Subject Chemistry

Paper No and Title 10, Physical Chemistry- III (Classical Thermodynamics, Non-Equilibrium Thermodynamics, Surface chemistry, Fast kinetics) Module No and 11, Fugacity and Third law of thermodynamics Title Module Tag CHE_P10_M11

CHEMISTRY Paper No. 10: Physical Chemistry- III (Classical Thermodynamics, Non-Equilibrium Thermodynamics, Surface chemistry, Fast kinetics) Module No. 11: Fugacity and Third law of thermodynamics

TABLE OF CONTENTS 1. Learning Outcomes 2. Fugacity 3. Fugacity coefficient 4. Determination of fugacity of gas 5. Physical significance of fugacity 6. Third law of thermodynamics 7. Determination of entropy for solids, liquids and gases 8. Residual entropy 9. 9.Summary

CHEMISTRY Paper No. 10: Physical Chemistry- III (Classical Thermodynamics, Non-Equilibrium Thermodynamics, Surface chemistry, Fast kinetics) Module No. 11: Fugacity and Third law of thermodynamics

1. Learning Outcomes

After studying this module you shall be able to:  Know the concept of fugacity  Determine the coefficient of fugacity  Learn about the third law of thermodynamics  Know the concept of residual entropy

2. Fugacity

The concept of fugacity was introduced by an American chemist, G.N Lewis. It is used for representing the real behavior of Vander Waal’s gases which is different from behavior of ideal gases. At constant temperature, the variation of free energy with pressure is given by: ( / )G   P V …(1) T For one mole of an ideal gas, V stands for the molar volume and thus we can write it as: ()/dGRTdPP …(2) T Similarly for n number of moles of gas, the equation turns into: ()/lndGnRTdPPnRTP …(3) T Integrating the equation (3) G G* nRT ln P …(4) in which G* (the integration constant) is the free energy of n moles of the gas when pressure P is unity. The equation (4) gives the free energy of ideal gas having n number of moles at temperature T and pressure P. Now integrating the above equation between the limits PPand while keeping temperature 12 constant, P2 dP P G  nRT  nRT ln 2 …(5) PP P1 1 Similarly if n = 1, then P G RT ln 2 …(6) P 1 The above equations are not valid for real gases. For making these equations valid for real gases the fugacity function was introduced. It substitutes P in equation (3) giving, (dG ) nRT d (ln f ) …(7) T Then equation (4) becomes; G G* nRTln f …(8) When f = 1 then G = G* that is, G* is the free energy of n moles of a gas when its fugacity becomes 1. Thus, fugacity act as the fictitious pressure i.e. it plays the role of pressure and is not equal to the actual pressure of the real gas. Therefore, it can be regarded as the corrected pressure for the real gas. CHEMISTRY Paper No. 10: Physical Chemistry- III (Classical Thermodynamics, Non-Equilibrium Thermodynamics, Surface chemistry, Fast kinetics) Module No. 11: Fugacity and Third law of thermodynamics

Now, integrating equation (7) between fugacities f1 and f2 while keeping temperature T constant, we get: f GnRT ln 2 …(9) f 1 Similarly, for 1 mole of the gas the equation becomes: f G R T ln 2 …(10) f 1 The above two equations are valid for real gases.

3. Fugacity coefficient (γ)

Fugacity coefficient is defined as the ratio of fugacity of a real gas to the actual pressure P of the gas. It is represented as: 푓 = 훾 …(11) 푝 It is the measure of deviations of a real gas from ideal gas behavior. The units of fugacity and pressure are same therefore fugacity coefficient will be a pure number. At extremely low pressure (when P approaches to zero) fugacity coefficient approaches to unity. All the gases approach ideality under this pressure range. f i.e. lim1  …(12) P  0 P Therefore at low pressure, fugacity becomes equal to pressure. Fugacity becomes different from pressure only in high pressure range.

