Ideal and Real Gas Isotherms This problem demonstrates the use of equations of s

Question

Ideal and Real Gas Isotherms This problem demonstrates the use of equations of state for both ideal and real gases to understand isotherms, critical behavior, and molar volumes. Recall that the equation of state for an ideal gas is given by

PVm = RT Where Vm is the molar volume.

The most common equation of state that represents the behavior of real gases is the van der Waals equation,

a. Use a spreadsheet program such as Microsoft Excel to generate and plot isotherms of the ideal gas equation. Remember that an isotherm is a plot of P (on the y-axis) vs. Vm (on the x-axis) for a specific fixed temperature. Plot five isotherms on the same graph, for T = 150, 300, 400, 600, and 900 K. For convenience, you might want to set the x- and y-axis ranges in your calculation to run from 0 to 1 L/mol for the molar volume and 0 to 500 atm for the pressure. Choose reasonable increments in the x and y variables that enable you to generate fairly smooth curves. Include the plot when you turn in your assignment. Discuss whether or not critical behavior is observed for the ideal gas.

b. Use a spreadsheet program such as Microsoft Excel to generate and plot isotherms of the van der Waals equation for ammonia. You will have to obtain the van der Waals constants a and b for NH3. Plot the same five isotherms as you did in part (a). Make sure to use the same range for the x and y axes as given in part (a). Include the plot when you turn in your assignment. Compare and contrast the van der Waals isotherms to the ideal gas isotherms. In what regions are they similar? In what regions are differences observed? Discuss whether or not critical behavior is observed in this case, and if so, indicate the region of the plot where critical behavior occurs.

c. The final step is to calculate the critical point for NH3, assuming the gas obeys the van der Waals equation of state. As discussed in class, the critical point appears as an inflection point on 2 the pressure vs. molar volume plot. At an inflection point, both the first and second derivatives are zero, which yields two equations,

To find the pressure, volume, and temperature of the critical point (labeled Pc ,Vc , and Tc ), solve the van der Waals equation for P. Then, evaluate the first and second partial derivatives given in the equations above. Setting those partial derivatives equal to zero yields two equations that can be solved for the temperature and volume of the critical point. Substitution of these two values back into the van der Waals equation gives the critical pressure. Report the values of the critical constants that you obtained and show your work.

Please I want the answer quickly.

Explanation / Answer

V is assumed and P is calculated

for ideal gas P= RT/V

for Vanderwaal gas P= RT/(V-b)- a/V2

where a =4.16 Atm.L2/mole2 b= 0.03713 L/mol

The plots are shown below

Ideal gas plot

Vanderwaal plot (series-1 : 150K, 2-200K, 3-300K, 4-400K, 5-600, 6-900

the plots suggest ideal gas law is obeyed at high temperatures and low pressures

critical temperature is seen to be at 400 K

Vandewaal Equation is

(P+a/V2)*(V-b)= RT

P= RT/(V-b)- a/V2

At critical point, VC, TC

dp/dV=0 and d2P/dV2=0

dp/dV= -RTC/(VC-b)2 +2a/Vc3=0    (1)

2a/VC3= RTC/ (Vc-b)2   (2)

Differentiating Eq.1 again

d2P/dV2=2RTC/(Vc-b)3-6a/VC4= 0    (3)

2RTC/(VC-b)3= 6a/Vc4    (4)

Multiplying Eq.2 abd Eq.4 gives

Vc= 3b

Substitution of this eq. into Eq.1

Gives

RTC/4b2= 2a/27b3

TC= 8a/(27bR) (5)

But PC= RTC/(VC-b)- a/VC2 = R*8a/27bR/ (Vc-b)- a/9b2   =a/27b2    (6)

TC/PC= ( a/27b2)*8a/(27bR)

b= RTc/8Pc

Squaraing Eq. 5

Gives

TC2= 84a2/729b2R2   (7)

Eq.7/Eq.6

TC2/PC= 64a/27R2

Therefore a = 27R2TC2/64Pc

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