THERMODYNAMIC PROPERTY RELATIONS OBJECTIVE Develop fundamental relations between

THERMODYNAMIC PROPERTY RELATIONS

OBJECTIVE üDevelop fundamental relations between commonly encountered thermodynamic properties and express the properties that cannot be measured directly in terms of easily measurable properties. üDevelop the Maxwell relations, which form the basis for many thermodynamic relations. üDevelop the Clapeyron equation and determine the enthalpy of vaporization from P, v, and T measurements alone. üDevelop general relations for cv, cp, du, dh, and ds that are valid for all pure substances. üDevelop a method of evaluating the Δh, Δu, and Δs of real gases through the use of generalized enthalphy & entrophy departure chart

PARTIAL DERIVATIVES & ASSOCIATED RELATIONS The state postulate The state of a simple, compressible substance is completely specified by any two independent, intensive properties. All other properties at that state can be expressed in terms of those two properties. The derivative of a function f(x) with respect to x represents the rate of change of f with x.

PARTIAL DERIVATIVES of z The variation of z(x, y) with x when y is held constant with respect to x, and it is expressed as The symbol δ differential changes Difference: Symbol d the total differential change of a function and reflects the influence of all variables, Symbol δ the partial differential change due to the variation of a single variable. The changes indicated by d and δ are identical for independent variables, but not for dependent variables.

To obtain a relation for the total differential change in z(x, y) for simultaneous changes in x and y, consider a small portion of the surface z(x, y) [Fig 3]. When the independent variables x and y change by Δx and Δy, respectively, the dependent variable z changes by Δz, which can be expressed as Adding & subtracting z(x, y + Δy), we get or Taking the limits as Δx 0 & Δy 0, using definitions of partial derivatives, we obtain E 1 Fig 3: Geometric representation of total derivative dz for a function z(x, y). This is the fundamental relation for the total differential of a dependent variable in terms of its partial derivatives with respect to the independent variables.

PARTIAL DIFFERENTIAL RELATIONS Same form as E 1 The order of differentiation is immaterial for properties since they are continuous point functions and have exact differentials. Thus, Demonstration of the reciprocity relation for the function z+2 xy – 3 y 2 z = 0 RECIPROCIT Y RELATION CYCLIC RELATION

THE MAXWELL RELATIONS The equations that relate the partial derivatives of properties P, v, T, and s of a simple compressible system to each other. Obtained from the four Gibbs equations by exploiting the exactness of the differentials of thermodynamic properties Maxwell relations provide means of determining the change in entropy, which cannot be measured directly, by simply measuring the changes in properties P, v, and T, limited to simple compressible systems.

THE CLAPEYRON EQUATION 3 rd Maxwell relation ØDuring a phase-change process, P = Psat, which depends on T only, is independent of the specific volume. Psat = f(Tsat). ØThe partial derivative (δP/δT)v can be expressed total derivative (d. P/d. T)sat = SLOPE of the saturation curve on a P-T diagram at a specified saturation state [Fig 4]. ØThis slope is independent of the specific volume, thus it can be treated as a constant during the integration of above Eq. between two saturation states at the same T for an isothermal liquid-vapor phase change process. For example, the integration yields Or The slope of the saturation curve on a P-T diagram is constant at a constant T or P.

During this process the P also remains constant. Therefore, from Eq. 12 -11, Substituting this result into Eq. 12 -21, we obtain CLAPEYRON EQUATION The Clapeyron equation enables us to determine the enthalpy of vaporization hfg at a given temperature by simply measuring the slope of the saturation curve on a P-T diagram and the specific volume of saturated liquid and saturated vapor at the given temperature. General form of the Clapeyron equation when the subscripts 1 and 2 indicate the two phases.