4. Determination of Fugacity of a gas

For one mole gas we can write, GGRTf* ln …(13)

Differentiating the above equation with respect to pressure keeping temperature and number of moles of various components be constant:

Gf(ln)   RT …(14) PPT And we know that G V …(15) P T Equating the above two equations, we get: (ln)fV  …(16) PRT T At specified temperature, the above equation becomes, CHEMISTRY Paper No. 10: Physical Chemistry- III (Classical Thermodynamics, Non-Equilibrium Thermodynamics, Surface chemistry, Fast kinetics) Module No. 11: Fugacity and Third law of thermodynamics

RT d (ln f) = VdP …(17) Let  be the quantity which defines the deviation from ideal behavior. Substituting for ideal gas (V = RT/P), the quantity is given by:  = RT/P – V …(18) Multiplying the equation by dP, we get:  dP = RT(dP/P) – VdP …(19) Combining equations (17) and (18), we may write: RTd(lnf) = RTdP/P –  dP Or d (lnf ) = d (ln P) – dP/RT …(20) Integrating the above equation between the pressure range 0 to P, we write: f 1 P ln dP …(21) PRT 0 Here  is an experimentally determined quantity measured at different pressures. Thus according to the above equation fugacity will be less than pressure P at low pressure range while it will be higher than pressure P at high pressure range.

Fugacity at low pressure At low pressure range, fugacity is found to have constant values. Under such conditions, therefore equation (21) ln//fPPRT …(22) At low pressure range, real gas tend to behave as ideal gas thus f = P. Or fP/1 …(23) Using the fact when x is small and approaches unity x becomes approximately equal to x-1, then we have: ln//1fPfP  Or fPfP/1ln/ …(24) Using the equation (22), we get: fPP/1/ RT …(25) By making the use of equation (18), we get: f / P= PV/RT or f p2 V/ RT …(26) Thus, by the equation obtained above fugacity can be calculated at moderately low pressures.

For one mole of pure gas, chemical potential becomes equal to Gibb’s free energy. The change in chemical potential for the gaseous component I is given by: dRT df (ln) …(27) ii Or it can be written as:   * RTfln iii …(28)  * Where i denotes the chemical potential of the gaseous component i having the unit fugacity.

CHEMISTRY Paper No. 10: Physical Chemistry- III (Classical Thermodynamics, Non-Equilibrium Thermodynamics, Surface chemistry, Fast kinetics) Module No. 11: Fugacity and Third law of thermodynamics

5. Physical significance of fugacity

Considering the system in which liquid is in contact with the vapor phase. In this water molecules in the liquid escape into the vapor phase by evaporation while molecules in the vapor phase have the tendency to escape into liquid phase by condensation. These two escaping tendencies become equal at equilibrium. Thus it is accepted that any substance has the tendency to escape from its state. Lewis termed this escaping tendency as Fugacity.

Now discussing the third law of thermodynamics

6. Third law of thermodynamics

The third law of thermodynamics is formulated by the German chemist, Walther Nernst during the years 1906-12.. This law is regarding the properties of the system in equilibrium at absolute zero temperature. The statement for the third law of thermodynamics is: At absolute zero temperature, the entropy of each perfectly crystalline solid becomes zero. At absolute zero temperature, the order of the pure crystal becomes perfect i.e. of zero disorder and hence of zero entropy. Thus at zero kelvin temperature, the system must possess minimum possible energy. The statement given by Nernst and Simon is: The change in entropy related to any condensed system which is undergoing any reversible isothermal process reaches zero as the temperature at which this process is performed approaches 0 K. Planck’s statement If one mole of a solid is heated from 0 K to some temperature T below its melting point, keeping its pressure constant then the change in entropy will be given by:

푇 퐶 ∆푆 = 푆 − 푆 = ∫ 푝,푚 푑푇 푇 0퐾 0퐾 푇 Or,

푇 퐶 푆 = 푆 + ∫ 푝,푚 푑푇 푇 0퐾 0퐾 푇 We can say that the entropy function increases with the temperature. Thus it can be minimum at absolute zero kelvin. Planck’s stated that the pure crystalline substance have the minimum entropy at absolute temperature. This is known as third law of thermodynamics.

