Subject Chemistry
Paper No and Title 10, Physical Chemistry- III (Classical Thermodynamics, Non-Equilibrium Thermodynamics, Surface chemistry, Fast kinetics) Module No and Title 9, Entropy
Module Tag CHE_P10_M9
CHEMISTRY Paper No. 10: Physical Chemistry- III (Classical Thermodynamics, Non-Equilibrium Thermodynamics, Surface chemistry, Fast kinetics) Module No. 9: Entropy
TABLE OF CONTENTS
1. Learning outcomes
2. Entropy
2.1 Theory
2.2 Mathematical form of entropy
2.3 Units of entropy
2.4 Characteristics of entropy
3. Entropy change
3.1 Change in entropy in an isothermal expansion of an ideal gas
3.2 Entropy change in reversible and irreversible processes
3.3 Entropy change of an ideal gas due to variation in P, V and T
3.4 Change in entropy of an ideal gas in different processes
3.5 Entropy of mixing
4. Physical significance of entropy
5. Questions
6. Summary
CHEMISTRY Paper No. 10: Physical Chemistry- III (Classical Thermodynamics, Non-Equilibrium Thermodynamics, Surface chemistry, Fast kinetics) Module No. 9: Entropy
1. Learning outcomes
After studying this module you shall be able to: Learn about the concept of entropy Know the dependence of entropy on temperature in reversible and irreversible expansion Know the effect of variation of temperature, pressure and volume on entropy Calculate the change in entropy in isothermal, isochoric and isobaric processes Learn about the relationship between entropy and thermodynamic probabilty
2. Entropy
2.1 Theory The concept of entropy is vital for thermodynamics. The concept of temperature was used in zeroth law of thermodynamics while the concept of internal energy was used in first law of thermodynamics. These are state variables which define the state of the system. The state variable which is related to second law of thermodynamics is entropy (S). The concept of entropy was introduced by Clausius in 1865. It is considered as degree of randomness or disorderness in a molecular system. More the randomness, larger is the value of entropy. The word entropy has been made from energy (en) and trope (tropy) which is for chaos. According to one of the statements of second law of thermodynamics, heat cannot be completely converted into work. This has been attributed to entropy. In thermodynamics, entropy is referred as the measure of the number of definite realizations or microstates that may realize a thermodynamic system, in a specified state defined by macroscopic variables. More precisely, the entropy is considered as a measure of molecular disorder within a macroscopic system. The second law of thermodynamics states that entropy of an isolated system never decreases. This kind of system spontaneously moves towards state with maximum entropy, i.e., at thermodynamic equilibrium. The system which is not isolated may lose entropy only when the entropy of environment is increased by same amount. As entropy is a state function, therefore the change in entropy of the system will remain constant for any process with known initial and final states. The change in entropy does not depend upon the nature of the process, i.e., whether the process is reversible or irreversible. But the irreversible processes raise the combined entropy of the system and its environment. As on increasing the temperature of the system, kinetic energy of the molecules increases; therefore the degree of randomness increases thereby increasing the entropy of the system.
2.2 Mathematical form of entropy The change in entropy is defined as
CHEMISTRY Paper No. 10: Physical Chemistry- III (Classical Thermodynamics, Non-Equilibrium Thermodynamics, Surface chemistry, Fast kinetics) Module No. 9: Entropy
dq dS r ev …(1) T In the above expression q is the heat transfer considered only during the reversible process. While the entropy change for an irreversible process is not related to the heat transfer during an irreversible process. q is a path function. It can be shown that entropy is a state function. For finite change and at constant temperature conditions, dS becomes S and dq becomes q. Thus equation (1) becomes = q/T …(2) Since entropy is a state function, thus change in entropy for change of state from A and B will be same whether the change is reversible or not. Mathematically, the above equation may be written as: B dq S …(3) A T This equation is valid only when the reversible change is brought. Thus we can write S qrev / T …(4)
2.3 Units of entropy Since change in entropy is given by dq dS= rev T i.e., heat term divided by the absolute temperature. Thus SI unit of entropy is given by Joules per degree Kelvin (J K-1). For molar entropy, the unit is J K-1mol-1.
2.4 Characteristics of entropy Entropy is refer to as degree of randomness. As the degree of randomness increases, the entropy of the system increases. Entropy can also be related to unavailable energy. For a given substance, the vapour phase has maximum randomness and solid phase has least randomness. Therefore, Svapour > Sliquid > Ssolid Entropy is a state function. Therefore, ΔS = Sfinal – Sinitial = S2 – S1 It depends upon temperature, pressure, volume and amount. Entropy is an extensive property. dS ≠ dqirr / T Entropy increases with increase in temperature. The entropy increases as the volume of the system increases.