The Clapeyron equation can be simplified for liquid–vapor and solid– vapor phase changes by utilizing some approximations. At low pressures Treating vapor as an ideal gas Substituting into Clapeyron Equation The Clapeyron–Clausius eq. 1. can be used to determine the variation of saturation pressure with temperature. 2. can also be used in the solid–vapor region by replacing hfg by hig (the enthalpy of sublimation) of the substance Integrating between 2 saturation states CLAPEYRONCLAUSIUS EQUATION

GENERAL RELATIONS FOR du, dh, ds, cv & cp üThe state of a simple compressible system is completely specified by two independent, intensive properties. üAble to calculate all the properties of a system (u, h, and s) at any state once two independent, intensive properties are available. üThe calculation of these properties from measurable ones depends on the availability of simple and accurate relations between the two groups. üDevelopment of general relations for Δu, Δh, Δs in terms of P, v, T, and cv, cp. üThe property values at specified states can be determined only after the selection of a reference state, the choice of which is quite arbitrary

INTERNAL ENERGY CHANGES Consider u = u(T, v) and take its total differential (Eq. 12 -3) Using the definition of cv, we have Consider entrophy to be a function of T and v s = s(T, v) and take its total differential Substituting this into the T ds relation du = T ds – P dv yields

Equating the coefficients of d. T and dv in Eqn. 12 -25 & 12 -27 gives Using the 3 rd Maxwell relation (Eq. 12 -28), we get Substituting this into Eq. 12 -25, we obtain the desired relation for du: Δu general relation The change in internal energy of a simple compressible system associated with a change of state from (T 1, v 1) to (T 2, v 2) is determined by integration of Eqn. 12 -29:

ENTHALPHY CHANGES Consider the enthalpy to be a function of T and P h = h(T, P), and take its total differential, Using the definition of cp, we have Consider again the entrophy to be a function of T and P s = s(T, P) and take its total differential, Substituting this into the T ds relation dh = T ds + v d. P gives

Equating the coefficients of d. T and d. P in Eq. 12 -31 & 12 -33, we obtain Using the 4 th Maxwell relation, we have Substituting this into Eq. 12 -31, we obtain the desired relation for dh: Δh general relation The change in enthalpy of a simple compressible system associated with a change of state from (T 1, P 1) to (T 2, P 2) is determined by integration: The other case can easily be determined by using the definition of enthalpy h = u + Pv:

ENTROPHY CHANGES The first relation is obtained by replacing the 1 st partial derivative in the total differential ds (Eq. 12 -26) by Eq. 12 -28 and the second partial derivative by 3 rd Maxwell relation (Eq. 12 -18), yielding And The 2 nd relation is obtained by replacing the 1 st partial derivative in the total differential of ds (Eq. 12 -32) by Eq. 12 -34 and the 2 nd partial derivative by the 4 th Maxwell relation (Eq. 12 -19), yielding And Δs general relation Either relation can be used to determine the entrophy change. The proper choice depends on the available data

Determine the change in the enthalpy of carbon dioxide (CO 2), in kg/kg, as it undergoes a change of state from 100 k. Pa and 20° C to 600 k. Pa and 300 ° C using the equation of state P(v-a)=RT where a = 0. 01 m 3/kg, and compare the result to the value obtained by using the ideal gas equation of state. SOLUTION: Using Eq. 12 -35, Assumptions: Constant specific heats for CO 2 can be used Properties: For CO 2, the average T (20+300)/2 = 160 C = 433 K Cp = 0. 965 k. J/kg. K Analysis: Solving the equation of state for v gives

Substituting, Integrating this result between the 2 states gives For an ideal gas, Which when integrated gives

SPECIFIC HEATS cv & cp üAt low pressures gases behave as ideal gases, and their specific heats depend on T only zero pressure or ideal gas, specifics heats (denoted cv 0 and cp 0)and they are relatively easier to determine. üIt is desirabe to have some general relations for determining the specific heats at higher P (or lower specific volumes) from a knowledge of cv 0 or cp 0 and P-v-T behaviour of the substance. üSuch relations are obtained by applying the test of exactness (Eq. 125) on Eq. 12 -38 and Eq. 12 -40, which yields üThe deviation of cp from cp 0 with increasing pressure is determined by integrating Eq. 12 -43 from zero pressure to any pressure P along an isothermal path: Cp, Cv general relation I

