Where Does the Second Term in the Pressure Equation for an Ideal Gas Come From?

[DanSandberg]

The following is a direct quote from Cramer's Essentials of Computational Chemistry:

Assuming ideal gas statistical mechanics and pairwise additive forces, pressure P can be computed as

P(t)=\frac{1}{V(t)}N(kb)(T(t))+(1/3)\sum\sumFF f(ij)r(ij)

My question is: I've always been taught P=NkT/V, where does the second term derive from?

EDIT: The double summation in the second term is supposed to be F(ij)r(ij) where F is the force between particles i and j and r is the distance. N is the number of particles, kb is boltzmann, T is temperature, V is volume.

 

[DanSandberg]

bump for desperate help

 

[PhaseShifter]

My question is: I've always been taught P=NkT/V, where does the second term derive from?

EDIT: The double summation in the second term is supposed to be F(ij)r(ij) where F is the force between particles i and j and r is the distance. N is the number of particles, kb is boltzmann, T is temperature, V is volume.

In an ideal gas, F=0, does it not?

(Also, is that supposed to be simple multiplication inside the summation, or the dot product of two vectors?)

 

[DanSandberg]

I mean the following seriously, not sarcastic or anything: Is your question really a question or a statement. The "net" force within a confined system has to be 0, I suppose, or a jar filled with an "ideal gas" would fall over due to a net force in one direction. However, that would be due to collisions between particles and the container wall. The second term here seems to indicate that these are forces between particles, inside the container.

Additional research is making me think that "pressure" as it relates to molecular dynamics, is actually "stress" within a system. Could this be the answer?

As for the dot product, Cramer does not specify it is a dot product but I would assume it is. Obviously it needs to be a scalar quantity and usually when two vectors are "multiplied" to be a scalar it is a dot product. So I think Cramer wanted us to assume dot product.

 

[the_house]

An ideal gas is defined to have no interactions between particles. That's how relations like P=NkT/V are derived. Interactions will change these relations from their ideal gas values.

 

[DanSandberg]

An ideal gas is defined to have no interactions between particles. That's how relations like P=NkT/V are derived. Interactions will change these relations from their ideal gas values.

Yahtzee. Agreed the second term MUST come from the fact that we have a set of interacting particles versus non-interacting. But where do we get the second term from? Is this some empirical lennard-jones treatment? Or can it be derived a priori, as they say. You know what I mean?

 

[the_house]

Sorry, that's all I can say at the moment. The equation doesn't look familiar at a glance and I have no time to investigate or think about it further (I really should be getting work done right now). Hopefully someone else can help.

 

[alxm]

As the_house says, that's the pressure of a non-ideal gas. An ideal gas has no interactions between the particles.

So all he's doing there is adding a generic interaction term in the form f(ij)r(ij) there. Could be an L-J potential, could be any potential really.
If you skip forward, can you see where he's going with this? I have a colleague with the book, I can check tomorrow otherwise.

 

[DanSandberg]

Guys thank you both immensely. Alxm - Cramer goes on to ensembles for molecular dynamics and thermostat and barostat algorithms. Although I have an eternal thirst for knowledge, I'm primarily focused on passing my oral general examination for my PhD at the moment. So I am preparing for questions related to that. I think it will suffice to say without derivation, the second term is a correction to the ideal gas law to address interactions between particles and if pressed ill have to say i'll get back to them... can't know everything, right? But I can re-read 8 years worth of textbooks in a week :-)