Answer:
[tex]a = \frac{24.79078- 1.7161b}{1.310 - b}[/tex]
[tex]b = 1.310 - \frac{22.5427}{a - 1.7161}[/tex]
Explanation:
Given
[tex]P = 1\ atm[/tex]
[tex]V = 1.310\ L[/tex]
[tex]T =160\ K[/tex]
Required
Solve for a and b
Van Der Waals equation is:
[tex]P = \frac{RT}{V - b} - \frac{a}{V^2}[/tex]
Substitute values for P, V and T, we have:
[tex]1 = \frac{R*160}{1.310 - b} - \frac{a}{1.310^2}[/tex]
R is a constant and the value is:
[tex]R = 0.0821[/tex]
So, the equation becomes:
[tex]1 = \frac{0.0821*160}{1.310 - b} - \frac{a}{1.310^2}[/tex]
Simplify the expression
[tex]1 = \frac{13.136}{1.310 - b} - \frac{a}{1.7161}[/tex] ----- (a)
Solving for (a):
[tex]1 + \frac{13.136}{1.310 - b} = \frac{a}{1.7161}[/tex]
Multiply both sides by 1.7161
[tex]a = [1 + \frac{13.136}{1.310 - b}] * 1.7161[/tex]
Take LCM
[tex]a = [\frac{1.310 - b+13.136}{1.310 - b}] * 1.7161[/tex]
Evaluate like terms
[tex]a = [\frac{14.446- b}{1.310 - b}] * 1.7161[/tex]
Open bracket
[tex]a = [\frac{24.79078- 1.7161b}{1.310 - b}[/tex]
Solving for (b), we have:
[tex]1 + \frac{13.136}{1.310 - b} = \frac{a}{1.7161}[/tex]
Subtract 1 from both sides
[tex]\frac{13.136}{1.310 - b} = \frac{a}{1.7161}-1[/tex]
Take LCM
[tex]\frac{13.136}{1.310 - b} = \frac{a-1.7161}{1.7161}[/tex]
Inverse both sides
[tex]\frac{1.310 - b}{13.136} = \frac{1.7161}{a - 1.7161}[/tex]
Multiply both sides by 13.136
[tex]1.310 - b = 13.136 * \frac{1.7161}{a - 1.7161}[/tex]
[tex]1.310 - b = \frac{22.5427}{a - 1.7161}[/tex]
Collect like terms
[tex]b = 1.310 - \frac{22.5427}{a - 1.7161}[/tex]
