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Wize University Physics Textbook (Master) > Thermodynamics

Entropy

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Entropy

Entropy is a quantitative measure of disorder or randomness in a system and is defined as:

ds=dQTds=\dfrac{dQ}{T}




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Entropy for different thermodynamic processes:

a) Isochoric Processes:

Δs=∫ds=∫dQT=∫nCvdTT=nCv∫dTT=nCvln⁡TfTi\Delta s=\displaystyle\int ds=\displaystyle\int\frac{dQ}{T}=\displaystyle\int nC_{v}\frac{dT}{T}= nC_{v}\displaystyle\int\frac{dT}{T}=nC_{v}\ln\frac{T_{f}}{T_{i}}







b) Isobaric Processes:

Δs=∫nCp dTT=nCpln⁡ TfTi\Delta s=\displaystyle\int_{ }^{ }nC_p\ \frac{dT}{T}=nC_p\ln\ \frac{T_f}{T_i}



c) Isothermal Processes:


Δs=∫dQT=1T∫dQ=QT=nRTln⁡(vfvi)T=nRln⁡ vfvi\Delta s=\displaystyle\int_{ }^{ }\frac{dQ}{T}=\frac{1}{T}\displaystyle\int_{ }^{ }dQ=\frac{Q}{T}=\frac{nRT\ln\left(\frac{v_f}{v_i}\right)}{T}=nR\ln\ \frac{v_f}{v_i}


d) Adiabatic Processes:

Δs=0\Delta s=0




More discussion on entropy!


ΔSuniverse=ΔSsystem+ΔSenvironment\Delta S_{universe}=\Delta S_{system}+\Delta S_{environment}


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Example: Internal Energy and Entropy

Two moles of a diatomic gas are expanding from 2 liters to 4 liters through an unknown process. During this process, the pressure of the gas is changed from 3atm to 5atm.
a) Find the change in the internal energy of the gas.
b) Find the change in the entropy of the system.

a) ΔU=nCvΔT=52nR(TB−TA)=52(PBVB−PAVA)\Delta U=nC_v\Delta T=\frac{5}{2}nR\left(T_B-T_A\right)=\frac{5}{2}\left(P_BV_B-P_AV_A\right)

=52(5×105×4×10−3−3×105×2×10−2)=3500J=\frac{5}{2}\left(5\times10^5\times4\times10^{-3}-3\times10^5\times2\times10^{-2}\right)=3500J

b) Since ΔS\Delta S does not depend on path, we consider combination of a isochoric and isobaric process which connects points A and B as it is shown in the PV-Curve

ΔSa→c=nCpln⁡(TcTa)=72nRln⁡(PcVcPaVa)\Delta S_{a\to c}=nC_p\ln\left(\frac{T_c}{T_a}\right)=\frac{7}{2}nR\ln\left(\frac{P_cV_c}{P_aV_a}\right)

ΔSc→b=nCvln⁡(TbTc)=52nRln⁡(PbVbPcVc)\Delta S_{c\to b}=nC_v\ln\left(\frac{T_b}{T_c}\right)=\frac{5}{2}nR\ln\left(\frac{P_bV_b}{P_cV_c}\right)

→ΔSab=72(2)(8.31)ln⁡(3(4)3(2))→52(2)(8.31)ln⁡(5(4)3(4))\to\Delta S_{ab}=\frac{7}{2}\left(2\right)\left(8.31\right)\ln\left(\frac{3\left(4\right)}{3\left(2\right)}\right)\to\frac{5}{2}\left(2\right)\left(8.31\right)\ln\left(\frac{5\left(4\right)}{3\left(4\right)}\right)

=61.545 JK=61.545\ \frac{J}{K}

Extra Practice
Entropy: Heat Engines
A cylinder contains Nitrogen gas (𝐶v = 20.8Jmol.K\frac{J}{mol.K} , 𝛾 = 1.4) at a pressure of 3 atm and a temperature of 250K. Its initial volume is 2.0 liters. The Nitrogen gas is next carried through the following processes:
a. It is heated in an isochoric process to a temperature of 375K
b. It is expanded at a constant temperature to a volume of 3.0 liters
Thermodynamics: Entropy
If 3.0 kg of water at 100.0℃ is mixed with 1.0kg of ice at −10.0℃ . Find the
final temperature and the total change in entropy of the system.
(Cw=4190 Jkg⋅K, Ci=2100 Jkg⋅K, Lf=334×103 JKg)\left(C_w=4190\ \frac{J}{kg\cdot K},\ C_i=2100\ \frac{J}{kg\cdot K},\ L_f=334\times10^3\ \frac{J}{Kg}\right)
Entropy: Heat Engines
A cylinder contains a diatomic gas at a pressure of 3 atm and a temperature of 250K. Its initial volume is 4.0 liters. The gas is next carried through the following processes:
a. It is expanded isobarically to a temperature of 375K.
b. It is cooled down to 250K in an isochoric process.
How does entropy differ from energy?
Predict the sign of the ΔH, ΔS,(+ or -) and identify when the following reactions are spontaneous at low temperature, high temperature, or all temperatures (low, high, all). Use commas to write your answers ie. +,+,low
For freezing of a popsicle: ΔH is ___________, ΔS is ___________. Reaction is spontaneous at ____________________ temperatures
For evaporation of liquid water: ΔH is ___________, ΔS is ___________. Reaction is spontaneous at ____________________ temperatures
Arrange the following compounds in order of increasing entropy, assuming 1 mol of each compound:
Ar(g)Fe(s)CH3CH2CH3(g)HOCH2CH2CH2OH(l)Ar(g) \hspace{15pt}Fe(s)\hspace{15pt} CH_3CH_2CH_{3(g)} \hspace{15pt} HOCH_2CH_2CH_2OH_{(l)}
Ar(g)Ar(g)