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THERMODYNAMICS RELATIONS Introduction Some thermodynamic properties can be measured directly, but many others cannot. Therefore, it is necessary to develop some relations between these two groups so that the properties that cannot be measured directly can be evaluated. The derivations are based on the fact that properties are point functions, and the state of a simple, compressible system is completely specified by any two independent, intensive properties. Some Mathematical Preliminaries Thermodynamic properties are continuous point functions and have exact differentials. A property of a single component system may be written as general mathematical function z = z(x,y). For instance, this function may be the pressure P = P(T,v). The total differential of z is written as Fundamental Property Relations: Apart from enthalpy, internal energy and entropy, there are other thermodynamic properties which are necessary to apply to phase equilibria and reaction equilibria. So far we have treated only closed systems which consist of single phase without any reaction. In order treat the systems we need to know other energy terms involved with reaction and phase equilibria. The use these properties will get clearer when we do the chapters on these areas. Let us derive the fundamental property relations from basics. Enthalpy: From I law, dU dQ dW dU TdS pdV H U pV dH dU d(pV) dU pdV Vdp TdS pdV pdV Vdp dH TdS Vdp Helmholtz free energy: It is defined as F = U – TS Differentiating, dF = dU-d(TS) = TdS – pdV – TdS – SdT, dF= – pdV – SdT Gibb’s Free energy: It is defined as G = H – TS Differentiating, dG= dH –TdS – SdT = TdS +Vdp – TdS – SdT, dG = Vdp - SdT These four equations are called fundamental property relations as many equations are derived from these equations. Maxwell’s relations As some of the thermodynamic properties are not directly measurable, we must write the equations in terms of measurable properties. The measurable and determinable (by experiment) properties are temperature, pressure, volume, heat capacities and in some cases even enthalpies. These relations are derived from fundamental property relations. The derivations are based upon a mathematical relation applied to exact differential equations. f f dx dy x y y x If z = f(x,y) then df This equation is also written as df = Mdx + Ndy For this differential equation to be exact, M y N x x y Applying these conditions to all the property relations, we get, df Mdx Ndy M y N x x y dU TdS pdV T p V S S V dH TdS Vdp T V p S S p dA pdV SdT p S T V V T dG Vdp SdT S V T p p T Generating function: Consider G d RT 1 G 1 RTdG Gd ( RT ) H TS dG dT ( Vdp SdT ) dT 2 2 2 2 RT RT RT RT RT Vdp SdT HdT SdT V H dp dT 2 RT RT RT RT RT RT 2 H G V d dp dT RT 2 RT RT From this equation, at constant temperature and at constant pressure we can write, ( G / RT ) V p T RT H ( G / RT ) ; T RT2 p If we know G/RT in terms of p and T, it can be used to evaluate other thermodynamic properties and hence it is called generating function. Entropy changes: Q Q ; CV T p T V We know that CP CP S S T p T p T Since dQ TdS , CP T CV S S T V T V T And CV T 2 T2 T dT dT Using these, ΔS p CP CP ln and ΔS V CV CV ln 2 T T T1 T1 T1 T1 T2 T S S dT dp T p p T If S is considered as S = f (T,p) then dS In this equation, the first partial derivative is replaced by CP/T which is derived and second by a Maxwell’s relation, we get, dS T, we get, dS CP V dT dp Similarly by taking S as function of V and T T p . CV p dT dV T T V If there is any relation between three variables, from mathematics we have, a relation V p T between p, V and T as 1 p T T V V p Using these relations, we can get, dS ( V / T )p CV dT dV T ( V / p )T ( V / T )p dU CV dT T p dV ( V / p )T V dp pdV T p Internal energy may also be written as dU C P dT T V dp T p Using dH TdS Vdp , we get dH C P dT V T Internal energy changes: The internal energy changes can be expressed in terms measurable quantities such as heat capacities, pressure, volume and temperature. We can use Maxwell’s relations to get these expressions. The following are fundamental property relations. dU TdS pdV dH TdS Vdp dA pdV SdT dG Vdp SdT dS CV p dT dV T T V ( V / T ) p C dS V dT dV T ( V / p )T dS We have derived the following relations. V p CP V dT dp T T p p T 1 T T V V p Let us derive the expression for internal energy in terms CP and CV. dU TdS pdV Replace dS in terms of CV, ( V / T )p C dU T V dT dV pdV ( V / p )T T Internal energy change in terms of Cp: ( V / T )p dU CV dT T p dV ( V / p )T dU TdS pdV Replace dS in terms of Cp, C V dU T P dT dp pdV T p T V dU CP dT T dp pdV T p Enthalpy changes: Starting from dH TdS Vdp in the same way we can get, V dH CP dT V T dp T p We can use these expressions and find the changes in enthalpy. These equations are applicable for any gas. P-V-T relations used should be for the gas for which you are determining the changes. Use these relations and derive the expressions for dH and dU for ideal gas. Effect of p, V and T on U, H and S: S V p T p T This is one of the Maxwell’s relations which gives the effect of pressure on entropy at constant temperature. C S P T T p This equation derived earlier gives the effect of temperature on entropy at constant pressure. ( V / T )p dU CV dT Vdp T dV ( V / p )T dU At constant volume, CV dT U CV T V Or This gives the effect of temperature on internal energy at constant volume. At constant ( V / T )p U pthe T This gives effect of volume of internal V T ( V / p )T temperature, since dT is zero, energy at constant temperature. Similarly from we can get, V dH CP dT V T dp T p At constant pressure, H CP T p H V V T T p p T At constant temperature, These expressions give the effect of temperature and pressure on enthalpy. Relationship between the heat capacities: The heat capacities are related by the following expression. V p CP CV T T p T V which can also be written as 2 V p CP CV T T p V T using the cyclic relation of partial derivatives of p, V and T. 1 V ; V T p k 1 V are V p T called Volume expansivity and Compressibility. The difference between heat capacities is written in terms of these as CP CV 2VT Effect of pressure and volume on heat capacities: These are given by the following equations. C P p 2V T T 2 T 2V C P T 2 V T T p p V T p 2 p CV T 2 V T T V CV p 2 p V T T 2 p T p T isothermal