Thermodynamics of Multicomponent Systems

Multicomponent systems and composition variables

A closed system with variable composition can, under some changing conditions, behave in much the same way as an open system. Here “closed” does not mean absolutely isolated; it only means that total matter is conserved.

  • A closed system with variable composition can be treated analogously to an open system.

State description for a variable-composition system:

  • Extensive variables: V=(T,p,nB,nC)V = (T, p, n_B, n_C), giving a total of k+2k + 2 variables.
    • These are the mole numbers of each substance together with temperature and pressure.
  • Intensive variables: ρ=ρ(T,p,xB,xC)\rho = \rho(T, p, x_B, x_C), giving a total of k+1k + 1 variables.
    • Since these are relative amounts, one variable is effectively removed.

Partial molar quantities

Definition
  • These describe how a thermodynamic quantity changes when the mole number of a component changes.
  • Because the definition is based on mole number, partial molar quantities are associated with extensive properties.
  • A partial molar quantity is itself a state function, so its value no longer depends directly on the total amount nn.
  • Common examples:

VB=(∂V∂nB)T,p,bCV=∑BnBVBV_B=\left( \frac{\partial V}{\partial n_B} \right)_{T,p,b_C}\qquad V=\sum_{B}n_{B}V_{B}

UB=(∂U∂nB)T,p,bCU=∑BnBUBU_B=(\frac{\partial U}{\partial n_B})_{T,p,b_C}\qquad U=\sum_{B}n_{B}U_{B}

HB=(∂H∂nH)T,p,bCH=∑BnBHBH_B=(\frac{\partial H}{\partial n_H})_{T,p,b_C}\qquad H=\sum_{B}n_{B}H_{B}

FB=(∂F∂nB)T,p,bCF=∑BnBFBF_B=(\frac{\partial F}{\partial n_B})_{T,p,b_C}\qquad F=\sum_{B}n_{B}F_{B}

SB=(∂S∂nB)T,p,bCS=∑BnBSBS_B=(\frac{\partial S}{\partial n_B})_{T,p,b_C}\qquad S=\sum_{B}n_{B}S_{B}

GB=(∂G∂nB)T,p,bCG=∑BnBGBG_B=(\frac{\partial G}{\partial n_B})_{T,p,b_C}\qquad G=\sum_{B}n_{B}G_{B}

  • The same relations between thermodynamic functions for pure substances or fixed-composition systems also hold for partial molar quantities:
    • HB=UB+pVBH_B = U_B + pV_B
    • FB=UB−TSBF_B = U_B - TS_B
    • GB=HB−TSB=UB+pVB−TSB=FB+pVBG_B = H_B - TS_B = U_B + pV_B - TS_B = F_B + pV_B
    • In other words, the usual thermodynamic identities still hold on a per-mole basis.
Additive formula

The sum over all components gives the total quantity:

  • ∑nBVB=V\sum n_B V_B = V
Gibbs-Duhem formula

If we differentiate a state description with respect to TT, pp, and the component amounts, we obtain:

dV=(∂V∂T)p,nB,nC,…dT+(∂V∂p)T,bB,nC,…dp+∑VBdnBdV=(\frac{\partial V}{\partial T})_{p,n_B,n_C,\dots}dT+(\frac{\partial V}{\partial p})_{T,b_B,n_C,\dots}dp+\sum V_Bdn_B

Chemical potential

Definition

The chemical potential is the partial derivative of GG with respect to nn. Since Gibbs free energy combines the system’s internal energy with its capacity to do useful work on the surroundings, it is especially useful when describing chemical reactions and phase changes.

μB=(∂G∂nB)T,p,bC,…\mu_B=(\frac{\partial G}{\partial n_B})_{T,p,b_C,\dots}

  • Similar to other partial molar quantities, it satisfies the same general formal properties.

  • To be honest, some textbook discussions around it feel more verbose than enlightening.

  • The absolute value of μB\mu_B depends on the chosen reference, which is why it is called a potential.

