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Unformatted text preview: The Gibbs free energy and non
PV work So far in our discussion of the work we have assumed it is associated with the expansion or compression of some material, in which case it is given by the formula δW =
PdV. We will call it a PV
type of work. In general, other types of work are possible. For example if our gas contains charged particles and is in an electric field, then we should add the term φ dQ where φ is the potential difference and dQ the transferred charge. Or, if we stretch a piece of rubber, the stretching force f will do the work f dl, where l is the extension. At the same time, the PV work done on the rubber will be –P dV, where P is the external (atmospheric) pressure. So generally we should write: δW = −PdV + δWnon
PV We now have: T dS ≥ dU + p dV
δWnon
PV add
(TS):
SdT ≥ d(U –TS) + pdV
δWnon
PV add V dp :
SdT + V dp ≥ dG
δWnon
PV or δWnon
PV ≥ dG + SdT – V dp Therefore, for a process with constant p and T we have: Wnon
PV ≥ G(2)
G(1) The minimum amount of non
PV work required to perform a process at constant pressure and temperature is the work done in a reversible process and is equal to the change of the system’s Gibbs energy ΔG (provided that it is positive. If it is negative, no work is needed. On the contrary, the system can perform work equal to
ΔG) Obtaining thermodynamic functions A,U,H,S,G from measurable quantities (P,V,T, CP, CV). If we know the free energies A, G, or the entropy S, we can say many useful things about our system (such as the state of the system in thermodynamic equilibrium). Unfortunately, A, U, H, S, G etc. cannot be measured directly so before we can start taking advantage of our findings concerning these functions we need to figure out how to relate them to what can be measured. P, V, and T are obviously measurable. In addition, heat capacities can be measured through calorimetry (which is essentially based on bringing a body with an unknown heat capacity in contact with a reference object of known heat capacity and observing the temperature, at which they equilibrate. If we know the initial temperatures of each object we can deduce the unknown heat capacity quite easily – see the equilibration example discussed previously). So we’d like to relate the thermodynamic functions A, G, U, S, H to the measurable quantities (heat capacities and the equation of state). To make progress, we will need some identities that follow from the 1st and the 2nd laws. Here and below we will be talking about equilibrium processes so that the 2nd law has the form: dS = δ q / T . What happens to A or G if the state is changed in such an equilibrium process? We have dA = dU − TdS − SdT = (− PdV + TdS ) − TdS − SdT = − PdV − SdT similarly (1a) dG = dA + PdV + VdP = VdP − SdT (1b) On the other hand, viewing A as a function of V and T or G as a function of P,T, we can write the formal identities: dA = ⎜ ⎛ ∂A ⎞ ⎛ ∂A ⎞ ⎟ dV + ⎜ ⎟ dT ⎝ ∂V ⎠T ⎝ ∂T ⎠V (2a) ⎛ ∂G ⎞ ⎛ ∂G ⎞ dG = ⎜ ⎟ dP + ⎜ ⎟ dT ⎝ ∂P ⎠T ⎝ ∂T ⎠ P
Comparing Eqs. 1a and 2a, we find (2b) ⎛ ∂A ⎞ P = −⎜ ⎟ ⎝ ∂V ⎠T (3) ⎛ ∂A ⎞ S = −⎜ ⎟ ⎝ ∂T ⎠V
(4) So if the function A(V,T) is known, P and S can be calculated for any V and T. Example: Suppose for some material A = − RT ⎢ ln V + ⎡ ⎣ 3 ⎤ ln T ⎥ . What is this material? 2 ⎦ 3 R ln T + constant . Therefore 2 the material is an ideal gas whose heat capacity is equal to CV = 3R / 2 Solution: Applying Eqs. 3
4 we find P = RT / V , S = R ln V + More generally, if the free energy A (or G) for a material is known as a function of V & T (or P & T) then its any other thermodynamic property can be calculated straightforwardly. Eq. 3, for example, gives us the equation of state (i.e., P as a function of V and T). To calculate the internal energy, we use the definition of A = U
TS and Eq. 4: ⎛ ∂A ⎞ U = A + TS = A − T ⎜ ⎟ ⎝ ∂T ⎠V
If we wish now to compute the heat capacity we use: ∂⎛ ⎛ ∂U ⎞ ⎛ ∂A ⎞ ⎞ CV = ⎜ ⎜ A−T ⎜ ⎟= ⎟⎟ ⎝ ∂T ⎠V ∂T ⎝ ⎝ ∂T ⎠V ⎠V
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This note was uploaded on 04/24/2011 for the course CH 52635 taught by Professor Makarov during the Spring '11 term at University of Texas at Austin.
 Spring '11
 Makarov

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