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# handout13 - 4.12 Gibbs Energies of Mixtures AIM Introduce...

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Unformatted text preview: 4.12 Gibbs Energies of Mixtures AIM: Introduce the concept of chemical potential. BACKGROUND: We are interested in developing a framework for free energies when the composition of a substance changes (chemical reactions, mixtures of particles or open systems). • In this case, the state functions are also dependent on the amount of the individual molecular species in the system. • For example, the Gibbs energy of the system would be written as G = G ( T , P , n 1 , n 2 , n 3 , . . . ) , (4.147) where n i is the number of moles of species i . • The total differential of this expression is d G = ∂ G ∂ T P , n 1 , n 2,... d T + ∂ G ∂ P T , n 1 , n 2,... d P + ∂ G ∂ n 1 T , P , n 2 , n 3 ,... d n 1 + ∂ G ∂ n 2 T , P , n 1 , n 3 ,... d n 2 + ··· . (4.148) • We would like an expression for d G that does not have the partial derivatives in it. • This is easy for red and blue terms. If the concentration of all 4–47 species is constant, then d n i = for all i , and this reduces to d G = ∂ G ∂ T P , n 1 , n 2,... d T + ∂ G ∂ P T , n 1 , n 2,... d P =- S d T + V d P , (4.149) where we have applied Eqs. 4.141 and 4.142. • For the green terms we now define a new thermodynamic quantity μ i = ∂ G ∂ n i T , P , n j 6 = n i (4.150) called the chemical potential of species i . Then we can write Eq. 4.148 as d G =- S d T + V d P + ∑ i μ i d n i . (4.151) NOTE: The units of chemical potential are Energy mol . QUESTION: How can we intuitively understand chemical potential? ANSWER: The Gibbs energy of a system will change by n i μ i if we add n i mol of substance i to the system at constant concentration (this is an infinitesimal change, therefore the concentration of particles does not change). 4–48 EXAMPLE: Lets perform an thought experiment to allow us to better understand this concept. • We have a system in which: ‹ we are at constant T and P › the composition is constant; i.e. the ratio of all n i n j is constant • Now we start from a very small system (all n s i very small, we assume 0 and G s = ) and add particles to system in the correct...
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handout13 - 4.12 Gibbs Energies of Mixtures AIM Introduce...

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