HOMEWORK 1

DUE: Friday, January 25

 

 

1. Show that the van der Waals equation of state can be rearranged to get:

 

                              

 

Now expand the denominator of this expression using :

 

                                        (1-x)-1 ~ 1+x+x2

 

and show that the second and third virial coefficients, B2V and B3V respectively, can be written in terms of the van der Waals constants.

 

2.   Show that the compression factor, Z,  can be written as:

 

                              

 

i.e: as the relative value of an observed quantity and that of an ideal gas.

 

a) Why would Z be greater than one?

b) Why could Z be less than one?

c) What values of Z are likely at higher temperatures?

d) Sketch Z vs P for a large molecule like ethane.

 

 

3. The coefficient of thermal expansion is defined to be:

 

                                

 

a) Obtain an expression for a   for an ideal gas

 

b)  What is a  for a Dieterici gas?

 

 

[A Dieterici gas has the equation of state (EOS):

 

                                , where a and b and constants     ]

 

c) For the same temperature change,  will a Dieterici gas expand more at higher pressure?

 

4. a) Show that the Redlich-Kwong EOS is a cubic function of the volume (like the van der Waals EOS we studied in class)

 

 

[The Redlich-Kwong equation is :

 

                                              ]

 

 

b)  Draw P-V diagrams for a Redlich-Kwong gas at decreasing temperatures.

i)                Will this gas have a critical temperature, Tc?

ii)              If so, explain how to find this temperature.

 

 

 

5. Using the equations in (16.16) and (16.18), show that the Redlich-Kwong equation can   be written in terms of reduced quantities as:

 

 

                              

 

 

to give a Law of Corresponding States.

 

 

6.  The isothermal compressibility k is defined as:

 

                                   

 

Show that k is inversely proportional to the pressure for an ideal gas.

 

 

7. Use the van der Waals and Redlich-Kwong equation to calculate the value of the

    molar density of one mole of methane at 500K and 500 bar.

 

      Compare these results to the experimental value of 10.06 mol L–1.