Exploring the science with CES Elements 273

12.11 Exploring the science with CES Elements 273

Exercise E12.17 When a solid melts, some of the bonds between its atoms are broken, but not all—liquids still have a bulk modulus, for example. You might then expect that the latent heat of melting, L m , should be less than the cohesive

energy, H c , since it is the basic measure of the strength of the bonding. Plot one against the other, using CES Elements (L m is called the heat of fusion in the database). By what factor is L m less than H c ? What does this tell you about cohesion in the liquid?

Exercise E12.18 The latent heat of melting (heat of fusion), L m , of a material is said to be about equal to the heat required to heat it from absolute zero to its melt- ing point, C p T m , where C p is the specific heat and T m is the absolute melt- ing point. Make a chart with L m on one axis and C p T m on the other. To make the comparison right we have to change the units of L m in making the chart to J/kg instead of kJ/mol. To do this multiply L m by

10 6 Atomic weight in kg/kmol

using the ‘Advanced’ facility when defining axes in CES. Is the statement true? Exercise E12.19 The claim was made in the text that the modulus E is roughly proportional

to the absolute melting point T m . If you use CES Elements to explore this cor- relation you will find that it is not, in fact, very good (try it). That is because T m and E are measured in different units and, from a physical point of view, the comparison is meaningless. To make a proper comparison, we use instead

k B T m and EΩ, where k B is Boltzmann’s (1.38 ⫻ 10 ⫺23 J/K) constant and Ωm 3 /atom is the atomic volume. These two quantities are both energies, the first proportional to the thermal energy per atom at the melting point and the second proportional to the work to elastically stretch an atomic bond. It makes better sense, from a physical standpoint, to compare these.

Make a chart for the elements with k B T m on the x-axis and EΩ on the y- axis to explore how good this correlation is. Correlations like these (if good) that apply right across the Periodic Table provide powerful tools for check- ing data, and for predicting one property (say, E) if the other (here, T m ) is known. Formulate an equation relating the two energies that could be used for these purposes.

Exercise E12.20 Above the Debye temperature, the specific heat is predicted to be 3R, where R is in units of kJ/kmol.K. Make a plot for the elements with Debye temperature on the x-axis and specific heat in these units on the y-axis to explore this. You need to insert a conversion factor because of the units. Here it is, expressed in the units contained in the database:

Specific heat in kJ/kmol.K ⫽ Specific heat in J/kg.K

⫻ Atomic weight in kg/kmol /1000.

274 Chapter 12 Agitated atoms: materials and heat

Form this quantity, dividing the result by R ⫽ 8.314 kJ/kmol.K, and plot it against the Debye temperature. The result should be 3 except for materials with high Debye temperatures. Is it? Fit a curve by eye to the data. At roughly what temperature does the drop-off first begin?

Exercise E12.21 Explore the mean free path of phonons ᐉ m in the elements using equation (12.10) of the text. Inverting it gives

λ m ⫽ 3 ρ Cc

po

in which the speed of sound c o ⫽ 兹苶 E/ρ. Use the ‘Advanced’ facility when defining the axes in CES to make a bar chart of ᐉ m for the elements. Which materials have the longest values? Which have the shortest?

Chapter 13

Running hot: using materials

at high temperatures

Shuttle flame. (Image courtesy of C. Michael Holoway, NASA Langley, USA.)