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Chapter 5: Q. 5.52 (page 185)

Plot the Van der Waals isotherm for T/Tc = 0.95, working in terms of reduced variables. Perform the Maxwell construction (either graphically or numerically) to obtain the vapor pressure. Then plot the Gibbs free energy (in units of NkTc) as a function of pressure for this same temperature and check that this graph predicts the same value for the vapor pressure.

Short Answer

Expert verified

The pressure of the phase transition is 0.812

Step by step solution

01

Step 1  Given information

From problem 5.51 we know that

p=8t3v-1-3v2

02

Substituting the value of t=0.95 and iterating the values

The new equation becomes

p=8(0.95)3v-1-3v2p=7.63v-1-3v2

03

Gibbs free energy

We know Gibbs energy is described as follows

G=NkTlnV-Nb+(NKT)(NB)(V-Nb)-2aN2V+c(T)

Converting the equation into reduced variables

t=TTCv=VVCVC=3NbTC=827ab

WE GET,


Substituting t=0.95 we get


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Most popular questions from this chapter

Functions encountered in physics are generally well enough behaved that their mixed partial derivatives do not depend on which derivative is taken first. Therefore, for instance,

鈭�鈭�V鈭�U鈭�S=鈭�鈭�S鈭�U鈭�V

where each 鈭�/鈭�Vis taken with Sfixed, each 鈭�/鈭�Sis taken with Vfixed, and Nis always held fixed. From the thermodynamic identity (forU) you can evaluate the partial derivatives in parentheses to obtain

鈭�T鈭�VS=-鈭�P鈭�SV

a nontrivial identity called a Maxwell relation. Go through the derivation of this relation step by step. Then derive an analogous Maxwell relation from each of the other three thermodynamic identities discussed in the text (for H,F,andG ). Hold N fixed in all the partial derivatives; other Maxwell relations can be derived by considering partial derivatives with respect to N, but after you've done four of them the novelty begins to wear off. For applications of these Maxwell relations, see the next four problems.

Effect of altitude on boiling water.

(a) Use the result of the previous problem and the data in Figure 5.11 to plot a graph of the vapor pressure of water between 50掳C and 100掳C. How well can you match the data at the two endpoints?

(b) Reading the graph backward, estimate the boiling temperature of water at each of the locations for which you determined the pressure in Problem 1.16. Explain why it takes longer to cook noodles when you're camping in the mountains.

(c) Show that the dependence of boiling temperature on altitude is very nearly (though not exactly) a linear function, and calculate the slope in degrees Celsius per thousand feet (or in degrees Celsius per kilometer).

Check that equations 5.69 and 5.70 satisfy the identityG=NA渭A+NB渭B (equation 5.37)

Plumber's solder is composed of 67% lead and 33% tin by weight. Describe what happens to this mixture as it cools, and explain why this composition might be more suitable than the eutectic composition for joining pipes.

Is heat capacity (C) extensive or intensive? What about specific heat (c) ? Explain briefly.
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