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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

The enthalpy and Gibbs free energy, as defined in this section, give special treatment to mechanical (compression-expansion) work, -PdV. Analogous quantities can be defined for other kinds of work, for instance, magnetic work." Consider the situation shown in Figure 5.7, where a long solenoid ( Nturns, total length N) surrounds a magnetic specimen (perhaps a paramagnetic solid). If the magnetic field inside the specimen is B鈫�and its total magnetic moment is M鈫�, then we define an auxilliary field H鈫�(often called simply the magnetic field) by the relation

H鈫�鈮�1渭0B鈫�-M鈫�V,

where 渭0is the "permeability of free space," 4蟺脳10-7N/A2. Assuming cylindrical symmetry, all vectors must point either left or right, so we can drop the -鈫�symbols and agree that rightward is positive, leftward negative. From Ampere's law, one can also show that when the current in the wire is I, the Hfield inside the solenoid is NI/L, whether or not the specimen is present.

(a) Imagine making an infinitesimal change in the current in the wire, resulting in infinitesimal changes in B, M, and H. Use Faraday's law to show that the work required (from the power supply) to accomplish this change is Wtotal=VHdB. (Neglect the resistance of the wire.)

(b) Rewrite the result of part (a) in terms of Hand M, then subtract off the work that would be required even if the specimen were not present. If we define W, the work done on the system, 鈥�to be what's left, show that W=渭0HdM.

(c) What is the thermodynamic identity for this system? (Include magnetic work but not mechanical work or particle flow.)

(d) How would you define analogues of the enthalpy and Gibbs free energy for a magnetic system? (The Helmholtz free energy is defined in the same way as for a mechanical system.) Derive the thermodynamic identities for each of these quantities, and discuss their interpretations.

Sulfuric acid, H2SO4,readily dissociates intoH+andHSO4-H+andHSO4-ions

H2SO4鉄�H++HSO4-

The hydrogen sulfate ion, in turn, can dissociate again:

HSO4-鉄�H++SO42-

The equilibrium constants for these reactions, in aqueous solutions at 298 K, are approximately 10 and 10*, respectively. (For dissociation of acids it is usually more convenient to look up K than 鈭�G掳. By the way, the negative base-10 logarithm of K for such a reaction is called pK, in analogy to pH. So for the first reaction pK = -2, while for the second reaction pK = 1.9.)

(a) Argue that the first reaction tends so strongly to the right that we might as well consider it to have gone to completion, in any solution that could possibly be considered dilute. At what pH values would a significant fraction of the sulfuric acid not be dissociated?

(b) In industrialized regions where lots of coal is burned, the concentration of sulfate in rainwater is typically 5 x 10 mol/kg. The sulfate can take any of the chemical forms mentioned above. Show that, at this concentration, the second reaction will also have gone essentially to completion, so all the sulfate is in the form of SOg. What is the pH of this rainwater?

(c) Explain why you can neglect dissociation of water into H* and OH in answering the previous question. (d) At what pH would dissolved sulfate be equally distributed between HSO and SO2-?

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.

Assume that the air you exhale is at 35掳C, with a relative humidity of 90%. This air immediately mixes with environmental air at 5掳C and unknown relative humidity; during the mixing, a variety of intermediate temperatures and water vapour percentages temporarily occur. If you are able to "see your breath" due to the formation of cloud droplets during this mixing, what can you conclude about the relative humidity of your environment? (Refer to the vapour pressure graph drawn in Problem 5.42.)

In this problem you will derive approximate formulas for the shapes of the phase boundary curves in diagrams such as Figures 5.31 and 5.32, assuming that both phases behave as ideal mixtures. For definiteness, suppose that the phases are liquid and gas.

(a) Show that in an ideal mixture of A and B, the chemical potential of species A can be written 渭A=渭A掳+kTln(1-x)where A is the chemical potential of pure A (at the same temperature and pressure) and x=NB/NA+NB. Derive a similar formula for the chemical potential of species B. Note that both formulas can be written for either the liquid phase or the gas phase.

(b) At any given temperature T, let x1 and xgbe the compositions of the liquid and gas phases that are in equilibrium with each other. By setting the appropriate chemical potentials equal to each other, show that x1and xg obey the equations =1-xl1-xg=e螖GA掳/RTandxlxg=e螖GB掳/RT and where 螖G掳represents the change in G for the pure substance undergoing the phase change at temperature T.

(c) Over a limited range of temperatures, we can often assume that the main temperature dependence of 螖G掳=螖H掳-T螖S掳comes from the explicit T; both 螖H掳and螖S掳are approximately constant. With this simplification, rewrite the results of part (b) entirely in terms of 螖HA掳,螖HB掳 TA, and TB (eliminating 螖Gand螖S). Solve for x1and xgas functions of T.

(d) Plot your results for the nitrogen-oxygen system. The latent heats of the pure substances are螖HN2掳=5570J/moland螖HO2掳=6820J/mol. Compare to the experimental diagram, Figure 5.31.

(e) Show that you can account for the shape of Figure 5.32 with suitably chosen螖H掳 values. What are those values?
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