Explanation Given: Aluminum silicate has three different crystalline state − kyanite, andalusite and sillimanite. The molar volume of kyanite is 44.09 × 10 − 6 m 3 , that of andalusite is 51.53 × 10 − 6 m 3 and that of sillimanite is 49.90 × 10 − 6 m 3 . Formula used: Write the expression for Gibbs free energy. G = − T S + P V + μ N …… (1) Here, G is Gibbs free energy, T is the absolute temperature, S is the entropy, P is the pressure, V is the volume, μ is the chemical potential and N is the number of molecules. Write the expression for the final temperature of the material. T 2 = T 1 + Δ G Δ S …… (2) Here, T 2 is the final temperature, T 1 is the initial temperature, Δ G is the change in Gibbs energy and Δ S is the change in entropy. Write the expression for the slope of phase boundary. d P d T = S 1 − S 2 V 1 − V 2 …… (3) Calculation: Write the expression for the infinitesimal change in G . d G = − S d T + V d P + μ d N Rearrange the above expression at constant number of molecules N and at constant temperature T , for which d N = d T = 0 . ( ∂ G ∂ P ) T , N = V Rearrange the above expression. d G = V d P …… (4) Integrate both sides of expression (4) for kyanite. G k = G k 0 + V k P …… (5) Integrate both sides of expression (4) for andalusite. G a = G a 0 + V a P …… (6) Integrate both sides of expression (4) for sillimanite. G s = G s 0 + V s P …… (7) Refer to the text at the back of the book. Substitute − 2443.88 × 10 3 J for G k 0 , 10 5 Pa and 44.09 × 10 − 6 m 3 for V k in expression (5). G k = − 2443875.591 J Substitute − 2442.66 × 10 3 J for G k 0 , 10 5 Pa and 51.53 × 10 − 6 m 3 for V a in expression (6). G a = − 2442654.847 J Substitute − 2440.99 × 10 3 J for G k 0 , 10 5 Pa and 49.90 × 10 − 6 m 3 for V s in expression (7). G s = − 2440985.01 J Refer to the back of the book. Substitute 298 K for T 1 , − 2442654.847 J for G a , − 2443875.591 J for G k , 93.22 J ⋅ K − 1 for S a and 83.81 J ⋅ K − 1 for S k in expression (2). T k a = 298 K + − 2442654.847 J + − 2443875.591 J 93.22 J ⋅ K − 1 − 83.81 J ⋅ K − 1 = 428 K Substitute 298 K for T 1 , − 2440985.01 J for G a , − 2443875.591 J for G k , 96.11 J ⋅ K − 1 for S s and 83.81 J ⋅ K − 1 for S k in expression (2). T k s = 298 K + − 2440985.01 J + − 2443875.591 J 96.11 J ⋅ K − 1 − 83.81 J ⋅ K − 1 = 533 K Substitute 298 K for T 1 , − 2440985.01 J for G a , − 2442654.847 J for G a , 96.11 J ⋅ K − 1 for S s and 93.22 J ⋅ K − 1 for S a in expression (2). T a s = 298 K + − 2440985.01 J + − 2442654.847 J 96.11 J ⋅ K − 1 − 93.22 J ⋅ K − 1 = 876 K Refer to the back of the book

1st Edition

ISBN: 9780201380279

Author: Daniel V. Schroeder

Publisher: Addison Wesley

See similar textbooks

Question

Chapter 5.3, Problem 39P

To determine

The slope of all three phase boundaries and the variation of the phase diagram.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 5.3, Problem 38P

Chapter 5.3, Problem 40P

Students have asked these similar questions

Problem 5.5. A superconducting alloy has Hea = 150 koe and K = 96. Find the energy of an isolated vortex in this alloy. Compare it with the condensation energy stored in the normal core of the vortex.

Statistical Physics. Microcanonical Ensemble.

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.

Chapter 5 Solutions

An Introduction to Thermal Physics

Ch. 5.1 - Prob. 1P Ch. 5.1 - Consider the production of ammonia from nitrogen...Ch. 5.1 - Prob. 3P Ch. 5.1 - Prob. 4P Ch. 5.1 - Consider a fuel cell that uses methane (natural...Ch. 5.1 - Prob. 6P Ch. 5.1 - The metabolism of a glucose molecule (see previous...Ch. 5.1 - Derive the thermodynamic identity for G (equation...Ch. 5.1 - Sketch a qualitatively accurate graph of G vs. T...Ch. 5.1 - Suppose you have a mole of water at 25C and...

