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Chapter 5: Q.5.80 (page 208)
Use the Clausius-Clapeyron relation to derive equation 5.90 directly from Raoult's law. Be sure to explain the logic carefully.
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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 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,
( 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 .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 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
Calculate K on both sides of the critical point in the van der Waals model, and show that 'Y = -y' in this model.
If expression 5.68 is correct, it must be extensive: Increasing both NA and NB by a common factor while holding all intensive variables fixed should increase G by the same factor. Show that expression 5.68 has this property. Show that it would not have this property had we not added the term proportional to In NA!.
Check that equations 5.69 and 5.70 satisfy the identity (equation 5.37)
By subtracting from localid="1648229964064" ,or,one can obtain four new thermodynamic potentials. Of the four, the most useful is the grand free energy (or grand potential),
(a) Derive the thermodynamic identity for , and the related formulas for the partial derivatives ofwith respect to, and
(b) Prove that, for a system in thermal and diffusive equilibrium (with a reservoir that can supply both energy and particles), tends to decrease.
(c) Prove that
(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 ) 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 and an electron concentration of about per cubic meter. Calculate 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.)
Sketch qualitatively accurate graphs of G vs. P for the three phases of H20 (ice, water, and steam) at 0掳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?
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