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Physics LibreTexts

8: Thermodynamics

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Mathematical introduction

If there exists a relation f(x,y,z)=0 between 3 variables, one can write: x=x(y,z), y=y(x,z) and z=z(x,y). The total differential dz of z is than given by:

dz=(zx)ydx+(zy)xdy

By writing this also for dx and dy it can be shown that

(xy)z(yz)x(zx)y=1

Because dz is a total differential dz=0.

A homogeneous function of degree m obeys: εmF(x,y,z)=F(εx,εy,εz). For such a function Euler’s theorem applies:

mF(x,y,z)=xFx+yFy+zFz

Definitions

  • The isochoric pressure coefficient: βV=1p(pT)V
  • The isothermal compressibility: κT=1V(Vp)T
  • The isobaric volume coefficient: γp=1V(VT)p
  • The adiabatic compressibility: κS=1V(Vp)S

For an ideal gas it follows that: γp=1/T, κT=1/p and βV=1/V.

Thermal heat capacity

  • The specific heat at constant X is: CX=T(ST)X
  • The specific heat at constant pressure: Cp=(HT)p
  • The specific heat at constant volume: CV=(UT)V

For an ideal gas : CmpCmV=R. Further, if the temperature is high enough to thermalize all internal rotational and vibrational degrees of freedom: CV=12sR. Hence Cp=12(s+2)R. From their ratio it now follows that γ=(2+s)/s. For a lower T one needs only to consider the thermalized degrees of freedom. For a Van der Waals gas: CmV=12sR+ap/RT2.

In general:

CpCV=T(pT)V(VT)p=T(VT)2p(pV)T0

Because (p/V)T is always <0, the following is always valid: CpCV. If the coefficient of expansion is 0, Cp=CV, and this is true also at T=0K.

The laws of thermodynamics

The zeroth law states that heat flows from higher to lower temperatures. The first law is the conservation of energy. For a closed system: Q=ΔU+W, where Q is the total added heat, W the work done and ΔU the difference in the internal energy. Notice that the work is taken as the work done by the system on the surroundings. In differential form this becomes: dQ=dU+dW, where d means that the it is not a differential of a state function. For a quasi-static process: dW=pdV. So for a reversible process: dQ=dU+pdV.

For an open (flowing) system the first law is: Q=ΔH+Wi+ΔEkin+ΔEpot. One can extract an amount of work Wt from the system or add Wt=Wi to the system. In chemistry, one uses the opposite convention, that positive work is work done on the system.

The second law states: for a closed system there exists an additive quantity S, called the entropy, the differential of which has the following property:

dSdQT

If the only processes occurring are reversible: dS=dQrev/T. So, the entropy difference after a reversible process is:

S2S1=21dQrevT

For a reversible cycle: dQrevT=0.

For an irreversible cycle: dQirrT<0.

The third law of thermodynamics is (Nernst's law):

limT0(SX)T=0

From this it can be concluded that the thermal heat capacity 0 if T0, so absolute zero temperature cannot be reached by cooling through a finite number of steps.

State functions and Maxwell relations

The state functions and their differentials are:

Internal energy: U dU=TdSpdV
Enthalpy: H=U+pV dH=TdS+Vdp
Free energy: F=UTS dF=SdTpdV
Gibbs free energy: G=HTS dG=SdT+Vdp

From this one can derive Maxwell’s relations:

(TV)S=(pS)V ,  (Tp)S=(VS)p ,  (pT)V=(SV)T ,  (VT)p=(Sp)T

From the total differential and the definitions of CV and Cp it can be derived that:

TdS=CVdT+T(pT)VdV  and  TdS=CpdTT(VT)pdp

For an ideal gas:

Sm=CVln(TT0)+Rln(VV0)+Sm0  and  Sm=Cpln(TT0)Rln(pp0)+Sm0

Helmholtzequations are:

(UV)T=T(pT)Vp  ,  (Hp)T=VT(VT)p

For a macroscopic surface: dWrev=γdA, with γ the surface tension. From this follows: 

γ=(UA)S=(FA)T

Processes

The efficiency η of a process is given by: η=Work doneHeat added

The cold factor ξ of a cooling down process is given by: ξ=Cold deliveredWork added

Reversible adiabatic processes

For adiabatic processes: W=U1U2. For reversible adiabatic processes using Poisson’s equation with γ=Cp/CV one gets that pVγ=constant. Also: TVγ1=constant. Also Tγp1γ=constant. Adiabats are steeper on a p-V diagram than isotherms because γ>1.

Isobaric processes

Here: H2H1=21CpdT. For a reversible isobaric process: H2H1=Qrev.

Throttle processes

This is also called the Joule-Kelvin effect and is the result of an adiabatic expansion of a gas through a porous material or a small opening. Here H is a conserved quantity, and dS>0. In general this is accompanied with a change in temperature. The quantity which is important here is the throttle coefficient:

αH=(Tp)H=1Cp[T(VT)pV]

The inversion temperature is the temperature where an adiabatically expanding gas maintains the same temperature. If T>Ti the gas heats up, if T<Ti the gas cools down. Ti=2TB, with for TB: [(pV)/p]T=0. The throttle process is, for example, used in refrigerators.

The Carnot Cycle

The system undergoes a reversible cycle consisting of two isotherms and two adiabats

  1. Isothermal expansion at T1. The system absorbs an amount of heat Q1 from the reservoir.
  2. Adiabatic expansion with a temperature drop to T2
  3. Isothermal compression at T2 removing an amount of heat Q2 from the system.
  4. Adiabatic compression with the temperature increasing to T1

The efficiency for a Carnot cycle is:

η=1|Q2||Q1|=1T2T1:=ηC

The Carnot efficiency ηC is the maximal efficiency at which a heat engine can operate. If the process is applied in reverse order and the system performs a work W the cold factor is given by:

ξ=|Q2|W=|Q2||Q1||Q2|=T2T1T2

The Stirling cycle

The Stirling cycle consists of 2 isotherms and 2 isochoric processes. The efficiency in the ideal case is the same as for a Carnot cycle.

