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-VlTc-Tβ.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-PcV-VcδCalculate 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

κT-Tc-γTc-T-γ'

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

Short Answer

Expert verified

(a) The van der wall force in Taylor seriesPcVcNkTe=127ab2·3NbN·278ba=38=0.375

(b) The two curves becoming indistinguishable over the range.

(c) The difference between volume between the gas and liquid phases at the vapor pressure.

vg-vl=(1+21-t)-(1-21-t)=41-t

(d) the predicted latent heat of the transformation as a function of temperature

LVkTc=38

Step by step solution

01

Part(a) Step 1: Given information

We have been givenp=8t(3v-1)-1-3v-2

02

Part(a) Step 2: Simplify

The terms used in:

3pv3=-1296t(3v-1)-4+72v-5

Therefore, to plot isotherm and perform the Maxwell construction

Series8t/(3v-1)-3/v~2,{v,1,3}

03

Part(b) Step 1: Given information

We have been given, As V-1even gets closer to. the two curves becoming indistinguishable over the range.

04

Part(b) Step 2; Simplify

Constant-pressure line that results in equal area enclosed by two loops

05

Part(c) Step 1: Given information

We have been givenvg-vl=(1+21-t)-(1-21-t)=41-t

The volume of the liquid and gas at transition pressure are just the values of vat transition pressure found in the above parts.

06

Part(c) Step 2: Simplify

The term we will get:

4t-3=4t-3-6(t-1)(v-1)-32(9t-8)(v-1)3
07

Part(d) Step 1;Given information

We have been givenL=TVg-VldPdT=PcVcvg-vldpdt=38NkTcvg-vldpdt

This equation describes a parabola opening to the left descresing to left as t1

08

Part(d) Step 2: Simplify

The term we get:

LNkTc=38·41-t·4=61-t

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Most popular questions from this chapter

Repeat the previous problem for the opposite case where the liquid has a substantial negative mixing energy, so that its free energy curve dips |below the gas's free energy curve at a temperature higher than TB. Construct the phase diagram and show that this system also has an azeotrope.

A mixture of one part nitrogen and three parts hydrogen is heated, in the presence of a suitable catalyst, to a temperature of 500° C. What fraction of the nitrogen (atom for atom) is converted to ammonia, if the final total pressure is 400 atm? Pretend for simplicity that the gases behave ideally despite the very high pressure. The equilibrium constant at 500° C is 6.9 x 10-5. (Hint: You'l have to solve a quadratic equation.)

Sketch qualitatively accurate graphs of G vs. P for the three phases of H20 (ice, water, and steam) at 0°C. Put all three graphs on the same set of axes, and label the point corresponding to atmospheric pressure. How would |the graphs differ at slightly higher temperatures?

Below 0.3 K the slope of the °He solid-liquid phase boundary is negative (see Figure 5.13).

(a) Which phase, solid or liquid, is more dense? Which phase has more entropy (per mole)? Explain your reasoning carefully.

(b) Use the third law of thermodynamics to argue that the slope of the phase boundary must go to zero at T = 0. (Note that the *He solid-liquid phase boundary is essentially horizontal below 1 K.)

(c) Suppose that you compress liquid *He adiabatically until it becomes a solid. If the temperature just before the phase change is 0.1 K, will the temperature after the phase change be higher or lower? Explain your reasoning carefully.

In the previous section I derived the formula (F/V)T=-P. Explain why this formula makes intuitive sense, by discussing graphs of F vs. V with different slopes.

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