Problem Set #1

[NOTE:  HTML does not lend itself easily to insertion of 
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	Therefore FORTRAN notation will be used for 
	exponents, i. e., x squared is written as x**2.  
	Subscripts are written on the same line, S 
	is used as the summation sign and partial 
	derivatives are written with a "d" instead of the
	Greek symbol delta. ]

1.  The van der Waals' equation-of-state for one mole of gas is 

	(1)	[P + a/V**2][V - b] = RT.

(A)  Let  y = 1/V.   Solve the van der Waals' equation-of-state for the 
pressure and obtain P as a function of y and T.

(B)  Write P as a power series expansion in y (a virial equation), i.e.,

		    	
	(2)	P    = 	S An y**n .
		   	n=0 

Use the standard Taylor series expansion formulae to obtain 
Ao , A1, A2, and A3 .

(C)  Prove that only the second virial coefficient, A2, depends 
upon the paramerter "a". 

The van der Waals constants for CH4 are  a = 2.253 liter2-atm/mol2
	and b = 0.0428 liter/mol. 

(D)  Compute the second and third virial coefficients for CH4 at 
	300 K.  Estimate the residual volume of CH4 at 300 K.


2.  (A)  A chemist measures y as a function of x and obtains 
	N data points.  She desires to fit the data to the form

	(1)	y = ao + a1 x  + a2 x**2  + a3 x**3 . 

Using a least-squares method, obtain expression for ao, a1, 
a2, and a3 in terms of the xi and yi for i = 1,2,3,...,N.   
You may leave the final result in determinant form, if you wish.  


3.  The following set of P-V data is obtained for one mole of 
	a nonideal gas at 300 K:

V(liters)  	P(atm)
----------------------------
2.000		11.297
2.590		8.903
3.179		7.342
3.769		6.246
4.359		5.435
4.949		4.809
5.538		4.313
6.128		3.909
6.718		3.575
7.308		3.293
7.897		3.052
8.487		2.844
9.077		2.663
9.667		2.503
10.256		2.362
10.846		2.235
11.436		2.122
12.026		2.019
12.615		1.926
13.205		1.841
13.795		1.763
14.385		1.692
14.974		1.626
15.564		1.565
16.154		1.508
16.744		1.456
17.333		1.407
17.923		1.361
18.513		1.318
19.103		1.278
19.692		1.240
20.282		1.204
20.872		1.170
21.462		1.138
22.051		1.108
22.641		1.079
23.231		1.052
23.821		1.026
24.410		1.002
-----------------------------

A plot of this data is given on the next page. 

(A)  Using the results of Problem 2, fit the above data to a 
	truncated virial expansion of the form

	   (1)	P = Ao + A1 y + A2 y**2 + A3 y**3  , 

where y = 1/V, and obtain the best values of Ao, A1, 
	A2, and A3.  Compare the pressure computed from 
	Eq.(1) with the data.  

(B)  Using the results obtained in Part B of Problem #1, 
	determine approximate values for "a" and "b" parameters 
	if the above nonideal gas is to be represented by a van der 
	Waals equation-of-state.

(C)  Estimate the residual volume for this nonideal gas.  


4.  (A)  In a different galaxy far removed from ours, let us assume
		 that the "ideal" gas laws are as follows:

	(a)  At constant temperature, pressure is inversely 
		proportional to the square of the volume.
	(b)  At constant pressure, the volume varies directly with 
		the temperature to the 2/3 power. 
	(c)  At 273.16 K and 1 atm pressure, one mole of an "ideal" 
		gas is found to occupy 22.414 liters.

Under these conditions, show that   V**6 P**3/T**4  =  a constant, 
and obtain the form of the "ideal" gas equation-of-state in this galaxy.  

   (B)  The coefficient of thermal expansion G is defined to be  
		G = (1/V)[dV/dT]p . 
	Obtain G for the above "ideal" gas as a function of T alone. 

5.  Problem 1.2 - Atkins - Fifth Edition - page 52.
6.  Problem 1.9 - Atkins - Fifth Edition - page 52.
7.  Problem 1.31 - Atkins - Fifth Edition - page 54.

8.  If Avagadro's approximation were "6.023 x 1023  molecules of 
	any gas occupies 45.00 liters at 273.15 K and 2.5 atm. 
	pressure", what value would the ideal gas constant have? 

9.  Consider a gas that obeys the equation of state

		P =  (RT/V) exp[-a/VRT]

	where a is a constant and the notation exp[x] means e**x. 

	(A)  Determine the second and third virial coefficients for this 
		gas as a function of "a" V, R, and T. 

	(B)  Determine the residule volume of this gas as a function 
		of "a", R, and T. 


10.  The coefficient of thermal expansion G  is defined to be  
	
		G  = (1/V)[dV/dT]p .

    	(A) Obtain an expression for G for an ideal gas.
	(B) Show that for a Dieterici gas, G  is given by 

                 G  =  [(RV + a/T)] / [PV**2 exp(a/VRT) - a] .


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