LECTURE ONTHE CONCEPT OF SECOND LAW OF THERMODYNAMICS
BY
DR. NNABUK OKON EDDY
Outline
Introduction The need for the second law Concept of entropy Statements of the second law Properties of entropy Derivation of equation for entropy Consequences of the second law
Introduction The second law explains the phenomenon of
irreversibility in nature The need for the second law arises because
the first law failed in some aspects. For example,
It fails to explain why natural processes have a preferred direction
The first law fails to produce thermodynamic functions that can be used to predict the direction of a spontaneous reaction
The second law deals with entropy
Entropy
The key concept for the explanation of phenomenon through the second law is the definition of a physical property called entropy
Entropy is a measure of the degree of disorderliness of a system.
A change in entropy of a system is the infinitesimal transfer of heat to a close system driving a reversible process divided by the equilibrium temperature (T) of the system, i.e dS = dqrev /T
Statements of the second law No process is possible whose sole result is the
transfer of heat from a body of lower temperature to a body of higher temperature (Clausius-Mussoti)
No process is possible in which the sole result is the absorption of heat from a reservoir and its complete conversion into work (Kelvin-Plank)
Equivalent ways of stating the laws are i. the entropy of a spontaneous reaction
increases and tends toward a maximum ii. After any spontaneous reaction, work must
be converted to heat in order to restore the system to its initial state
Properties of entropy
Entropy is a state function: its properties depends on the initial and final state of the system
Entropy is additive. i.e ST = S1 + S2 + S3 + -----
Entropy is a probability function
Derivation of expression for entropy change
If the probability of finding a system in state 1 and 2 are W1 and W2, then the probability of finding the system in the two parts is the total probability, W = W1 x W2,
S(W) = S(W1) x S(W2) = S(W1) + S(W2) (1)
Conditions set by Eq 1 can only be fulfilled if entropy is logarithm dependent, i.eS = log(W1 x W2) = logW1 x logW2 (2)
Consider an ideal gas expanding into two systems joined together, the probabilities for the first and second is proportional to their respective volumes, therefore, W1 = aV1, W2 = aV2 and since S is additive, S = S2 β S1
= log(aV2) β log(aV1) = log(V2/V1)
From first law of thermodynamics, it can be shown that the reversible work done = reversible heat absorbed = nRTln(V2/V1) and if we multiply S by the constants, 2.303R, we have, qads = T x S. It therefore follows that S can be expressed as follows
S = qads/T (3)
Consequence of the second law of thermodynamics
We shall consider the following consequences of the 2nd law of thermodynamics,
Entropy change for an ideal gas Entropy of mixing ideal gases Carnot cycle Free energy change
Entropy change for an ideal gasdS = β« πππ = πΆπ£πππ + RT πππ 4
= CVln((π2π1) + Rln(π2π1) 5
From the general gas equation, it can be shown that π2π1 = (π1π2)( π2π1) and by substituting for V2/V1, we
have,
S = CVln((π2π1) + R[(ln(π1π2 ) + ln(π2π1)] 6
The relationship between Cp, CV and R can be written as Cp = CV + R, therefore equation 6 can be written as follows
= CPln((π2π1) - Rln ln(π2π1 ) 7
Equation 7 can be applied to three special cases as follows,
i. Isothermal condition, S = 2.303RTlog(π2π1)
ii. Isobaric condition, S = 2.303Cplogn((π2π1)
iii. Isochoric condition, S = 2.303CVlog((π2π1)
Entropy of mixing ideal gasesConsider two gases, A and B mixed together. The work done in mixing the gases can be defined as follows,
dW = WA + WB = PAdVA + PBdVB 8
From the ideal gas equation, P = nRT/V, therefore, PA = nART/VA and PB = nART/VB. substituting for PA and PB in equation 8 yields equation 9 and upon simplication, equation 10 is obtained,
β«dW = nAπ π ππ΄ππ΄ππ΄+ππ΅ πππ΄ + RT ππ΅ππ΅
ππ΄+ππ΅ πππ΅ ) 9
W = RTln(nA + nB)/nA + RTln(nA + nB)/nB 10
= nARTln( 1π₯π΄) + nB RTln( 1π₯π΅) 11
Also, since W = dqads and S = πππππ π , S can be expresses as follows,
S = -2.303RT(nAlnXA + nBlnXB) 12
Reversible cycle and efficiency: Carnot cycle
A cycle process in which a succession of changes occurs as a results of which the system returns to its original state and all properties assume their original values. This findings was made by Carnot and the cycle is commonly called Carnot cycle. Carnot cycle consist of four major components
i. Reversible isothermal expansion ii. Adiabatic reversible expansion iii. Isothermal reversible compression iv. Adiabatic reversible compression
In step i., the change in internal energy is zero, and S = - π2π2. In step ii, q = 0 and S = 0. In
step iii, S = - π1π1 and the change in internal energy = 0. Finally in step iv, q=0 and S = 0
In a carnot cycle, the heat absorbed = q2 β q1 and the efficiency is defined as the fraction of the workdone to the heat absorbed at higher temperature. i.e Efficiency = W/q2. Also the total entropy change in the cycle is q2/T2 β q1/T1 = 0. Therefore, q2/q1 = T2/T1 hence
Efficiency = (q2 β q1)/q2 = (T2 β T1)/T2 13
Free energy Enthalpy and entropy are state functions obtained from the first and second law of thermodynamics respectively. Enthalpy measures the tendency towards orderliness while entropy represents the tendency towards disorderliness. The difference between orderliness and diorderliness leads to the concept of free energy, which can be expressed as follows,
G = H - TS 14
Note, H = E + PV , therefore,
G = E + PV - TS 15
Differentiating both sides of equation 15 yields equation 16
dG = dE + VdP + PdV - TdS + SdT 16
Note, qads = dE + PdV and qads = TdS and by substitution, equation 16 simplifies to equation 17
dG = VdP - SdT 17
Effect of pressure and temperature
At constant temperature,
dG = VdP and dGππ = V. Also, since PV = RT, V = π ππ , then
dG = RTπππ 18
integration of equation 18 yields
dG = RTlnπ2π1 19
At constant pressure, dG = -SdT and (dGππ)p = -S
Also, (dG2ππ)p - (dG1ππ)p = -S = S2 - S1 20
But G βHπ = -S 21
Then, πππ(βGπ) = (1π) πππ₯ (βG) - (βG) πππ(1π) = 1π(G βHπ ) - βGπ2 = -βHπ2 22
Since From equation 22, it can be deduced that a plot of βGπ versus 1/T should give a
straight line with slope equal to H
Note, πππ(βGπ ) = - βHπ2 and πππ(1π) = 1/T2, therefore πππ((G/T1/π) = H 23
S and spontaneousity of a reaction
When S is positive, spontaneous reaction
When S is zero, reaction at equilibrium When S is negative, non spontaneous Limitation is that we who measures the
entropy are part of the environment. Therefore S is not a unique parameter for predicting the direction of a chemical reaction
G and spontaneousity of a reaction
G > 0, non spontaneous (H > TS) G < 0, spontaneous (H < TS) G = 0, reaction at equilibrium (H = TSG is a state function obtained at constant pressure.
At constant volume the state function is work function expressed as
A = E - TSWhen A > 0, spontaneous
When A <0 , non spontaneous
When A = 0, at equilibrium
CONCLUSION
Thermodynamic function obtained from the second law is entropy
Entropy is a measure of disorderliness while enthalpy measures orderliness
Entropy data must be combined with enthalpy (or internal energy data) in order to predict the direction of a chemical reaction
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