Energy In A Mole Of A Crystalline Solid

Energy In A Mole Of A Crystalline Solid
TECTechnics Classroom TECTechnics Overview

The above photo is of a sample of crystalline scandium (Sc, atomic number 21).
The atoms and molecules of a solid are in constant motion at ordinary temperatures eventhough there is equilibrium spacing between them. Thermal agitation due to heat causes the atoms and molecules to oscillate about their equilibrium positions oftentimes at very high frequency. The resulting kinetic energy acquired by the atoms and molecules constitute the thermal energy of the substance.
Consequently, both kinetic energy (thermal) and potential energy (structure) consitute the total energy in solids and liquids (this potential energy is not in gases).

(a) Suppose an individual atom of a crystalline solid behaves as a point mass. Calculate the total energy of the atom at 25^oC.

(b) Calculate the total energy in one mole of the crystalline solid 25^oC.

The strings:

S₇P₃A₃₂ (Force - Push)

The math:
Crystalline Scandium
Pj Problem of Interest is of type force. Energy is the capacity for work. It is force that is the doer of the work. So energy and work problems are of type force. In this instance, the total energy consists of kinetic energy and potential energy about 50-50. The kinetic energy is due to atom collissions (more of a force-push), the potential energy is more of a force-pull. So either categorization is appropriate. force-push is selected here because of the influence of temperature.

(a) On average, the total energy of each atom of a crystalline solid = kT for each degree of freedom.
Where, T is the absolute temperature in degree Kelvin (K)
and k is Boltzmann's Constant = 3.30 x 10^-24 (calorie/^oK)/atom.
Since the atom is assumed to be a point mass, it has 3 degrees of freedom in translation and none in rotation.
So, average total energy of atom = 3kT = 3 x 3.30 x 10^-24 x 298
= 2.77 x 10^-22 cal.

(b) A mole of a crystalline substance contains N atoms.
Where N is Avogadro's Number = 6.02 x 10^-22
So, total energy in one mole of crystalline solid = 3NkT
The product, Nk is called the universal gas constant, R
R = 1.987 (cal/^oK)/mole
So, total energy in one mole of crystalline solid at 25^oC = 1788 cal.

Energy = 3RT is approximately correct for many materials at ordinary temperatures. It is not correct at low temperatures.

Math

The point . is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.
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