Maximum work

Consider a system that changes from state 1 into state 2, with the temperature and pressure of the surroundings given by T0 and p0. The maximum work which can be obtained from this change is, when all processes are reversible:

  1. Closed system: Wmax=(U1U2)T0(S1S2)+p0(V1V2).
  2. Open system: Wmax=(H1H2)T0(S1S2)ΔEkinΔEpot.

The minimum work needed to attain a certain state is: Wmin=Wmax.

Phase transitions

Phase transitions are isothermal and isobaric, so dG=0. When the phases are indicated by α, β and γ: Gαm=Gβm and

ΔSm=SαmSβm=rβαT0

where rβα is the heat of transition from β to phase α and T0 is the transition temperature. The following holds: rβα=rαβ and rβα=rγαrγβ. Further

Sm=(GmT)p

so G has a kink in the transition point and the derivative is discontinuous. In a two phase system Clapeyron’s equation is valid:

dpdT=SαmSβmVαmVβm=rβα(VαmVβm)T

For an ideal gas one finds for the vapor line at some distance from the critical point:

p=p0erβα/RT

There also exist also phase transitions with rβα=0. For those there will only be a discontinuity in the second derivatives of Gm. These second-order transitions appear as organization phenomena.

A phase-change of the 3rd order, with e.g. [3Gm/T3]p non continuous arises e.g. when ferromagnetic iron changes to the paramagnetic state.

Thermodynamic potential

When the number of particles within a system changes this number becomes a third state function. Because addition of matter usually takes place at constant p and T, G is the relevant quantity. If a system has many components this becomes:

dG=SdT+Vdp+iμidni where μ=(Gni)p,T,nj

is called the thermodynamic potential. This is a partial quantity. For V:

V=ci=1ni(Vni)nj,p,T:=ci=1niVi

where Vi is the partial volume of component i. The following holds:

Vm=ixiVi0=ixidVi

where xi=ni/n is the molar fraction of component i. The molar volume of a mixture of two components can be a concave line in a V-x2 diagram: the mixing leads to a contraction of the volume

The thermodynamic potentials are not independent in a multiple-phase system. It can be derived that inidμi=SdT+Vdp, this gives at constant p and T: ixidμi=0 (Gibbs-Duhmen).

Each component has as many μ’s as there are phases. The number of free parameters in a system with c components and p different phases is given by f=c+2p which is called the Gibbs phase rule

Ideal mixtures

For a mixture of n components (the index 0 is the value for the pure component):

Umixture=iniU0i  ,  Hmixture=iniH0i  ,  Smixture=nixiS0i+ΔSmix

where for ideal gases: ΔSmix=nRixiln(xi).

For the thermodynamic potentials: μi=μ0i+RTln(xi)<μ0i. A mixture of two liquids is rarely ideal: this is usually only the case for chemically related components or isotopes. In spite of this Raoult’s law holds for the vapour pressure for many binary mixtures: pi=xip0i=yip. Here xi is the fraction of the ith component in liquid phase and yi the fraction of the ith component in gas phase.

A solution for one component in a second gives rise to an increase in the boiling point ΔTk and a decrease of the freezing point ΔTs. For x21:

ΔTk=RT2krβαx2  ,  ΔTs=RT2srγβx2

with rβα the heat of evaporation and rγβ<0 the melting heat. For the osmotic pressure Π of a solution: ΠV0m1=x2RT.

These are called collegative properties

Conditions for equilibrium

When a system evolves towards equilibrium the only changes that are possible are those for which: (dS)U,V0 or (dU)S,V0 or (dH)S,p0 or (dF)T,V0 or (dG)T,p0. In equilibrium for each component: μαi=μβi=μγi.

Statistical basis for thermodynamics

The number of possibilities P to distribute N particles on n possible energy levels, each with a g-fold degeneracy is called the thermodynamic probability and is given by:

P=N!igniini!

The most probable distribution, that with the maximum value for P, is the equilibrium state. When Stirling’s equation, ln(n!)nln(n)n is used, one finds the Maxwell-Boltzmann distribution for a discrete system . The occupation numbers in equilibrium are then given by:

ni=NZgiexp(WikT)

The state sum Z is a normalization constant, given by: Z=igiexp(Wi/kT). For an ideal gas:

Z=V(2πmkT)3/2h3

The entropy can then be defined as:  S=kln(P). For a system in thermodynamic equilibrium this becomes:

S=UT+kNln(ZN)+kNUT+kln(ZNN!)

For an ideal gas, with U=32kT then: S=kN+kNln(V(2πmkT)3/2Nh3)

Application to other systems

Thermodynamics can be applied to other systems than gases and liquids. To do this the term dW=pdV has to be replaced with the correct work term, for example dWrev=Fdl for the stretching of a wire, dWrev=γdA for the expansion of a soap bubble or dWrev=BdM for a magnetic system.

A rotating, non-charged black hole has a temperature T=c/8πkm. It has an entropy S=Akc3/4κ with A the area of its event horizon. For a Schwarzschild black hole A is given by A=16πm2. Hawkings area theorem states that dA/dt0.

Hence, the lifetime of a black hole m3.


This page titled 8: Thermodynamics is shared under a CC BY license and was authored, remixed, and/or curated by Johan Wevers.

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