Chemistry 331 Lecture 12 Free Energy Functions System and surroundings both play in role in the entropy In an isolated system the criterion d S &gt; 0 indicates that a process is spontaneous. In general, we must consider dSsys for the system and dSsurr for surroundings. Since we can think of the entire universe as an isolated system d Stotal &gt; 0. The entropy tends to increase for the universe as a whole. If we decompose dStotal into the entropy change for the system and that for the surroundings we have a criterion for spontaneity for the system that also requires consideration of the entropy change in the surroundings. The free energy functions will allow us to eliminate consideration of the surroundings and to express a criterion for spontaneity solely in terms of parameters that depend on the system. NC State University Free Energy at Constant T and V Starting with the First Law dU = w + q At constant temperature and volume we have w = 0 and dU = q Recall that dS q /T so we have dU TdS which leads to dU - TdS 0 Since T and V are constant we can write this as d(U - TS) 0 The quantity in parentheses is a measure of the spontaneity of the system that depends on known state functions. Definition of Helmholtz Free Energy We define a new state function: A = U -TS such that dA 0. We call A the Helmholtz free energy. At constant T and V the Helmholtz free energy will decrease until all possible spontaneous processes have occurred. At that point the system will be in equilibrium. The condition for equilibrium is dA = 0. A time Definition of Helmholtz Free Energy Expressing the change in the Helmholtz free energy we have A = U T S for an isothermal change from one state to another. The condition for spontaneous change is that A is less than zero and the condition for equilibrium is that A = 0. We write A = U T S 0 (at constant T and V) If A is greater than zero a process is not spontaneous. It can occur if work is done on the system, however. The Helmholtz free energy has an important physical interpretation. Noting the q rev = T S we have A = U qrev According to the first law U qrev = w rev so A = wrev (reversible, isothermal) A represents the maximum amount of reversible work that can be extracted from the system. Definition of <a href="/keyword/gibbs-free-energy/" ><a href="/keyword/gibbs-free/" >gibbs free</a> energy</a> Most reactions occur at constant pressure rather than constant volume. Using the facts that qrev = TdS and w rev = -PdV we have: dU TdS PdV which can be written dU - TdS + PdV 0. The = sign applies to an equilibrium condition and the &lt; sign means that the process is spontaneous. Therefore: d(U - TS + PV) 0 (at constant T and P) We define a state function G = U + PV TS = H TS. Thus, dG 0 (at constant T and P) The quantity G is called the Gibb's free energy. In a system at constant T and P, the Gibb's energy will decrease as the result of spontaneous processes until the system reaches equilibrium, where dG = 0. 1 Comparing Gibbs and Helmholtz The quantity G is called the Gibb's free energy. In a system at constant T and P, the Gibb's energy will decrease as the result of spontaneous processes until the system reaches equilibrium, where dG = 0. Comparing the Helmholtz and Gibb's free energies we see that A(V,T) and G(P,T) are completely analogous except that A is valid at constant V and G is valid at constant P. We can see that G = A + PV which is exactly analogous to H = U + PV the relationship between enthalpy and internal energy. For chemical processes we see that G = H T S 0 (at constant T and P) A = U T S 0 (at constant T and V) Conditions for Spontaneity We will not use the Helmholtz free energy to describe chemical processes. It is an important concept in the derivation of the Gibbs energy. However, from this point we will consider the implications of the Gibbs energy for physical and chemical processes. There are four possible combinations of the sign of H and S in the <a href="/keyword/gibbs-free-energy/" ><a href="/keyword/gibbs-free/" >gibbs free</a> energy</a> change: H &gt;0 &lt;0 &lt;0 &gt;0 S &gt;0 &lt;0 &gt;0 &gt;0 Description of process Endothermic, spontaneous for T &gt; H/ S Exothermic, spontaneous for T &lt; H/ S Exothermic, spontaneous for all T Never spontaneous Gibbs energy for a phase change For a phase transition the two phases are in equilibrium. Therefore, G = 0 for a phase transition. For example, for water liquid and vapor are in equilibrium at 373.15 K (at 1 atm of pressure). We can write Gibbs energy for a phase change However, if we wer...