State Functions, Exact Differentials, and Maxwell Relations 1. A Theorem From Calculus Consider the differential form: df ≡ M(x, y)dx + N(x, y)dy. (1) If can we define a single-valued, differentiable function f(x,y) which satisfies Eq. (1), then M(x, y)dx + N(x, y)dy is said to be an exact differential. Of course, we can always define f(x,y) by integrating the right hand side of Eq. (1) along some path; however, we require that the func- tion be single-valued (i.e., that it be a state function); hence, different paths must give the same answer. THEOREM: If M and N have continuous first partial derivative satify all points of some open rectangle, the differential form, (1), is exact at each point of the rectangle if and only if the condition ( ∂M /dx)y=( ∂N /dy)x (2) is satisfied throughout the rectangle. When this holds, the function f(x,y) is given by the line integral f (x, y) =∫ M(s, t)ds + N(s, t)dt (3) 2. Applications to Thermodynamics: Maxwell Relations We hav e been able to combine the first and second laws of thermodynamics to write dE = TdS − PdV , (a) (d2E /dv.dt )= (dt/dv) –( dp/ds) d2E = 0 thus Eq. (4) gives (dT/dV)s= (dP/dS)t. ( 1) (d2E /dt.dv )= (ds/dv) –( dp/dt) 0 = (ds/dv)t – (dp/dt)v So (ds/dv)t = (dp/dt)v (2) This is called a Maxwell relation and is a powerful tool for relating different quantities thermo- dynamics. Another Maxwell relation can be obtained from the enthalpy for which dH = TdS + VdP Hence, Eq. (2) gives d2H/(dT.dP) =(dT/dp)+(dv/dt) 0 = (dt/dp)s + (dv/dt)p hence (dT/dP)s =- (dV/dT)p (3) And d2H/(dT.dP) =(ds/dp)+(dv/dt) hence (ds/dp)t = -(dv/dt)p (4) Clearly every state function will generate one or more Maxwell relations. The trick is to know which ones to use in any given application. With entropy: Entropy is a function to, T,P,V,N dS(t,p,v,n)=(dS/dT)p,v,n+(dS/dP)t,v,n+(dS/dV)t,p,n+(dS/dN)tpv dS= nCp+(dS/dP)t,v,n+(dS/dV)t,p,n+(dS/dN)tpv S=Q/T=Q/P=Q/V=Q/N ds = (dq/dt)dt+(ds/dp)dp+(ds/dv)dv+(ds/dn)dn (dq/dt)+(ds/dp)+(ds/dv)+(ds/dn)=d2s Cp+dS/dP+dS/dV+dS/dN =0 dS/dP = -dS/dV = Q/dV nCp dT = Smol*dN
Posted on: Thu, 01 Jan 2015 15:08:34 +0000
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