3. Entropy change
Now we shall find the change in entropy in different processes:
3.1 Change in entropy in an isothermal expansion of an ideal gas
CHEMISTRY Paper No. 10: Physical Chemistry- III (Classical Thermodynamics, Non-Equilibrium Thermodynamics, Surface chemistry, Fast kinetics) Module No. 9: Entropy
When an ideal gas is expanded isothermally and reversibly, then the internal energy does not changes, i.e., U = 0 thus from first law of thermodynamics (viz., U = q + w), we get: 푛푅푇푑푉 qwPdV = …(5) rev 푉 Thus change in entropy will be given by:
푉 푞 ∆푆 = ∫ 2 푟푒푣 …(6) 푉1 푇
푉 푑푉 ∆푆 = ∫ 2 푛푅 ( ) …(7) 푉1 푉
푉 ∆푆 = 푛푅 푙푛 ( 2) …(8) 푉1
3.2 Entropy change in reversible and irreversible processes Suppose an ideal gas is expanded isothermally in vacuum keeping temperature to be constant. This process will proceed irreversibly. As no opposing force is applied, therefore the work done (w) by the system will be zero. As temperature is kept constant, thus change in internal energy will also be zero. According to first law equation then q = 0, i.e., no heat is exchanged between the system and surroundings during the process. Thus, the entropy of the surroundings does not change. The entropy of the system is defined by the state of the system, i.e., temperature and pressure (or volume). During this process let the volume changes from V1 to V2. Therefore, the increase in entropy is given by (for one mole of gas):
S R ln (V21 / V ) …(9)
푉 Or ∆푆 = 푛푅 푙푛 2 (for n moles of an ideal gas) 푉1
Thus total increase in entropy of the system and its surroundings during isothermal expansion is R ln (V/2121 V )0R ln V/ V …(10)
As VV21 , thus we can say that there is increase in entropy of the system and the surroundings during the irreversible isothermal expansion. Now considering the reversible isothermal expansion of an ideal gas at the temperature T. Suppose the volume changes from V1 to V2 during infinitesimally slow expansion. Expansion is done such that the pressure on the frictionless piston remains always less than that of the gas by an infinitesimally small amount. Therefore some work is done by the system, i.e., w = -P V . Thus at this temperature T, equivalent amount of heat is absorbed by the system. Accordingly the increase in entropy is given by qrev/T. Similarly entropy of the surroundings decrease by qrev /T. Conventionally applying the negative sign for decrease of entropy and positive for increase of entropy, thus the net entropy change of the system and the surroundings is given by:
CHEMISTRY Paper No. 10: Physical Chemistry- III (Classical Thermodynamics, Non-Equilibrium Thermodynamics, Surface chemistry, Fast kinetics) Module No. 9: Entropy
q/Tq/T0revrev …(11) Thus total entropy change during isothermal expansion of a gas is zero. Conclusion: During an irreversible isothermal expansion there is always increase in an entropy of the system and surroundings taken together while during reversible isothermal expansion the entropy of the system and surroundings remains same. This can be written as
(SS)0syssur (for reversible process)
(SS)0syssur (for irreversible process)
Or combination of the above tow equations can be written as:
SS0syssur …(12)
in which equal to sign stands for the reversible process and ‘ greater than’ sign stands for irreversible process. Thus, the above equation helps in determining whether the process occurs spontaneously or not. Since all the naturally occurring processes are spontaneous in nature thus we can say that the entropy of the universe increases continuously. This statement is also regarded as another statement of the second law. The combination of first law and second law of thermodynamics is given by Clausius which is: The energy of the universe remains same while the entropy of the universe increases continuously towards the maximum. 3.3 Entropy change of an ideal gas due to variation in P, V and T Entropy of the system depends upon the state of the system, i.e., on T, P and V. For a pure gaseous substance the value of entropy is dependent only on two of the three variables P,V and T. Generally T is taken as one of the variable therefore two variables to be considered are either T and V or T and P. Case 1: When the two variables are T and V
