Solutions to Unit 1 Homework No



Solutions to Unit 1 Homework No. 1

1. (a) [pic]

A plot of Z versus :

The two partial derivatives are:

[pic], and [pic]

[pic]

2. The total differential of P is:

[pic]

We can determine each partial derivative. The total differential is then:

[pic], which can then be approximated as:

[pic]

[pic]

We have used the fact that 1 J = 1 Nm.

3. We need to solve each second derivative. For the ideal gas equation:

[pic]

For the van der Waals equation:

[pic]

4. Book problems.

H-3. For an ideal gas [pic]

[pic], and [pic]

For the equation of state [pic], the partial derivative of P with respect to V is:

[pic]

The partial derivative of V with respect to P is:

[pic]

H-5. The total differential of P is:

[pic]

For a Redlich-Kwong gas: [pic]

And, [pic]

Then, [pic]

H-8

[pic]

H-11

[pic]

Because these derivatives are equal, dx/T is an exact differential.

H-13. We begin with:

[pic], which can be written as: [pic]

Solving for pressure gives: [pic]

Also in terms of molar volume, this equation is: [pic]

[pic], and [pic]

5. Using the van der Waals equation of state [pic], where [pic] = 0.05 dm3 mol-1.

[pic]

Ideal gas: [pic]

Redlich-Kwong: [pic]

With A = 64.597 dm6 bar mol-2 K1/2, and B = 0.029677 dm3 mol-1, P = 136.38 bar

6. We use the Newton-Raphson method.

The cubic form of the van der Waals equation is:

[pic]

For ethane a = 5.5088 dm6 atm mol-2

b = 0.065144 dm3 mol-1

Putting these values into the above cubic equation yields the following:

[pic]

[pic]

We can get an initial guess for the molar volume using the ideal gas equation.

[pic]

[pic]

Continuing the iterations will lead to the molar volume converging to [pic]

7. [pic]

8. [pic], [pic]

[pic]

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