How can I derive the Van der Waals equation?

Answer 1

To begin with, the van der Waals equation appears as follows:

#\mathbf(P = (RT)/(barV - b) - a/(barV^2))#

where

#\mathbf(a)# and #\mathbf(b)# depend on the gas itself .
For example, #a = 13.888# and #b = 0.11641# for butane (Physical Chemistry: A Molecular Approach, McQuarrie).

It can be obtained by beginning with the ideal gas law:

#PbarV = RT#
#P = (RT)/(barV)#

However, the actual derivation is fairly involved, so we will approach it more conceptually rather than fully.

By assuming the gas is a hard sphere, we say that it takes up the space it has available to move, i.e. it decreases #barV# by #b#. Again, this depends on the exact gas.

Consequently, the initial portion becomes:

#P = (RT)/(barV - b) pm ?#
Now we examine #a#; #a# essentially accounts for the different extent of the attractive intermolecular forces present in each gas. Basically, when accounting for that we subtract #a/(barV^2)#.
#color(blue)(P = (RT)/(barV - b) - a/(barV^2))#
If we note that the compressibility factor #Z = (PbarV)/(RT)# is #1# for an ideal gas, it is less for a real gas if it is easily compressible and greater for a real gas if it is hardly compressible. As such, we have these two relationships:

This should demonstrate that:

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To derive the Van der Waals equation, start with the ideal gas law, then incorporate corrections for the volume occupied by gas molecules and intermolecular forces. Begin with the ideal gas law ( PV = nRT ), where ( P ) is pressure, ( V ) is volume, ( n ) is the number of moles, ( R ) is the gas constant, and ( T ) is temperature. Then introduce corrections for the volume occupied by gas molecules and intermolecular forces using ( V - nb ) instead of ( V ) and ( P + \frac{an^2}{V^2} ) instead of ( P ), where ( a ) and ( b ) are constants specific to each gas. This yields the Van der Waals equation: ( \left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT ).

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7