CHAPTER 8 PROBLEMS

H. CHAPTER 8 PROBLEMS

Problem H1 Helmholtz function A is generally a function of A = A (T, V, N 1 .... N n ) ; a) Write

down the Euler equation for A. Then obtain a , b) Find the differential da , c) Write down the Gibbs-Duhem equation for A. Express it on a unit kmol basis, d) Use (c) in (b) to obtain simplified expression for da , e) What is (da / dx ) 2 at constant v , x 3 ,

x 4 …x K .

Problem H2 One wishes to prepare a mix of 60% acetylene and 40% CO 2 (mole basis) at a pres- sure of 100 bar and at a temperature of 47 °C. Your boss asks you to determine the

number of kmol of acetylene and CO 2 required to form the mixture. Assume tank volume to be 1 m 3 . Determine the kmol using the following method: a) Ideal gas law,

b) Kay’s rule and compressibility charts, c) Law of additive pressures. and RK equa- tion for pure component, d) Law of additive volumes and RK equation for pure com- ponent, e) Empirical equation for m a and b and RK equation for the mixtures By m

looking at the answers you must report to your boss regarding the expected minimum and maximum requirements.

Problem H3 Consider a mixture of methane species (1) and propane species (2) (40: 60, Kmol ba- sis). Assume that Kay's rule is applicable for the mixture and the mixture follows VW

eq. of state. Assume P = 30 bar and T = 300 K. a) Are a and b functions of mole fractions?, b) Determine v for the mixture (m 3 /kmol), c) Determine the partial molal volume of species (1) (v ˆ) 1 , d) Determine the molal volumes in the pure state for spe- cies (1) and (2). e) Comment on the molal volumes of species (1) in the pure states (v 1 ) , f) If you assume a hypothetical ideal gas state for species (2) (v ) 2 in the pure state, what is the ideal gas volume of species (2)?, g) Determine the ideal molal vol- umes of mixture (v ) id based on answers for i) part (d), and ii) part (e).

Problem H4 Consider the VW equation: P = RT/(v-b) - a/v 2 . Neglect body volume "b". Solve for v. Suppose this equation is valid for two component mixtures (say H2O vapor- spe- cies 1 and air-species 2) at T = 300 K, P = 200 bar. a) Plot ^v 1 , ^v 2 vs. x 1 using Kay’s rule and a spreadsheet program. Compare the solution for (v ) with ideal solu-

tion model following LR rule and HL. Problem H5

Consider the equation of state for a mixture: P V = N Z ¯R T where N = N 1 +N 2 + ....N K . Test whether Z (T, P, N 1 ,N 2... ) is an extensive property ? Hint: Use the defini- tion of partial molal property b 1 = (∂B/∂T) T,P,N2 ,.., and show that N 1 ∂Z/∂N 1 + N 2 ∂Z/∂N 2 + ....... = 0 using the Euler equation.

Problem H6 Consider the approximate virial equation of state valid at low to moderate pressures: Z = 1 + BP/RT. This equation can be used for mixtures with n components

B= ΣΣY i Y j B ij , i=1,..n, j=1... n,

B ii ,B jj : virial coefficient of pure species i,

B ij = (R T c,ij /P c,ij ) = (B o +w ij B 1 )

w ij = (w i +w j )/2 T c,ij = (T c,i T c,j ) 1/2 (1 - k ij ) (kij = 0 when i =j, kij >0 when i is not equal to j; assume kij =0) P c,ij =Z c,ij RT c,ij /v c,ij Z c,ij = (Z c,i +Z c,j )/2

v c,ij = ((v 1/3 c,i +v 1/3 c,j ) /2) 3

B o = 0.083 - 0.422/Tr 1.6

B 1 = 0.139 - 0.172/Tr 4.2 Obtain an expression for partial molal volume of species 1.

Problem H7 Consider a 60:40 NH 3 -H 2 O mixture at 10 bar, 400 K. a) Obtain the partial molal vol- ume of H 2 O at 10 bar and 400 K. Use the VW relation and Kay’s rule. Since ln( ) = (Z - 1)dP / P P φ ∫

0 , treating ln ( φ) as an intensive property and (N ln(φ)) as an extensive property, ln ˆ ( ) = ( / N )[N ( )] φ 1 ∂∂ 1 ln φ . Show that for any real gas, ln ˆ φ 1 = ln φ+ P (Z - Z)dP / P ∫ˆ 0 1 , where ˆ φ 1 is the partial molal fugacity coefficient of species 1.

