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Multivariable Calculus In thermodynamics, we will frequently deal with functions of more than one variable e.g., PPT,,Vn, UUT,V,n, UUT,,Pn U = energy n = # moles extensive variable: depends on the size of the system intensive variable: independent of the size of the system V, n are extensive, P, T, molar volume V/n are intensive change volume at fixed T, n pressure changes P dP dV V nT, these variables are cylinder with piston held constant Suppose we have an ideal gas PV nRT PnRT 2 VV nT, subscripts refer PnR to variables that TV are held constant Vn, PRT nV VT, now suppose we want to see how Pchanges when both V and Tchange PP Note that this is fully consistent with dPdV dT Taylor‐series expansions V T Tn,,Vn If n changes as well P dn need to add a term n TV, In general, yy x,,xx 12 n n y dx x dy i, ' hold fixed all n i1 x' variables except x i Ideal gas PPP dP dT dV dn TVn nV,,Tn T,V nR nRT RT dPdT dV dn 2 VVV for small finite changes nR nRT RT PT V n 2 VVV These equations are most useful when we don't have an analytical function for the quantity of interest. Consider UU (,TP,n) UUU dU dT dP dn TPn Pn,,Tn T,P The derivatives are often available experimentally. We can also write UU T,,Vn UUU dU dT dV dn TVn Vn,,Tn T,V UU Note, in general, TT P,,nVn
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