CHEMISTRY Paper No. 10: Physical Chemistry- III (Classical Thermodynamics, Non-Equilibrium Thermodynamics, Surface chemistry, Fast kinetics) Module No. 11: Fugacity and Third law of thermodynamics

From the above equation, the third law entropy will be given as:

푇 퐶 푆 = ∫ 푝,푚 푑푇 푇 0퐾 푇

Here 푆푇 is the entropy of the substance at temperature T and pressure p. If the pressure of the ° system is 1 bar then the entropy is termed as standard entropy 푆푇.

CHEMISTRY Paper No. 10: Physical Chemistry- III (Classical Thermodynamics, Non-Equilibrium Thermodynamics, Surface chemistry, Fast kinetics) Module No. 11: Fugacity and Third law of thermodynamics

7. Determination of entropy for solids, liquids and gases

For solids: For determining the entropy, the heat capacity of solids should be known. The heat capacity of the solids is determined from Debye T- cubed law which is given by:

3 퐶푝,푚 = 푇 where  is the constant. The entropy will be:

푇 푑( ) 푇푚푖푛 퐶푝,푚 푇 퐶푝,푚 푇푚푖푛 퐶푝,푚 푇 퐾 푆푇 = ∫ 푑푇 + ∫ 푑푇 = ∫ 푑푇 + ∫ 퐶푝,푚 푇 0 퐾 푇 푇푚푖푛 푇 0 퐾 푇 푇푚푖푛 ( ) 퐾

푇푚푖푛 퐶푝,푚 푇 푇 = ∫ 푑푇 + 2.303 ∫ 퐶푝,푚푑 log( ) 0 퐾 푇 푇푚푖푛 퐾

where Tmin is the minimum temperature to which the value of heat capacity is available. For liquids: The entropy for liquids at temperature T is determined as follows:

 The solid is heated from temperature 0 K to its melting point Tm.

 This solid is transformed to liquid at its melting point Tm.  Then the liquid is heated to temperature T. Thus, by adding these three processes we get:

푇푚 퐶푝,푚(푠) ∆푓푢푠퐻푚 푇 퐶푝,푚(푙) 푆푇 = ∫ 푑푇 + + ∫ 푑푇 0 퐾 푇 푇푚 푇푚 푇

For gases: Similarly for gases, the entropy is given by:

푇푚 퐶푝,푚(푠) ∆푓푢푠퐻푚 푇푏 퐶푝,푚(푙) ∆푣푎푝퐻 푇 퐶푝,푚(푔) 푆푇 = ∫ 푑푇 + + ∫ 푑푇 + + ∫ 푑푇 0 퐾 푇 푇푚 푇푚 푇 푇푏 푇푏 푇

8. Residual entropy

The entropies which are calculated using the Third law of thermodynamics are known as thermal entropies. But the expression for entropy suggested by Boltzmann is given by S = k ln W. These entropies are known as statistical entropies. Statistical entropies are larger in value as compared to thermal entropies. The difference ranges between 3.1 to 4.8 JK–1mol–1. It was CHEMISTRY Paper No. 10: Physical Chemistry- III (Classical Thermodynamics, Non-Equilibrium Thermodynamics, Surface chemistry, Fast kinetics) Module No. 11: Fugacity and Third law of thermodynamics

founded that entropy of substances like H2, D2, CO, NO, N2O, H2O etc. are not zero at 0 K. Such entropies are known as residual entropies. Their existence is mainly due to the alternative arrangements of molecules in solid.

9. Summary

 Fugacity is used for representing the real behavior of Vander Waal’s gases which is different from behavior of ideal gases.  Fugacity coefficient is defined as the ratio of fugacity of a real gas to the actual pressure P of the gas. It is represented as: 푓 = 훾 푝  The third law of thermodynamics is formulated by the German chemist, Walther Nernst. This law states that at absolute zero temperature, the entropy of each perfectly crystalline solid becomes zero.  According to third law of thermodynamics the entropy is given by following expression: 푇 퐶 푆 = ∫ 푝,푚 푑푇 푇 0퐾 푇

CHEMISTRY Paper No. 10: Physical Chemistry- III (Classical Thermodynamics, Non-Equilibrium Thermodynamics, Surface chemistry, Fast kinetics) Module No. 11: Fugacity and Third law of thermodynamics