Thermodynamics and Molar Specific Heats

Thermodynamics and Molar Specific Heats is the sixth lecture within the Properties of Matter section of PH1011. It covers thermodyamics, the interactions of heat with energy in various systems, rotational motion in terms of kinetic energy, equipartition (degrees of freedom), and a comparison of solids, liquids and gases in this context.

Previous: Statistical Mechanics

Next: Heat Transfer

Thermodynamics
The first law of thermodynamics states that energy cannot be created or destroyed within a system; this is mathematically described by ∆Eint = Q - W, where these mean change in internal energy, energy/heat input and work done by the system respectively. The work done is given by W = P dV. This equation changes depending on factors affecting the system:


 * Adiabatic conditions (no energy input; Q=0) give ∆Eint = - W
 * Constant volume (dV=0) gives ∆Eint = Q
 * Constant pressure (dP) gives W=PdV = nR(dT)
 * Constant temperature (dT=0) gives W = nRT ln(V1-V2)
 * Closed cycle (∆Eint=0) gives Q = W
 * Free expansion (Q=W=0) gives ∆Eint = 0

To find the heat required to cause a given rise in temperature, use the average kinetic energy: for monatomic gases this is 3/2kT. If the internal energy is therefore E = nNAK, k being R/NA, E = 3/2nRT. At constant volume, W=0, ∆Eint = Q = 3/2nR∆T. The molar specific heat (at constant volume) can be defined here as Cv = 3/2R (with ∆Eint = Q = nCv∆T).

At constant pressure the equation is similar - as Q = ∆Eint + W = 3/2nR∆T + P∆V = 3/2nR∆T + nR∆T (ideal gas law). This gives Cp = 3/2R +R = 5/2R

Degrees of Freedom
The equipartition theorem states that the average kinetic energy of a stem is relevent because the total kinetic energy is shared equally between its independent parts.

As atoms have 1/2kT energy per degree of freedom (direction of motion), a diatomic molecule will have more degrees of freedom than a monatomic molecule (it gains two rotational motion degrees of freedom). Therefore a diatomic gas will have Eint = 5/2kT, and a polyatomic molecule will have 3kT. This gives Cv = f/2R and Cp = f/2R+R, with f being the degrees of freedom. Changing temperature can also cause change in degrees of freedom - increasing temperature will cause the molecules to vibrate, adding degrees of freedom, whereas very low temperatures will cause rotational motion to cease.

Solids and Liquids
Solids and liquids are more difficult to compare in terms of degrees of freedom - in a rough average their Eavg = 3kT; Cv = 3R. Calculating the necessary increase in temperature for solids and liquids usually requires measurement of the specific heat capacity.

Summary
As the energy in a system must remain constant, the first law of thermodynamics (∆Eint = Q - W) applies. This varies depending on the exact conditions of the system but allows calculation of specific molar heats in constant volume and pressure systems; these are dependent on the degrees of freedom that the molecules in question have. Monatomic gases have three degrees of freedom at standard temperatures, diatoic gases that do not oscillate have five and polyatomic molecules have six. Oscillation adds further degrees of freedom but in practice does not occur until very high temperatures.