Ch. 5.1 - Suppose that a hydrogen fuel cell, as described in...Ch. 5.1 - Prob. 12P Ch. 5.1 - Prob. 13P Ch. 5.1 - Prob. 14P Ch. 5.1 - Prob. 15P Ch. 5.1 - Prob. 16P Ch. 5.1 - Prob. 17P Ch. 5.2 - Prob. 18P Ch. 5.2 - In the previous section 1 derived the formula...Ch. 5.2 - Prob. 20P Ch. 5.2 - Is heat capacity (C) extensive or intensive? What...Ch. 5.2 - Prob. 22P Ch. 5.2 - Prob. 23P Ch. 5.3 - Go through the arithmetic to verify that diamond...Ch. 5.3 - Prob. 25P Ch. 5.3 - How can diamond ever be more stable than graphite,...Ch. 5.3 - Prob. 27P Ch. 5.3 - Calcium carbonate, CaCO3, has two common...Ch. 5.3 - Aluminum silicate, Al2SiO5, has three different...Ch. 5.3 - Sketch qualitatively accurate graphs of G vs. T...Ch. 5.3 - Sketch qualitatively accurate graphs of G vs. P...Ch. 5.3 - The density of ice is 917kg/m3. (a) Use the...Ch. 5.3 - An inventor proposes to make a heat engine using...Ch. 5.3 - Below 0.3 K the Slope of the 3He solidâ€“liquid...Ch. 5.3 - Prob. 35P Ch. 5.3 - Effect of altitude on boiling water. (a) Use the...Ch. 5.3 - Prob. 37P Ch. 5.3 - Prob. 38P Ch. 5.3 - Prob. 39P Ch. 5.3 - The methods of this section can also be applied to...Ch. 5.3 - Suppose you have a liquid (say, water) in...Ch. 5.3 - Ordinarily, the partial pressure of water vapor in...Ch. 5.3 - Assume that the air you exhale is at 35C, with a...Ch. 5.3 - Prob. 44P Ch. 5.3 - Prob. 46P Ch. 5.3 - Prob. 47P Ch. 5.3 - Prob. 48P Ch. 5.3 - Prob. 49P Ch. 5.3 - The compression factor of a fluid is defined as...Ch. 5.3 - Prob. 51P Ch. 5.3 - Prob. 52P Ch. 5.3 - Repeat the preceding problem for T/Tc=0.8.Ch. 5.3 - Prob. 54P Ch. 5.3 - Prob. 55P Ch. 5.4 - Prove that the entropy of mixing of an ideal...Ch. 5.4 - In this problem you will model the mixing energy...Ch. 5.4 - Suppose you cool a mixture of 50% nitrogen and 50%...Ch. 5.4 - Suppose you start with a liquid mixture of 60%...Ch. 5.4 - Suppose you need a tank of oxygen that is 95%...Ch. 5.4 - Prob. 62P Ch. 5.4 - Everything in this section assumes that the total...Ch. 5.4 - Figure 5.32 shows the phase diagram of plagioclase...Ch. 5.4 - Prob. 65P Ch. 5.4 - Prob. 66P Ch. 5.4 - Prob. 67P Ch. 5.4 - Plumbers solder is composed of 67% lead and 33%...Ch. 5.4 - What happens when you spread salt crystals over an...Ch. 5.4 - What happens when you add salt to the ice bath in...Ch. 5.4 - Figure 5.35 (left) shows the free energy curves at...Ch. 5.4 - Repeat the previous problem for the diagram in...Ch. 5.5 - If expression 5.68 is correct, it must be...Ch. 5.5 - Prob. 74P Ch. 5.5 - Compare expression 5.68 for the Gibbs free energy...Ch. 5.5 - Seawater has a salinity of 3.5%, meaning that if...Ch. 5.5 - Osmotic pressure measurements can be used to...Ch. 5.5 - Because osmotic pressures can be quite large, you...Ch. 5.5 - Most pasta recipes instruct you to add a teaspoon...Ch. 5.5 - Use the Clausiusâ€“Clapeyron relation to derive...Ch. 5.5 - Prob. 81P Ch. 5.5 - Use the result of the previous problem to...Ch. 5.6 - Prob. 83P Ch. 5.6 - Prob. 84P Ch. 5.6 - Prob. 85P Ch. 5.6 - Prob. 86P Ch. 5.6 - Sulfuric acid, H2SO4, readily dissociates into H+...Ch. 5.6 - Prob. 88P Ch. 5.6 - Prob. 89P Ch. 5.6 - When solid quartz dissolves in water, it combines...Ch. 5.6 - When carbon dioxide dissolves in water,...Ch. 5.6 - Prob. 92P

Knowledge Booster

The slope of all three phase boundaries and the variation of the phase diagram.

Solution Summary: The author explains the slope of all three phase boundaries and the variation of the phase diagram.

The slope of all three phase boundaries and the variation of the phase diagram.

Want to see the full answer?

Chapter 5 Solutions