üAnother desirable general relation involving specific heats which relates the two specific heats cp and cv. üThe advantage of such a relation need to determine only one specific heat (usually cp) and calculate the other one using that relation and the P-v-T data of the substance. üThis development of such relation starts by equating the two ds relations (Eq. 12 -38 & Eq. 12 -40) and solving for d. T: Choosing T = T(v, P) and differentiating, we get Equating the coefficient of either dv or d. P of the above two equations gives the desired result: Cp, Cv general relation II

An alternating form of this relation is obtained by using the cyclic relation: Substituting the result into Eq. 12 -45 gives This relation can be expressed in terms of two othermodynamic properties called the volume expansivity β and the isothermal compressibility α, which are defined as [Fig 10] And Volume expansivity Isothermal compressibility Substituting these two relations into Eq. 12 -46, we obtain a third general relation for cp – cv Cp, Cv general relation III

MAYER RELATION CONCLUSIONS FROM MAYER RELATION: 1. The right hand side of the equation is ALWAYS GREATER THAN/ EQUAL to zero. Therefore, we conclude that 2. The difference between cp and cv approaches zero as the absolute T approaches zero 3. The two specific heats are identical for truly incompressible substances since v constant. The difference between the two specific heats is very small and is usually disregarded for substances that are nearly incompressible, such as liquids & solids The volume expansivity (also called the coefficient of volumetric expansion) is a measure of the change in volume with T at constant P

THE ∆h, ∆u, AND ∆s OF REAL GASES Ø Gases at low pressures behave as ideal gases and obey the relation Pv = RT. The properties of ideal gases are relatively easy to evaluate since the properties u, h, cv, and cp depend on temperature only. Ø At high pressures, however, gases deviate considerably from idealgas behavior, and it becomes necessary to account for this deviation. Ø To evaluate the changes in the enthalpy, internal energy, and entropy of non ideal (real) gases, using the general relations for du, dh, and ds developed earlier. 24

ENTHALPHY CHANGES OF REAL GASES The enthalpy of a real gas, in general, depends on P & T. The enthalpy change of a real gas during a process can be evaluated from the general relation for dh For an isothermal process d. T = 0, and the first term vanishes. For a constant-pressure process, d. P = 0, and the second term vanishes. An alternative process path to evaluate the enthalpy changes of real gases. 25

Using a superscript asterisk (*) to denote an ideal-gas state, we can express the enthalpy change of a real gas during process 1 -2 as ENTHALPHY DEPARTURE The difference between h and h* is and it represents the variation of the enthalpy of a gas with pressure at a fixed temperature. Can use the relation Pv = ZRT, where Z is the compressibility factor. Substituting, 26

Enthalpy departure factor The values of Zh are presented in graphical form as a function of PR (reduced pressure) and TR (reduced temperature) in the generalized enthalpy departure chart. Zh is used to determine the deviation of the enthalpy of a gas at a given P and T from the enthalpy of an ideal gas at the same T. For a real gas during a process 1 -2 Taken from ideal gas tables INTERNAL ENERGY CHANGES OF REAL GASES Using the definition 27

ENTROPHY CHANGES OF REAL GASES General relation for ds Using the approach in the figure During isothermal process An alternative process path to evaluate the entropy changes of real gases during process 1 -2. 28

Entrophy departure factor Entrophy departure The values of Zs are presented in graphical form as a function of PR (reduced pressure) and TR (reduced temperature) in the generalized entropy departure chart. Zs is used to determine the deviation of the entropy of a gas at a given P and T from the entropy of an ideal gas at the same P and T. For a real gas during a process 1 -2 from the ideal gas relations 29

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Consider air at 300 K and 0. 86 m 3/kg. The state of air changes to 302 K and 0. 87 m 3/kg as a result of some disturbance. Estimate the change in the pressure of air. SOLUTION: The temperature and specific volume of air changes slightly during a process. The resulting change in pressure is to be determined. Assumption: Air is an ideal gas.