dS dqrev / T …(13)
where dqrev is the infinitesimal amount of heat absorbed by the system at temperature T. According to first law of thermodynamics U = q + w, we get:
dqdUdwrev …(14)
CHEMISTRY Paper No. 10: Physical Chemistry- III (Classical Thermodynamics, Non-Equilibrium Thermodynamics, Surface chemistry, Fast kinetics) Module No. 9: Entropy
If work is done due to expansion of the gas , then infinitesimal change in volume will be given by: – dw = PdV …(15)
Also C(U/T)VV …(16) Substituting equation (15) and (16) in equation (14), we get
dqCdTPdVrevV …(17) Assuming the gas to be ideal, thus applying ideal gas equation in equation (17) we get, (for one mole of gas)
dqCdTRTrevV dV / V …(18) On rearranging the above equation we get,
dq/revV TdSCdT / TR dV / V …(19) Therefore, the entropy change for finite change of state of the system is given by:
TV22dTdV SSSCR21V …(20) TV11TV
In above equation, integration is done between initial state 1 and final state 2. And CV is
assumed to be constant within the temperature range T1 and T2. After integration, we get :
SCln T/V2121 TR ln V/ V …(21) Similarly, the change in entropy for n moles of the gas is given by:
S nCV ln(T 2 /T) 1 nRln(V 2 /V) 1 …(22) Thus from the above equation it is evident that the change in entropy depends on the initial and final volume and temperature of the system. Case 2: When two variables are T and P
Let pressure of the gas at initial and final state of an ideal gas be P1 and P2 respectively. Thus, accordingly
At initial state, P111 VRT …(23)
At final state , P2 V 2 RT 2 …(24) CHEMISTRY Paper No. 10: Physical Chemistry- III (Classical Thermodynamics, Non-Equilibrium Thermodynamics, Surface chemistry, Fast kinetics) Module No. 9: Entropy
Dividing equation (24) by (23), we get:
V/VPT/(PT)2 1 1 2 2 1 …(25) Substituting in equation (20) for one mole of the gas, we get:
S CV lnT/T 2 1 RlnT/T 2 1 RlnP/P 2 1 …(26)
Since C CPV R , substituting this in above equation we get,
S CP lnT/T 2 1 RlnP/P 2 1 …(27) Similarly for n moles of the ideal gas,
3 SnCln (T/ T )nR ln (P/ P ) …(28) P2121 2 Thus from the above equation it is evident that the change in entropy of an ideal gas depends upon initial and final pressure as well as initial and final temperature.
3.4 Change in entropy of an ideal gas in different processes Considering three different processes i.e. (1) Isothermal processes (2) Isobaric processes (3) Isochoric processes
Considering each process on detail: (1) Isothermal process Since in isothermal process, the temperature remains constant. Thus from equation (21) an d (27), we get:
SRln(VT212112 /V)Rln(P /P)Rln(P /P ) …(29) The subscript T indicates that the temperature of an ideal gas does not change during the
process. If VV21 , in that case the change in entropy will be positive. Thus during isothermal expansion, the entropy of the system increases. While during isothermal compression, the entropy of the system decreases.
CHEMISTRY Paper No. 10: Physical Chemistry- III (Classical Thermodynamics, Non-Equilibrium Thermodynamics, Surface chemistry, Fast kinetics) Module No. 9: Entropy
(2) Isobaric process
During this process, the pressure of the system remains constant. Accordingly, PP . 12 Thus equation (27) changes to
SCln(T/T) …(30) PP21 Subscript P is indicating that the pressure of the system remains constant. Thus from equation (30) we can say that on increasing the temperature of an ideal gas, entropy of an ideal gas increases.
During this process the volume of the gas remains constant. Accordingly, VV Applying 12 this, the equation (21) turns to:
SCln(T/T) …(31) VV21 Thus, from above equation it is evident that on increasing the temperature of the gas, entropy of the gas increases.
