005)

The RTln Xi term in Eq. (30) or, for that matter, the RTlnci and RTlnpi terms in (28) and (29) do not contain any variable pertaining to the types of substances in the mixture. Hence, these terms are generic. Interpretation of the

RTlnXi term follows from

$$
\left(\frac{\partial \mu\_i}{\partial T}\right)\_{p\_r = n\_{i,s}} = \frac{\partial \mu\_i^0}{\partial T} + R \ln X\_i \tag{31}
$$

which, because of (23), gives

$$s\_i = s\_i^0 - R \ln X\_i \tag{32}$$

The partial molar entropy of i is composed of a part 0 *<sup>i</sup> s* ,which is independent of the configurations of i in the mixture but dependent on the interactions of i with the other components, and a part Rln Xi, which takes into account the possible configurations of *i*. It follows that the RTlnXi (or RTlnci or RTlnpi) term in the expressions for µi stems from the configurationally possibilities as well.

chemical potentials equalize, at equilibrium. It is easy to see from this why the chemical potential is so useful in mixtures and solutions in matter transfer (open) processes (Norde, W., 2003). This is especially clear when it is understood that m*i* is a simple function of

0

for dilute mixtures, where m*i* o is the standard chemical potential of component '*i*', usually 1 M for solutes and 1 atm for gas mixtures. This equation is based on the entropy associated with a component in a mixture and is at the heart of why we generally plot measurable changes in any particular solution property against the log of the solute concentration, rather than using a linear scale. Generally, only substantial changes in concentration or pressure produce significant changes in the properties of the mixture. (For example,

The First Law of Thermodynamics is the law of conservation of energy; it simply requires that the total quantity of energy be the same both before and after the conversion. In other words, the total energy of any system and its surroundings is conserved. It does not place any restriction on the conversion of energy from one form to another. The interchange of heat and work is also considered in this first law. In principle, the internal energy of any system can be changed, by heating or doing work on the system. The First Law of Thermodynamics requires that for a closed (but not isolated) system, the energy changes of the system be exactly compensated by energy changes in the surroundings. Energy can be exchanged between such a system and its surroundings in two forms: heat and work. Heat and work have the same units (joule, J) and they are ways of transferring energy from one entity to another. A quantity of heat, Q, represents an amount of energy in transit between a system and its surroundings, and is not a property of the system. Heat flows from higher to lower temperature systems. Work, W, is the energy in transit between a system and its surroundings, resulting from the displacement of external force acting on the system. Like heat, a quantity of work represents an amount of energy and is not a property of the system. Temperature is a property of a system while heat and work refer to a process. It is important to realize the difference between temperature, heat capacity and heat: temperature, T, is a property which is equal when heat is no longer conducted between bodies in thermal contact and can be determined with suitable instruments (thermometers) having a reference system depending on a material property (for example, mercury thermometers show the density differences of liquid mercury metal with temperature in a capillary column in order to visualize and measure the change of temperature). Suppose any closed system (thus having a constant mass) undergoes a process by which it passes from an initial state to a final state. If the only interaction with its surroundings is in the form of transfers of heat, Q, and work, W, then only the internal energy, U, can be changed, and the First Law of Thermodynamics is expressed mathematically as (Lyklema, J. ;2005 & Keller J.U.;2005)

where Q and W are quantities inclusive of sign so that when the heat transfers from the system or work is done by the system, we use negative values in Equation (11). Processes

*ii i kT C* ln (41)

*UU U QW final* initial (42)

 

consider the use of the pH scale.) (Koopal L.K., and et al. 1994).

**3.1 Thermodynamics for closed systems** 

concentration, that is:
