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Unformatted text preview: 1/1 CHEM350: Lecture 4, Jan 11 2010 Gibbs free energy for nonideal gases (section 7.5 in textbook) We know from thermodynamics that the total differential for the molar Gibbs free energy G (T , P ) has the form dG = − SdT + VdP
from which we obtain the relation ⎛ ∂G ⎞ ⎜ ∂P ⎟ = V ⎝ ⎠T
If we consider a process at constant temtperature T, we obtain, dGT = VdP
and integrating from initial pressure Pi to final pressure Pf yields, ∫ P = Pf P = Pi dGT = ∫ P = Pf P = Pi V dP Using the ideal gas equation of state for V , we get ( ΔG ) T ⎛ Pf ⎞ ≡ G (T , Pf ) − G (T , Pi ) = RT ln ⎜ ⎟ . ⎝ Pi ⎠ If we let the initial pressure Pi be 1 bar, i.e., Pi ≡ P o , we obtain ⎛ Pf ⎞ G (T , Pf ) − G (T , P o ) = RT ln ⎜ o ⎟ ⎝P ⎠
or ⎛ Pf ⎞ G (T , Pf ) = G o (T ) + RT ln ⎜ o ⎟ ⎝P ⎠
How can we proceed for a nonideal gas? We can replace V with a more realistic expression such as the virial equation of state, PV = 1 + B '2 (T )P + B '3 (T )P 2 + ... RT
and integrate from pressure Pideal to P. This yields a virial expansion for the molar Gibbs free energy: ∫ P Pideal dGT = RT ∫ P Pideal P P dP + RT B '2 (T ) ∫ dP + RT B '3 (T ) ∫ P dP + ..., Pideal Pideal P so that 2/2 ⎛P⎞ 1 2 2 G (T , P ) − G (T , Pideal ) = RT ln ⎜ ⎟ + RT B '2 (T )( P − Pideal ) + 2 RT B '3 (T )( P − Pideal ) + ... ⎝ Pideal ⎠
or ⎛P⎞ 1 G (T , P ) = G (T , Pideal ) + RT ln ⎜ + RT B '2 (T )( P − Pideal ) + RT B '3 (T )( P 2 − Pid2eal ) + ... 2 ⎝ Pideal ⎟ ⎠
We may also simplify the result further by using the ideal gas free energy expression,
⎛P ⎞ G (T , Pideal ) = G o (T ) + RT ln ⎜ ideal ⎟ o ⎝P ⎠ so that ⎛P⎞ 1 G (T , Pideal ) + RT ln ⎜ + RT B '2 (T )( P − Pideal ) + RT B '3 (T )( P 2 − Pid2eal ) + ... 2 ⎝ Pideal ⎟ ⎠ = 1 ⎛ P⎞ G o (T ) + RT ln ⎜ o ⎟ + RT B '2 (T )( P − Pideal ) + RT B '3 (T )( P 2 − Pid2eal ) + ... ⎝P ⎠ 2
If we choose the lower integration limit as Pideal = 0, we obtain,
1 ⎛ P⎞ G (T , P ) = G o (T ) + RT ln ⎜ o ⎟ + RT B '2 (T )P + RT B '3 (T )P 2 + ... ⎝P ⎠ 2 as our expression for the Gibbs free energy of a nonideal gas described in terms of the virial equation of state. This expression is somewhat problematic because the Gibbs free energy is no longer universal. This is because the second and third virial coefficients differ from one substance to another. Such an expression is not convenient for the calculation of equilibrium constants for chemical reactions. How do we overcome this problem? We generalize the concept of pressure by defining a new thermodynamic function, f(T,P), called fugacity, via ⎛f⎞ G (T , P ) ≡ G o (T ) + RT ln ⎜ o ⎟ ⎝f ⎠
so that all gas nonideality is assigned to the fugacity f(T,P). We have the condition lim f (T , P ) = P
P→0 in order that the generalized expression reduces to the ideal gas expression in the limit of an ideal gas. If we rewrite our virial expression for G (T , P ) in the form 3/3 ⎧ P ⎡ B '2 (T ) P + 1 B '3 (T ) P 2 + ...⎤ ⎫ ⎢ ⎥⎪ ⎪ 2 ⎦ G (T , P ) = G (T ) + RT ln ⎨ o e⎣ ⎬ P ⎪ ⎪ ⎩ ⎭
o and compare to the expression involving fugacity above, we obtain f (T , P ) P = o exp B '2 (T )P + B '3 (T )P 2 + ... o f P { } This result suggests that fo =Po, and that f(T,P) has the form f (T , P ) = P exp B '2 (T )P + B '3 (T )P 2 + ... { } ...
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This note was uploaded on 04/11/2010 for the course CHEM 1101 taught by Professor Leroy during the Spring '10 term at University of Toronto Toronto.
 Spring '10
 Leroy

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