1

CBE 141- Chemical Engineering Thermodynamics

Spring 2015

Homework Set # 5

Deadline: Friday 13 Mar. 2015 (In class by 1:10 pm)

Strict adherence to these rules:

1) Solve each question on a new page.

2) Put your final answer in a box.

3) DO NOT staple your problems into one document; instead, make sure your name is

on each page.

4) No homework regrades.

5) Show all work and cite all sources/references (even if from the textbook, name the

table).

Problem 1

We defined an ideal gas as:

(a) P-V-T-n behavior described by the equation of state PVt =nRT

(b) Internal energy, Ut is a function of temperature and mass Ut =Ut(T, m)

Denbigh (Principles of Chemical Equilibrium, 4th ed., New York: Cambridge University,

1981) suggests an alternative definition: a single-component ideal gas is a substance

whose chemical potential is given by:

G = RT ln(P) + (T)

Prove that properties (a) and (b) follow from Denbighs definition.

(Hint: Start with the differential form dG = -SdT +VdP. Also note is just some generic

function of T )

2

Problem 2

The thermal expansion coefficient, , and the isothermal compressibility, , of a gas are

given by the following expressions:

Where k1 and k2 are constants. Cp and Cv may be assumed to be independent of

temperature.

(a) Derive a volume explicit equation of state, i.e V=V(T,P)

(b) Does this fluid have a critical point? Why?

Problem 3

(a) Obtain expressions for the derivatives (T/V)H and (T/V)U .Your expressions

should include only P, V, T, their derivatives [e.g. (T /P)V and CP or CV as

needed; no H, U or S should appear in the derivatives]

(b) Can you propose experiments to measure these properties directly?

Problem 4

You have already seen that you can take Legendre transforms of dU to get dH, dA, and dG

(where A and G are the Helmholtz energy and the Gibbs energy, respectively). Legendre

transforms are a mathematical tool to transform any function while retaining information

content. In this problem, you will see how to take the Legendre transform of S, called

(the Greek letter Xi, pronounced zi or ksi, with a long i" sound.)

(a) Write a differential expression for dS as a function of U and V

(b) Show that (

)=

1

and (

)=

(c) The function is defined as the Legendre transformation of S(U ,V) with respect

to U and V. Write the macroscopic (as opposed to the differential) expression for

in terms of S, U, V, P, and T, using your answer from part (b). What are the natural

variables of ?

/P V VC C CP

P V

CVk T

V T C

1

1

T

kV

V P P

21

3

(d) Simplify your macroscopic expression for so that it is only a function of

thermodynamic potentials and temperature.

(e) Using your expression for in part (c), take the derivative of to find the

differential expression d.

(f) Calculate the derivatives

[

(1/)]/

and

[

(/)]1/

in terms of familiar thermodynamic functions such as U, H, A, T, P.

Problem 5

Consider a process where carbon dioxide is throttled from 1600kPa and 60C to

atmospheric pressure.

(a) What is the residual enthalpy (HR /RTc) and residual entropy (SR /R) for the inital

condition?

Assume the truncated Virial equation describes the state of carbon dioxide initially such

that

GR/RT=Z-1=BP/RT

for the initial state where B is the first Virial coefficient.

(b) Find the final temperature of the gas. Note that the process is carried out in an

isenthalpic manner and we assume carbon dioxide to be an ideal gas at the final

state

(c) What is the total change in entropy for this process?

4

Problem 6 (not graded)

You saw during the first weeks of class that = for an ideal gas. Using thermodynamic derivatives, derive for the general case as a function of P, T, and V only. Show that this simplifies to R using the ideal gas EOS.

Hint: Start by taking the total derivative of U with respect to T and V.

Note: Though this question is not graded, there are tricks involved in the derivation that

you will be expected to be familiar with, so please work through it!