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Chapter 5: Q 5.9 (page 158)

Sketch a qualitatively accurate graph of G vs. T for a pure substance as it changes from solid to liquid to gas at fixed pressure. Think carefully about the slope of the graph. Mark the points of the phase transformations and discuss the features of the graph briefly.

Short Answer

Expert verified

The slope of the curve is most stepper for gaseous state than liquid state than solid state for a pure substance.

Step by step solution

01

Introduction

Write the expression for Gibbs free energy.

G=U-TS+PV

Here, G is Gibbs free energy, T is the absolute temperature, S is the entropy, P is the pressure and V is the volume.

02

Explanation

A plot between G and T will simply be a line graph where the slope of the graph is the negative entropy -S. The entropy is lowest for the solid and since the slope is negative, the line for the solid has the highest value in the vertical axis. The substance then melts into liquid which results larger value of entropy than solid, making the slope of the graph more stepper and finally it evaporates into a gas, the value of entropy for which is largest. It makes the slope of the graph most stepper.

Draw a graph to show the variation of G and T for a pure substance.

The slope of the curve is most stepper for gaseous state than liquid state than solid state for a pure substance.

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Most popular questions from this chapter

Prove that the entropy of mixing of an ideal mixture has an infinite slope, when plotted vs. x, at x = 0 and x= 1.

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?

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 S fixed, each鈭�/鈭�S is taken with V fixed, and N is always held fixed. From the thermodynamic identity (for U ) 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, and G ). 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.

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.

Figure 5.35 (left) shows the free energy curves at one particular temperature for a two-component system that has three possible solid phases (crystal structures), one of essentially pure A, one of essentially pure B, and one of intermediate composition. Draw tangent lines to determine which phases are present at which values of x. To determine qualitatively what happens at other temperatures, you can simply shift the liquid free energy curve up or down (since the entropy of the liquid is larger than that of any solid). Do so, and construct a qualitative phase diagram for this system. You should find two eutectic points. Examples of systems with this behaviour include water + ethylene glycol and tin - magnesium.
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