Heat Capacity of Mono-atomic Solids

By: Chris Hayes, Tammy Johnson, Rich Parry

            

 

   

In the early 1800s Dulong and Petit both French scientists found several heat capacities of monoatomic solids.  Using their data, which was later to be found less than reliable, stated that all heat capacities of monoatomic solids are the same and are close to 25J/Kmol.  Heat capacity is defined as Cv= (dU/dT)v

   

If classic physics could hold water the equipartion principle could be used to calculate heat capacity of a solid.  This theory states that "the average energy of an atom as it oscillates about its mean position in a solid is (kT) for each direction of displacement."  Since atoms can only oscillate in three dimensions the mean energy is 3kT for 1 atom.  For all atoms present in the solid the average energy is 3NkT.  This motion contributes to the molar energy internal energy is

Um=3NkT=3RT

This is because Na*k=R.  R being the gas constant.  Using this the molar heat capacity should be...

Cv,m=(dUm/dT)v = 3R           3R=24.9 J/K*mol

This is very close to Dulong's and Petit's data.

  Using this principle, the average energy of a 3-dimensionally oscillating atom in a solid is 3kT; where k is the Boltzmann constant and T is the temperature.  For N number of atoms in the system, the total amount of energy from the oscillating atoms becomes 3NkT.  This contributes to the internal energy, U, of the system. 

                                               

                                                U= 3NkT

 

We know that the molar gas constant, R, is equal to NAk, where NA is Avogadro’s number.  Thus, the internal energy can be rewritten as

 

                                                U= 3RT

 

            As the temperature of a system is raised, the internal energy also increases.  The conditions under which this heating takes place determines the increase.  Assuming we are working under constant volume, the plot of internal energy versus temperature gives us a tangent to the curve whose slope is the heat capacity of the system.  The heat capacity at constant volume is defined as

 

                                                CV= (dU/dT)V = 3R

 

for a solid at high temperatures. 

            The molar heat capacity is what we are trying to determine.  Since we know that the oscillating motion contributes 3NkT to the internal energy, we can write

 

                                                Um= Um(0) + 3NkT

 

where Um is the molar internal energy and Um(0) is the molar internal energy at T=0.  Since we know that NAk=R, the above equation becomes

 

                                                Um= Um(0) + 3RT

 

Thus, from the above equation, upon the differentiation of the equation with respect to T, the molar heat capacity becomes

 

                                                CV,m= d/dT (Um(0) +3RT) = 3R

 

 

  

Once scientists could measure heat capacities at lower temperatures major deviations from this data were found.  Scientists found that ALL metals heat capacities were lower than 3R and that the values of the heat capacities approached 0 as T in Kelvin approaches 0.

           

             In 1905, Einstein assumed that every atom would oscillate with a single frequency, n, about its equilibrium position.  Using Planck’s hypothesis of the energy of oscillation possessing discrete values, nhn, where n is an integer assigned to the discrete energy levels, h is Planck’s constant, and n, is the frequency of oscillation, Einstein calculated the contribution to the total internal energy of the metal from the oscillating atoms, and came up with

 

                                    U= 3N(hn/(eh n/kT – 1))

 

The above equation replaced the classical equation 3RT.  Through the differentiation of U with respect to T, Einstein determined the heat capacity.  To make the equation easier to work with, we rewrite it as

           

                         U= 3Nhn(1/( eh n/kT – 1)) = 3Nhn(( eh n/kT – 1)-1)

 

We then differentiate U with respect to T and get

 

                        dU/dT= 3Nhn d/dT (( eh n/kT – 1)-1)

 

                          = 3Nhn(-1) (( eh n/kT – 1)-2)(-hn/kT2)( eh n/kT )

 

                           =(3Nh2n2/kT2) ( eh n/kT /(eh n/kT – 1)2)

 

Recall that R=NAk and multiply the equation by k/k.  This gives Einsteins formula for the heat capacity of a monoatomic solid.

 

                             CV,m= 3R(hn/kT)2 (eh n/kT /( eh n/kT – 1)2)

 

It is known that Einstein’s temperature, qE, is defined as hn/k.  This is a way of expressing the frequency of oscillation of atoms as a temperature.  Substituting qE into the above equation gives

 

                             CV,m= 3R(qE /T)2 (eqE/T  /( eQE/T – 1)2)

 

From which we can say that CV,m= 3Rf2 and

 

                                                f=(qE /2T) (eqE/T  /( eQE/T – 1))

 

At high temperatures, upon expansion of f, the equation of f yields

 

                f=(qE/T)((1+(qE/2T)+( qE/2T)2+…)/((1+(qE/T)+ (qE/T)2+…)~1

 

Thus, this reduces to CV,m= 3R at high temperatures, which is the classical approach to heat capacity. Proving once again that quantum reduces to classical if under the right conditions.  

We used maple to explain part A and B on problem 11.3 in our text books.

../pchem 1.mws

Adkins, Peter, and Julio de Paula. Physical Chemistry. New York: Freeman, 2002.

Lalidler, Kevin and John Meiser. Physical Chemistry. Boston: Houghton, 1999.

Sibley, Mike and Tim Alberty. Physical Chemistry. New York: Wiley & sons, 2001.

Storm, David. Physics web page. 9 February 2003.

http://faculty.uvi.edu/users/dstorm/classes/00phy241/mods/lect_44.html

Veytsman, Boris. Equipartion Theorem. 9 February 2003.

http://www.plmsc.psu.edu/~www/matsc597c-1997/systems/Lecture4/node1.html