PLEASE SHOW WORK NOT JUST ANSWER The ideal gas law is one example of an equation

Question

PLEASE SHOW WORK NOT JUST ANSWER

The ideal gas law is one example of an equation of state. It relates the pressure, volume, temperature and amount of gas. Another equation of state which better models real gases is the van der Waals equation: [P+a(n/V)2 ] (V – nb) = nRT. In the van der Waals equation, a and b are constants which depend on what type of gas you have. The constant “a” is determined by the strength of intermolecular forces and is quite large for water vapor compared to the noble gases. The constant “b” is determined by the size of the molecules. Larger molecules have larger values for “b”.

(a) What are the SI units for “a” and “b”?

(b) Solve the van der Waals equation for P.

(c) Determine the work done on n moles of a van der Waals gas which expands isothermally at a temperature T from a volume V1 to a volume V2.

Explanation / Answer

The equation relates three state variables: the pressure of the fluid p, the total volume of the fluid's container V, the number of particles N, and the absolute temperature of the system T.

a) a = Pascal

b = m^3

b) P + (an^2) / (V^2) = (nRT) / (V-nb)

P = ((nRT) / (V-nb)) - ((an^2) / (V^2))

c) We can start from the definition of work,

w = Pext dV

Since the process is reversible, Pext = P, and so we have

w = P dV

An expression for P is required – this comes from the van der Waals equation. Solving the van der Waals equation for P leads to

P = nRT / (V nb) an^2 / V^2 .

Substituting this relation into the equation for work yields

w = [nRT / (V nb) an^2 / V^2] dV from V1 to V2

w = [nRT / (V nb)] dV + [an^2 / V^2] dV from V1 to V2

Since the process is isothermal, T can be pulled out of the integral along with the other constants to give

w = nRT * [1 / (V nb)] dV + an^2 * [1 / V^2] dV from V1 to V2

Integrating leads to the result

w = nRT n [(V2 - nb) / (V1 nb)] - an^2 [1/V2 - 1/V1]

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