Why are the constants a and b independent of temperature in gas properties?

[subhradeep mahata]

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.

Homework Equations

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.

 

[BvU]

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.

 

[subhradeep mahata]

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 ?

 

[BvU]

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]

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]

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.

Homework Equations

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?