Problem H8 Determine u,h and f of H 2 O( Ρ) at T = 90 C and P =100 kPa., b) Determine u,h and f

of H 2 O( Ρ) at T = 90 C and P = 50 kPa. Assume that u sat (90 C), v sat (90 C) are avail- able. Problem H9 Determine the chemical potential of CO 2 at P = 34 bar, 320 K. Assume real gas be-

havior. For ideal enthalpy use h 0 = c p0 (T- 273), s 0 c p0 ln (T/273) - R ln (P/1), c p0 =

10.08 kJ/ k mole. Use a) charts, b) RK equation. Problem H10

Using the relations for s fg for RK equation of state (Chapter 07) for pure component, obtain the relations for a) ^s fg,1 and b) ^h fg,1 using RK mixing rule. Note that ^h fg,1 , enthalpy of vaporization when component 1 is inside the mixture.

Problem H11 Obtain the relations for a) ˆ u k − u k, 0 , and b) h ˆ k − h ˆ k, 0 for a gas mixture following Berthelot equation and Kay’s rule for critical constants.

Problem H12

A mixture of of 60% acetylene and 40% CO 2 (mole basis) is compressed isothermally from 1 bar, 47ºC to a pressure of 100 bar. Determine the amount of work in kJ/kmol of mixture if a) a closed system is used, b) an open system is used assuming ideal gas law and Kay’s rule and RK equation of state.

Problem H13

A piston-cylinder assembly with a weight at the top consists of a wet H 2 O mixture of 20% quality at 1.5 bar. The initial total volume is 20 L. The whole system is im- mersed in a bath maintained at 111.4ºC. Through a hole in the vapor phase section of

cylinder we inject inert gas say N 2 until mole fraction of N 2 in vapor phase is 25%. The N 2 does not dissolve in liquid. Do you believe there will be more liquid or more vapor, or will the mixture remain as before? You can use either arguments or calcula- tions.

Problem H14 Ammonia is manufactured using hydrogen and nitrogen. A mixture having a molar ratio of H 2 to N 2 equal to 3 is compressed to 400 atm and heated to 573 K. Determine the specific volume at this condition using the following methods for RK mixture: a) Ideal gas. b). Law of additive pressures and generalized Z charts. c). Law of Additive Volumes and generalized Z charts. d). Kay’s rule.

Problem H15 Obtain an expression for partial molal volume of component 1 in a mixture following RK equation of state and the mixing rule a =( Σ X a m 1/2 k k k ) 2 , b m = Σ k X k b k .

Problem H16

A real gaseous mixture of acetylene (species 2) and CO 2 (species 1) is considered. The mole fraction of (1) is x 1 . Assume that Kay’s rule applies for the critical pressure A real gaseous mixture of acetylene (species 2) and CO 2 (species 1) is considered. The mole fraction of (1) is x 1 . Assume that Kay’s rule applies for the critical pressure

a = 0.4275 R 2 T 2.5 c /P c , and b = 0.08664 R T c /P c , where T cm denotes the critical temperature and P ) cm the critical pressure of the mixture. )

a) Obtain an expression for v 2 and then reduce the expression for v 2 when x 2 goes to a very small value (say 0.01). )

b) Determine v 2 when x 2 is small (say, 0.01) at 320 K and 100 bar. )

c) If x 1 = 0.6, what is the value of v 2 at T = 320 K and P = 100 bar? Compare with the answer from part b.

Problem H17 Obtain the relations for ˆ u k − u k, 0 , ^s 10 (T,P) - ^s 1 (T,P) and ˆ h k − h ˆ k, 0 for a gas mixture following RK equation and RK mixing rule ¯a

m =( ΣY k ¯a k ) 2 , ¯b m = ΣY k ¯b k . Problem H18

A gas mixture containing CO 2 and acetylene exists at 100 bar and 0.0938 K. If the mass fraction of CO 2 is 0.4, determine the temperature. Use LAV, LAP and RK.