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# Chemical Engineering Thermodynamics ## Chapter 3 ### Volumetric Properties of Pure Fluids ### 3.1 PVT Behavior of Pure Substances * **The object of thermodynamics**: is to describe quantitatively the interrelation of various physical properties of macroscopic systems. * **Properties of inter...

# Chemical Engineering Thermodynamics ## Chapter 3 ### Volumetric Properties of Pure Fluids ### 3.1 PVT Behavior of Pure Substances * **The object of thermodynamics**: is to describe quantitatively the interrelation of various physical properties of macroscopic systems. * **Properties of interest**: are such directly measurable quantities as temperature, pressure, and volume, as well as certain other quantities. * **This chapter**: is largely devoted to a discussion of the PVT behavior of pure materials. ### 3.2 The Virial Equation of State * **For gases at low to moderate densities**, the data show that the PV isotherms are nearly linear $$ P=\frac{RT}{V} $$ * **For a real gas** $$ Z=\frac{PV}{RT} $$ * **where**: Z is the **compressibility factor** and is a function of temperature and pressure for a given species. * **The virial equation of state** $$ Z=1+\frac{B}{V}+\frac{C}{V^2}+\frac{D}{V^3}+... $$ * **where**: the virial coefficients B, C, and D are functions of temperature only and are different for different gases. * **An alternative expression**: is $$ Z=1+B'P+C'P^2+D'P^3+... $$ * **statistical mechanics provides the theoretical basis**: for the virial equations and shows that the coefficients are related to the interactions between molecules. * **B**: is related to interactions between pairs of molecules. * **C**: is related to interactions among triplets, and so on. * **Because**: the contributions of triplets, quadruplets, etc., are much smaller than the contribution of pairs, the series converges rapidly. ### 3.3 Cubic Equations of State * **The simplest equation of state**: that is cubic in molar volume is $$ P=\frac{RT}{V-b} $$ * **where**: b is a positive constant. This is known as the **van der Waals equation of state**. * **A more accurate cubic equation of state**: is the **Redlich/Kwong equation of state** $$ P=\frac{RT}{V-b}-\frac{a}{T^{1/2}V(V+b)} $$ * **where**: a and b are positive constants. ### 3.4 Generalized Correlations for the Volumetric Properties of Gases * **Principle of corresponding states**: all fluids when compared at the same reduced temperature and reduced pressure, have approximately the same compressibility factor and all deviate from ideal-gas behavior to about the same degree $$ T_r=\frac{T}{T_c} \quad P_r=\frac{P}{P_c} $$ * **where**: $T_c$ and $P_c$ are the critical temperature and critical pressure. * **Therefore**: $$ Z=f(T_r,P_r) $$ ### 3.5 Determination of Pressure and Volume * **Example 3.6**: Estimate the volume of 1 kg of $n$-butane at 423.15 K ($150^\circ C$) and 15 bar by each of the following: * **(a)** The ideal-gas equation. * **(b)** The virial equation, with the following experimental value of the second virial coefficient: $B = -0.000269 m^3 mol^{-1}$. * **(c)** The Redlich/Kwong equation. * **(d)** The generalized compressibility-factor correlation. * **Solution**: * **(a)** $V^{ig}=\frac{RT}{P}=\frac{8.314*423.15}{15*10^5}=0.002345 m^3 mol^{-1}$ * **(b)** $Z=1+\frac{B}{V}$ $$ V=V^{ig}+B=0.002345-0.000269=0.002076 m^3 mol^{-1} $$ * **(c)** $$ P=\frac{RT}{V-b}-\frac{a}{T^{1/2}V(V+b)} $$ $$ a=\Omega_a \frac{R^2T_c^{2.5}}{P_c}=0.42748*\frac{8.314^2*425.1^{2.5}}{37.96*10^5}=15.973*10^{-2} m^6 bar mol^{-2} $$ $$ b=\Omega_b \frac{RT_c}{P_c}=0.08664*\frac{8.314*425.1}{37.96*10^5}=0.000807 m^3 mol^{-1} $$ $$ 15*10^5=\frac{8.314*423.15}{V-0.000807}-\frac{15.973*10^{-2}}{423.15^{1/2}V(V+0.000807)} $$ $$ V=0.002101 m^3 mol^{-1} $$ * **(d)** $T_r=\frac{T}{T_c}=\frac{423.15}{425.1}=0.995 \quad P_r=\frac{P}{P_c}=\frac{15}{37.96}=0.395$ From generalized compressibility-factor charts $Z=0.88$ $$ V=Z*V^{ig}=0.88*0.002345=0.002064 m^3 mol^{-1} $$ * All results are converted to a basis of 1 kg | Method | Volume / $m^3 kg^{-1}$ | | :-------------------------------- | :---------------------- | | Ideal-gas equation | 0.0403 | | Virial equation | 0.0357 | | Redlich/Kwong equation | 0.0361 | | Generalized correlation | 0.0355 | | Experimental | 0.0363 | | % Error (Generalized correlation) | -2.2% |