# Van der waal's constants

## Homework Statement

My book says that the constants a and b are characteristic properties of a gas, and they do not depend on temperature. But i am having trouble understanding it.

## The Attempt at a Solution

We know, a = ΔP*V2 / n2 , but volume is temperature dependent, so how can a be temperature independent? One explanation can be that as V increases then ΔP will decrease, so that the ratio remains constant, but i am not sure.
Similarly as I increase the temperature, volume will increase, hence b decreases, as the gas approaches ideality. Please help me out.

## Answers and Replies

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BvU
Homework Helper
2019 Award
Hello,

but volume is temperature dependent
indeed it is. But that is a 'different volume'.

the $b$ in the van der Waals equation of state refers to the 'actual' volume of the molecules themselves. In other words: the equivalent volume of the molecules if they are considered as hard spheres. The volume the gas assumes under a given pressure and at a given temperature is much greater: most of that is simply empty space trough which the molecules move at considerable speeds. $b$ follows when you go to zero Kelvin and classically the molecules don't move any more.

$a$ is a first correction on the pressure: in the ideal gas law there are no intermolecular forces and $pV = nRT$. In a real gas there is a small, almost always attractive, force between the molecules that reduces the pressure.

Okay, i understood the case of b.
But i need some clarifications regarding a.
If temperature increases, won't the intermolecular force and hence 'a' decrease ?

Last edited:
BvU
Homework Helper
2019 Award
The forces decrease because the intramolecular distances increase. The effect on the pressure in lowest order is best approximated with $a\displaystyle \left ( n\over V \right )^2 \ .$

Google intramolecular forces, van der Waals force, Lennard Jones potential

Borek
Mentor
As long as all electrons are in their ground state "volume" of the molecules and strength of intermolecular forces can be assumed constant (that's not entirely true, but it is quite a good approximation). In temperatures where the VdV equation is used electrons don't get excited.

Chestermiller
Mentor

## Homework Statement

My book says that the constants a and b are characteristic properties of a gas, and they do not depend on temperature. But i am having trouble understanding it.

## The Attempt at a Solution

We know, a = ΔP*V2 / n2 , but volume is temperature dependent, so how can a be temperature independent? One explanation can be that as V increases then ΔP will decrease, so that the ratio remains constant, but i am not sure.
By your mathematical rationale, in the ideal gas equation PV=nRT, the ideal gas constant R should be a function of the pressure, the volume, the number of moles, and the temperature. Is that correct?