μ = μ⁰ + RTlnf................ (1)
Differentiating the equation (1) write P At constant T,
(∂μ/∂P)T = RT ( ∂lnf/∂P)T
We know that ,
(∂μ/∂P)/T = V................. (2)
Hence Equation (2) can be written as,
RT ( ∂lnf)T = V
RT ∫ ∂lnf/∂P)T = V/P1 dp. = vdp
RT ∫ ∂lnf = V/P2 dp. = vdp
RTln f₂/f₁ = ∫ p₁ᴾ² vdp.
The equation on to the determination of the fugacity at one pressure from that of another, if 'V' is known as a function of pressure.
VARIATION OF FUGACITY WITH TEMPERATURE :-
Consider two states of the same gas in which the chemical potential are μ and μ*. The corresponding fugacities being f and f* it's related equation,
μ* − μ = RTln f*/f
Dividing by T and rearranged
Rln f*/f = ( μ*/T) − (μ/T)
Differentiating write T at constant P
R (∂ln f*/∂T)P − R(∂lnf/∂T) = [ ∂(μ*)]
T −
∂T
[∂(μ)]
( T ) P
∂T
We know that,
[∂( μ)]
( T )P = − μ̄/T²
T
( gibb's Helmoltz equation)
(∂lnf* ) p* − (∂lnf) p = H̄*/RT² + H̄/RT²
∂T ∂T
When P* → low P, or Zero
P* →0 f* = 1
P*
For a real gas at very low P,
Fugacity becomes equal to P.
Hence f* doesn't change with T at constant P.
(ie) (∂lnf)P = H̄* − H̄
∂T RT²
H̄* is the partial molal enthalpy in the state of almost zero pressure and H̄ is the value at P.
H* − H̄ is the increase in partial molal enthalpy accompanying the isothermal expansion of the gas from a pressure ' P' into vacuum.
H̄* − H̄ = − ∫₀p ( ∂H )T dp
∂P
since ( ∂H) T = − μⱼ.T cp.
∂P
From above the equation,
RT² [∂lnf ] P = ∫₀p J,T . Cp. dp
∂T
If Cₚ is treated as a constant and μJ, T is known as a function of P. The integral of the above equation can be evaluated.
Good
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