| |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

[NOTE: HTML does not lend itself easily to insertion of subscripts , superscrips, and Greek symbols. 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] .

Page 1 of Solutions to Set 1

Page 2 of Solutions to Set 1

Page 3 of Solutions to Set 1

Page 4 of Solutions to Set 1

Page 5 of Solutions to Set 1

Page 6 of Solutions to Set 1

Page 7 of Solutions to Set 1

Page 8 of Solutions to Set 1

Page 9 of Solutions to Set 1

Page 10 of Solutions to Set 1

Page 11 of Solutions to Set 1

Page 12 of Solutions to Set 1

Page 13 of Solutions to Set 1

Page 14 of Solutions to Set 1

Page 15 of Solutions to Set 1

Page 16 of Solutions to Set 1

Page 17 of Solutions to Set 1