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

In summary, Cramer discusses the pressure of a non-ideal gas and how it is corrected to account for interactions between particles.
  • #1
DanSandberg
31
0
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)=[tex]\frac{1}{V(t)}[/tex]N(kb)(T(t))+(1/3)[tex]\sum\sumFF f(ij)r(ij)[/tex]

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.
 
Physics news on Phys.org
  • #2
bump for desperate help
 
  • #3
DanSandberg said:
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?)
 
Last edited:
  • #4
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.
 
  • #5
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.
 
  • #6
the_house said:
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?
 
  • #7
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.
 
  • #8
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.
 
  • #9
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 :-)
 

What is pressure of an ideal gas?

The pressure of an ideal gas is the force exerted per unit area by the gas on the walls of the container it is in. It is a measure of how much the gas molecules are colliding with the walls of the container.

How is pressure of an ideal gas calculated?

The pressure of an ideal gas can be calculated using the ideal gas law, which states that pressure is equal to the number of moles of gas multiplied by the gas constant, temperature, and divided by the volume of the container. The equation is P = (nRT)/V, where P is pressure, n is number of moles, R is the gas constant, T is temperature, and V is volume.

What is the relationship between pressure and volume of an ideal gas?

According to Boyle's Law, the pressure of an ideal gas is inversely proportional to its volume at a constant temperature. This means that as pressure increases, the volume of the gas decreases, and vice versa.

How does temperature affect the pressure of an ideal gas?

According to Charles's Law, the pressure of an ideal gas is directly proportional to its temperature at a constant volume. This means that as temperature increases, the pressure of the gas also increases, and vice versa.

What is the significance of the pressure of an ideal gas?

The pressure of an ideal gas is an important property in understanding the behavior of gases. It helps us to determine how gases will behave in different conditions and how they will interact with their surroundings. It is also used in various industrial and scientific applications, such as in gas cylinders and in gas law experiments.

Similar threads

Replies
3
Views
914
Replies
23
Views
989
Replies
3
Views
876
  • Thermodynamics
Replies
19
Views
1K
  • Atomic and Condensed Matter
Replies
4
Views
2K
Replies
2
Views
479
  • Introductory Physics Homework Help
Replies
2
Views
648
  • Introductory Physics Homework Help
Replies
2
Views
740
  • Introductory Physics Homework Help
Replies
10
Views
948
Replies
6
Views
823
Back
Top