Consider a dilute solution of Ns molecules of solute B dissolved into Na molecules of solvent A(NB << NA). The chemical potentials µa and He of each species are:HA = Ho(T, P) – NakT/NAHB = f(T, P) + kT In(NB/NA)P-P2where Tis the temperature, P is the pressure, k is Boltzmann's constant, Ho(T, P) is the chemicalpotential of the solvent in the limit of no solute (NB = 0), and f(T, P) is the chemical potential ofthe solute extrapolated to the equal solute-solvent limit (NB = NA). Imagine that a semi-permeable membrane (indicated by the vertical dashed line) that can only be crossed bysolvent molecules A separates a pure solution of volume Vi at pressure P1 from a dilute solutionof volume V2 at pressure P2 as indicated in the figure above.(a) Is P1 or P2 higher in equilibrium? Explain your answer!(b) Derive an expression for the pressure difference AP = P2- P1. Show your work! Hint: youmight want to make use of the thermodynamic identity dG = µ dN-S dT + V dP where G isthe Gibbs free energy and S is the entropy.(c) If the temperature is increased by a small amount AT while the volumes V1 and V2 andpressure P2 of the dilute solution remain fixed, what will be the change AP1 in theequilibrium value of the pressure on the left side of the membrane?(d) If the temperature is increased by a small amount 4T while the volumes V1 and V2 andpressure P2 of the dilute solution remain fixed, what will be the fractional change ANA2/Na2in the equilibrium value of the number of solvent molecules on the right side of themembrane? Hint: the answer will involve the entropies per particle sAi = SA/NAi and densitiesPAi = NA/Vi.

The objective of this question is to analyse the equilibrium conditions, deriving a pressure…

Consider a dilute solution of No molecules of solute B dissolved into Na molecules of solvent A (NB << NA). The chemical potentials μA and μB of each species are: HAMO(T, P)-NBKT/NA HB = f(T, P) + kT In (NB/NA) P₁- P2 where T is the temperature, P is the pressure, k is Boltzmann's constant, μo(T, P) is the chemical potential of the solvent in the limit of no solute (NB = 0), and f(T, P) is the chemical potential of the solute extrapolated to the equal solute-solvent limit (NB - NA). Imagine that a semi- permeable membrane (indicated by the vertical dashed line) that can only be crossed by solvent molecules A separates a pure solution of volume V₂ at pressure P₁ from a dilute solution of volume V₂ at pressure Ro as indicated in the figure above

Consider a dilute solution of No molecules of solute B dissolved into Na molecules of solvent A (NB << NA). The chemical potentials μA and μB of each species are: HAMO(T, P)-NBKT/NA HB = f(T, P) + kT In (NB/NA) P₁- P2 where T is the temperature, P is the pressure, k is Boltzmann's constant, μo(T, P) is the chemical potential of the solvent in the limit of no solute (NB = 0), and f(T, P) is the chemical potential of the solute extrapolated to the equal solute-solvent limit (NB - NA). Imagine that a semi- permeable membrane (indicated by the vertical dashed line) that can only be crossed by solvent molecules A separates a pure solution of volume V₂ at pressure P₁ from a dilute solution of volume V₂ at pressure Ro as indicated in the figure above

Chemistry

10th Edition

ISBN:9781305957404

Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste

Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste

Chapter1: Chemical Foundations

Section: Chapter Questions

Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...

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Question

Consider a dilute solution of Ns molecules of solute B dissolved into Na molecules of solvent A
(NB << NA). The chemical potentials µa and He of each species are:
HA = Ho(T, P) – NakT/NA
HB = f(T, P) + kT In(NB/NA)
P-
P2
where Tis the temperature, P is the pressure, k is Boltzmann's constant, Ho(T, P) is the chemical
potential of the solvent in the limit of no solute (NB = 0), and f(T, P) is the chemical potential of
the solute extrapolated to the equal solute-solvent limit (NB = NA). Imagine that a semi-
permeable membrane (indicated by the vertical dashed line) that can only be crossed by
solvent molecules A separates a pure solution of volume Vi at pressure P1 from a dilute solution
of volume V2 at pressure P2 as indicated in the figure above.
(a) Is P1 or P2 higher in equilibrium? Explain your answer!
(b) Derive an expression for the pressure difference AP = P2- P1. Show your work! Hint: you
might want to make use of the thermodynamic identity dG = µ dN-S dT + V dP where G is
the Gibbs free energy and S is the entropy.
(c) If the temperature is increased by a small amount AT while the volumes V1 and V2 and
pressure P2 of the dilute solution remain fixed, what will be the change AP1 in the
equilibrium value of the pressure on the left side of the membrane?
(d) If the temperature is increased by a small amount 4T while the volumes V1 and V2 and
pressure P2 of the dilute solution remain fixed, what will be the fractional change ANA2/Na2
in the equilibrium value of the number of solvent molecules on the right side of the
membrane? Hint: the answer will involve the entropies per particle sAi = SA/NAi and densities
PAi = NA/Vi.

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