  • Combining it with the basic thermodynamic identities gives:

    • μ=(∂U∂nB)S,V,nc,⋯\mu=(\frac{\partial U}{\partial n_{B}})_{S,V,n_{c},\cdots}
    • μ=(∂H∂nB)S,p,nc,⋯\mu=(\frac{\partial H}{\partial n_{B}})_{S,p,n_{c},\cdots}
    • μ=(∂H∂nB)T,V,nc,⋯\mu=(\frac{\partial H}{\partial n_{B}})_{T,V,n_{c},\cdots}

  • Under different constraints, the chemical potential may be written as:

μB=(∂G∂nB)T,p,nC,…\mu_B=(\frac{\partial G}{\partial n_B})_{T,p,n_C,\dots}

μB=(∂U∂nB)S,V,nC,…\mu_B=(\frac{\partial U}{\partial n_B})_{S,V,n_C,\dots}

μB=(∂H∂nB)S,p,nC,…\mu_B=(\frac{\partial H}{\partial n_B})_{S,p,n_C,\dots}

μB=(∂F∂nB)T,V,nC,…\mu_B=(\frac{\partial F}{\partial n_B})_{T,V,n_C,\dots}

Dependence on temperature and pressure

μB=f(T,p,xB,xC,⋯ )\mu_{B}=f(T,p,x_{B},x_{C},\cdots)

For the simpler case μB=f(T,p)\mu_B = f(T, p):

  • Dependence on TT:

(∂μ∂T)p,nB,nC,⋯=[∂∂T(∂G∂nB)T,p,nC,⋯]p,nB,nC(\frac{\partial \mu}{\partial T})_{p,n_{B},n_{C},\cdots}=[\frac{\partial}{\partial T}(\frac{\partial G}{\partial n_{B}})_{T,p,n_{C},\cdots}]_{p,n_{B},n_{C}}

=[∂∂nB(∂G∂T)p,nB,nC,⋯]T,p,nC,⋯\qquad \qquad \qquad \qquad=[\frac{\partial}{\partial n_{B}}(\frac{\partial G}{\partial T})_{p,n_{B},n_{C},\cdots}]_{T,p,n_{C},\cdots}

=[∂(−S)∂nB]T,p,nC,⋯\qquad \quad=[\frac{\partial(-S)}{\partial n_{B}}]_{T,p,n_{C},\cdots}

T↑,μB↓T \uparrow, \mu_{B}\downarrow

  • Dependence on pp:

(∂μ∂p)T,nB,nC,⋯=[∂∂p(∂G∂nB)T,p,nC,⋯]p,nB,nC(\frac{\partial \mu}{\partial p})_{T,n_{B},n_{C},\cdots}=[\frac{\partial}{\partial p}(\frac{\partial G}{\partial n_{B}})_{T,p,n_{C},\cdots}]_{p,n_{B},n_{C}}

=[∂∂nB(∂G∂p)T,nB,nC,⋯]T,p,nC,⋯\qquad \qquad \qquad \qquad=[\frac{\partial}{\partial n_{B}}(\frac{\partial G}{\partial p})_{T,n_{B},n_{C},\cdots}]_{T,p,n_{C},\cdots}

=[∂(V)∂nB]T,p,nC,⋯\qquad \quad=[\frac{\partial(V)}{\partial n_{B}}]_{T,p,n_{C},\cdots}

T↑,μB↑T \uparrow, \mu_{B}\uparrow

  • Note: this uses relations such as
    • V=(∂G∂p)T=(∂G∂p)SV=(\frac{\partial G}{\partial p})_{T}=(\frac{\partial G}{\partial p})_{S}
    • S=−(∂F∂T)V=−(∂G∂T)pS=-(\frac{\partial F}{\partial T})_{V}=-(\frac{\partial G}{\partial T})_{p}
Criterion based on chemical potential
  • For a system of fixed composition at constant TT and pp, with W′=0W' = 0:

dGT,p,W′=0≤0dG_{T,p,W'=0}\le 0

  • For a multicomponent homogeneous system:

dG=−SdT+Vdp+∑μBdnBdG=-SdT+Vdp+\sum\mu_{B}dn_{B}

  • Therefore, under constant TT and pp with W′=0W' = 0, the criterion for direction and limit of a process is:

∑μBdnB≤0\sum\mu_{B}dn_{B}\le 0

  • In short, the conclusion is still the same: check whether dG≤0dG \le 0.