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Chapter 5 - Chemical

Potential and Fugacity

Gibbs Energy and Temperature

Gibbs Energy and Pressure

Chemical Potential

Real Gases - Fugacity

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Variation of G with T The variation of Gibbs

energy with tempera-ture is determined by

the entropy.

(G/T)p= -S The most stable phase

is the one with the

lowest Gibbs energy.

This also has the lowest

chemical potential, as we

shall see.

Solid

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Variation of G withp

The variation of Gibbs energy with pressureis determined by the volume.

(G/p)T= V

In integral form, G(p) - G(po

) =I

Vdp [limits:po

top]

For a perfect gas, use V = nRT/pto get

G(p) = G(po) + nRT ln (p/po) For liquids and solids, use kT= -(1/V) (MV/Mp)T,

and consider kTto be constant; get ln[V(p)/V(po)] =-

kTDp or V(p) = V(po)exp[- kTDp]. Two approx.:

Dp.p for p>>poand exp[- kTp] .(1- kTp)Y

V(p).

V(po

)[1-kTp] G - G o = -V o - o - k /2 2 - o2

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Chemical Potential

The chemical potential of a pure substanceis defined as

m= (G/n)T,p For a pure substance, the Gibbs energy =

G = nGm, so

m= (nGm/n)T,p= Gm

For instance, the Gibbs energy of a perfectgas at pressurepwas given as

G(p) = G(po) + nRT ln (p /po),which means that

m= mo+ RT ln (p /po) [Note mis an intensivequantity.]

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Introducing Fugacity m= mo+ RT ln (p /po) is for perfect gases.

What to do for imperfect (real) gases.

Can define a different chemical potential

function (call it m) such that

m = mo+ RT ln (p /po) Or can use the real gas pressure instead of

the perfect gas pressure, using some

equation of state other thanpV = nRT

Or can define a fugacity (effective

pressure), f, and m= mo+ RT ln (f /po)

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Using Fugacities

Fugacities should be used instead ofpressure for all functions of a real gas

involving chemical potential.

This includes equilibria and equilibriumconstants.

Example: For the formation of ammonia, N2+

3 H2--> 2 NH3, elementary chemistry courses

have used Kp=p2NH3 /pN2pH2

3

Now we shall use

K = f2NH3 /fN2fH23

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Interpreting Fugacities

Standard state of a real gase The standard state of a real gas is the

hypothetical state in which the gas is at a

pressurepoand behaving perfectly.

The relation between fugacity and pressure

is

f= fp

where fis the dimensionless fugacity

coeffient.

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min terms of f and f

Since m= mo+ RT ln (f /po),and f= p, this means that

m= mo+ RT ln (p /po) + RT ln The fugacity coefficient can be related to

the compression factor, Z = pVm/RT

ln = (Z - 1)/pdp [limits: 0 top]

For any gas, asp60, f61. (I.e., any

gas is perfect at zero pressure.)

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Fugacity of a van der Waals Gas If the repulsive term is dominant in a van

der Waals gas, thenp= RT/(Vm- b)YVm = RT/p+ b

So Z =pVm/RT = 1 +pb/RT

This leads to ln f = bp/RT If the attractive term is dominant in a van

der Waals gas, then

p= RT/Vm

- a/Vm

2,YpVm

2- RTVm

+ a

= 0

By the quadratic equation, Vm= RT [(RT)2- 4ap)].5

. 2p

Ifpis sufficiently low, (RT)2>> 4 ap, soVm= RT/p andZ = 1 - ap/(RT)

2

This leads to ln f = -ap/(RT)2

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Estimating Fugacities The figures below can be used to estimate

fugacities when the reduced temperatureand pressure are known.