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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

Derive the thermodynamic identity for G (equation 5.23), and from it the three partial derivative relations 5.24.

In this problem you will investigate the behavior of a van der Waals fluid near the critical point. It is easiest to work in terms of reduced variables throughout.

(a) Expand the van der Waals equation in a Taylor series in , keeping terms through order . Argue that, for T sufficiently close to Tc, the term quadratic in (V-VC)becomes negligible compared to the others and may be dropped.

(b) The resulting expression for P(V) is antisymmetric about the point V = Ve. Use this fact to find an approximate formula for the vapor pressure as a function of temperature. (You may find it helpful to plot the isotherm.) Evaluate the slope of the phase boundary,dP/dT

( c) Still working in the same limit, find an expression for the difference in volume between the gas and liquid phases at the vapor pressure. You should find Vg-Vl鈭�Tc-T尾.8, where (3 is known as a critical exponent. Experiments show that (3 has a universal value of about 1/3, but the van der Waals model predicts a larger value.

(d) Use the previous result to calculate the predicted latent heat of the transformation as a function of temperature, and sketch this function.

The shape of the T = Tc isotherm defines another critical exponent, called P-Pc鈭�V-Vc未Calculate 5 in the van der Waals model. (Experimental values of 5 are typically around 4 or 5.)

A third critical exponent describes the temperature dependence of the isothermal compressibility, K=-t This quantity diverges at the critical point, in proportion to a power of (T-Tc) that in principle could differ depending on whether one approaches the critical point from above or below. Therefore the critical exponents 'Y and -y' are defined by the relations

魏鈭�T-Tc-纬Tc-T-纬'

Calculate K on both sides of the critical point in the van der Waals model, and show that 'Y = -y' in this model.

Suppose that a hydrogen fuel cell, as described in the text, is to be operated at 75掳Cand atmospheric pressure. We wish to estimate the maximum electrical work done by the cell, using only the room temperature data at the back of this book. It is convenient to first establish a zero-point for each of the three substances, H2,O2,andH2O. Let us take Gfor both H2andO2to be zero at 25掳C, so that G for a mole of H2Ois -237KJat 25掳C.

(a) Using these conventions, estimate the Gibbs free energy of a mole of H2at 75掳C. Repeat for O2andH2O.

(b) Using the results of part (a), calculate the maximum electrical work done by the cell at 75掳C, for one mole of hydrogen fuel. Compare to the ideal performance of the cell at25掳C.

By subtracting 渭Nfrom localid="1648229964064" U,H,F,orG,one can obtain four new thermodynamic potentials. Of the four, the most useful is the grand free energy (or grand potential),

桅鈮�U-TS-渭N.

(a) Derive the thermodynamic identity for 桅, and the related formulas for the partial derivatives of桅with respect toT,V, and 渭N

(b) Prove that, for a system in thermal and diffusive equilibrium (with a reservoir that can supply both energy and particles), 桅tends to decrease.

(c) Prove that蠒=-PV.

(d) As a simple application, let the system be a single proton, which can be "occupied" either by a single electron (making a hydrogen atom, with energy -13.6eV) or by none (with energy zero). Neglect the excited states of the atom and the two spin states of the electron, so that both the occupied and unoccupied states of the proton have zero entropy. Suppose that this proton is in the atmosphere of the sun, a reservoir with a temperature of 5800Kand an electron concentration of about 2脳1019per cubic meter. Calculate 桅for both the occupied and unoccupied states, to determine which is more stable under these conditions. To compute the chemical potential of the electrons, treat them as an ideal gas. At about what temperature would the occupied and unoccupied states be equally stable, for this value of the electron concentration? (As in Problem 5.20, the prediction for such a small system is only a probabilistic one.)

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.
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