3.5 Entropy of mixing Entropy of mixing can be defined as the difference between the entropy of the mixture of gases and the sum of the entropies of individual gases which are at pressure P. Thus,
Sn (C ln T R ln P R ln xS )n(C ln T R ln PS ) …(32) mixP0P0
= R n ln x R(n ln x n ln x n ln x ....) …(33) 1 1 2 2 3 3
= R n ln x …(34) ii
where n stands for the number of moles and x stands for the mole fraction of each constituent i i of the mixture. Total number of moles is given by :
n n n n .... …(35) 1 2 3 Dividing the equation (35) by n, we get
1 n/n n/n n/n .... x x x .... x …(36) 1 2 3 1 2 3 i CHEMISTRY Paper No. 10: Physical Chemistry- III (Classical Thermodynamics, Non-Equilibrium Thermodynamics, Surface chemistry, Fast kinetics) Module No. 9: Entropy
Thus, for 1 mole of mixture, the entropy of mixing is given by:
SRxlnx …(37) mixii
As x is the mole fraction thus entropy of mixing is always positive. i
4. Physical significance of entropy
Entropy as a Measure of the Disorder of the System. All the spontaneous processes occur with an increase in randomness of the system. Such spontaneous processes include flow of electricity, solute diffusion from concentrated to dilute solution, expansion of gas in vacuum, etc. Thus in spontaneous process, entropy as well as disorder of the system increases. Similarly in the melting of solid, the entropy of the system increases. This is because the solid has a definite crystal lattice in which atoms or molecules are arranged in definite order, while in liquid, order is much less thereby increasing the disorder. Thus entropy is referred to as measure of disorder of the system. Entropy as a Measure of Probability In spontaneous process, the less probable state (less stable state) is converted into more probable state (more probable state). Therefore entropy can also be related to the thermodynamic probability W of the particular state of the system. Both entropy and thermodynamic probability are directly proportional. This relationship was expressed by Boltzmann as S = k ln W + constant ...(38)
where k stands for Boltzmann constant which is equal to R/NA. Boltzmann also regarded thermodynamic probability as ratio of the probability of the actual state to the probability of the state in which there is complete order for the same energy and volume. But according the Planck, constant term in equation 38 is zero. Hence, S= k ln W …(39) The above equation is known as Boltzmann entropy equation. Thus, a solid is regarded as a most ordered state at absolute zero temperature. This implies
that thermodynamic probability should be unity, thus making entropy equal to zero, i.e., S0 = 0. Accordingly, the entropy of all crystalline solids at absolute zero temperature is taken to be zero.
5. Question CHEMISTRY Paper No. 10: Physical Chemistry- III (Classical Thermodynamics, Non-Equilibrium Thermodynamics, Surface chemistry, Fast kinetics) Module No. 9: Entropy
Ques 1: Explain the concept of entropy and discuss the nature of change in entropy for the following:
(i) H2O(l) H2O(s)
(ii) MgCO3(s) MgO(s) +CO2 (g)
(iii) 2SO2 (g)+ O2 (g) 2SO3 (g)
Solution: The extent of randomness of a system is called entropy (i) Entropy will decrease because solids have smaller number of random molecules. (ii) Entropy will increase because a substance in gaseous state is formed which has greater degree of randomness. (iii) Entropy decreases because the total number of molecules decrease.
Ques 3: The enthalpy change for the transition of liquid water to steam is Calculate the entropy change for the process. Solution:
as there is increase in the randomness of the system. Ques: Calculate the entropy change when one mole of an ideal gas expands reversibly from an initial volume of 1 dm3 to a final volume of 10 dm3 at a constant temperature of 298 K.
푉 Solution: ∆푆 = 푛푅 푙푛 ( 2) 푉1
푉 = 2.303 푛푅 푙표푔 ( 2) 푉1 CHEMISTRY Paper No. 10: Physical Chemistry- III (Classical Thermodynamics, Non-Equilibrium Thermodynamics, Surface chemistry, Fast kinetics) Module No. 9: Entropy
3 3 Here n = 1, V1= 1 dm , V2 = 10 dm ∆푆 = 푅 ln 10
= 2.303 푅 log 10 = 2.303 ∗ (8.314 퐽 퐾−1푚표푙−1) = 19.15 퐽 퐾−1푚표푙−1
6. Summary
The concept of entropy was introduced by Clausius in 1865. It is considered as degree of randomness (also referred as disorder) of a system. The change in entropy is defined as dq dS r ev T SI unit of entropy is given by Joules per degree Kelvin (J K-1)
Entropy is a state function. Therefore, ΔS = Sfinal – Sinitial = S2 – S1 It depends upon temperature, pressure, volume and amount of a substance Entropy is an extensive property. During an irreversible isothermal expansion, there is always increase in an entropy of the system and surroundings taken together while, during a reversible isothermal expansion, the entropy of the system and surroundings remains same. This can be written as ( Ssys S sur ) 0 (for reversible process)
(SS)0syssur (for irreversible process) During isothermal process the change is entropy is given by:
ST Rln(V/V) 2 1 Rln(P/P) 2 1 Rln(P/P) 1 2 During isobaric process, the change in entropy is given by: S C ln (T / T ) P P 2 1 During isochoric process, the change is entropy is given by: S C ln (T / T ) V V 2 1
CHEMISTRY Paper No. 10: Physical Chemistry- III (Classical Thermodynamics, Non-Equilibrium Thermodynamics, Surface chemistry, Fast kinetics) Module No. 9: Entropy
CHEMISTRY Paper No. 10: Physical Chemistry- III (Classical Thermodynamics, Non-Equilibrium Thermodynamics, Surface chemistry, Fast kinetics) Module No. 9: Entropy