Derivation of pV = 1/3 Nmc^2

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In summary, the conversation discusses the calculation of time intervals between collisions with one wall of a box and the potential issue of double counting collisions at the beginning and end of a larger period of time. One argument suggests that the large numbers involved support a statistical approach, while another argument considers the random distribution of particles.
  • #1
etotheipi
Homework Statement
Question regarding derivation of pV = 1/3 Nmc^2
Relevant Equations
$$\Delta t = 2L/v_{x}$$
This is such a small point but I just wondered if anyone could clarify.

It is easy to work out the time between successive collisions with one wall of the box, namely $$\Delta t = 2L/v_{x}$$However, if this time interval is for instance 1 second, then we could have either 1 or 2 collisions in this time interval depending on whereabouts the particle starts (i.e. could be at t=0 and t=1). These would result in different values for exerted force, but evidently this only is a problem at the beginning and end since double counting is definitely wrong.My thinking was that when taking an average over a larger period of time - suppose n time intervals - the total time becomes [itex]n\Delta t[/itex] and if n becomes fairly large the calculated time interval for each collision [itex]\frac{n\Delta t}{n}[/itex] or [itex]\frac{n\Delta t}{n+1}[/itex] approach each other, so it doesn't matter if we count the first/last collisions.

Am I overcomplicating this?
 
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  • #2
etotheipi said:
Homework Statement: Question regarding derivation of pV = 1/3 Nmc^2
Homework Equations: $$\Delta t = 2L/v_{x}$$

This is such a small point but I just wondered if anyone could clarify.

It is easy to work out the time between successive collisions with one wall of the box, namely $$\Delta t = 2L/v_{x}$$However, if this time interval is for instance 1 second, then we could have either 1 or 2 collisions in this time interval depending on whereabouts the particle starts (i.e. could be at t=0 and t=1). These obviously result in different values for exerted force.My thinking was that when taking an average over a larger period of time - suppose n time intervals - the total time becomes [itex]n\Delta t[/itex] and if n becomes fairly large the calculated time interval for each collision [itex]\frac{n\Delta t}{n}[/itex] or [itex]\frac{n\Delta t}{n+1}[/itex] approach each other, so it doesn't matter if we count the first/last collisions.

Am I overcomplicating this?

That's one argument. Another is that if the particles are randomly distributed at ##t=0##, then the time to the first collision averages out. In any case, it's the large numbers that support a statistical approach.
 
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  • #3
PeroK said:
That's one argument. Another is that if the particles are randomly distributed at ##t=0##, then the time to the first collision averages out. In any case, it's the large numbers that support a statistical approach.

Ahh I hadn't thought of that, that's quite a nice way of thinking of it.
 

1. What is the equation "pV = 1/3 Nmc^2" used for?

The equation "pV = 1/3 Nmc^2" is used to describe the relationship between pressure (p), volume (V), number of particles (N), mass (m), and the speed of light (c). It is commonly used in thermodynamics and gas laws to calculate the energy of a system.

2. How was the equation "pV = 1/3 Nmc^2" derived?

The equation "pV = 1/3 Nmc^2" was derived from the famous equation E=mc^2, which relates energy (E) to mass (m) and the speed of light (c). By incorporating the ideal gas law (pV = NkT) and the kinetic energy equation (E=1/2mv^2), the final equation is obtained.

3. What are the units for the variables in the equation "pV = 1/3 Nmc^2"?

The units for pressure (p) are typically measured in pascals (Pa) or atmospheres (atm). Volume (V) is measured in cubic meters (m^3) or liters (L). The number of particles (N) is measured in moles (mol). Mass (m) is measured in kilograms (kg). And the speed of light (c) is measured in meters per second (m/s).

4. Is the equation "pV = 1/3 Nmc^2" applicable to all types of gases?

No, the equation "pV = 1/3 Nmc^2" is only applicable to ideal gases, which follow the ideal gas law. Real gases may deviate from this law at high pressures and low temperatures, and the equation may not accurately describe their behavior.

5. Can the equation "pV = 1/3 Nmc^2" be modified for different systems?

Yes, the equation "pV = 1/3 Nmc^2" can be modified to incorporate different variables or to describe different systems. For example, the ideal gas law can be modified to include the effect of intermolecular forces, resulting in the van der Waals equation. Additionally, the equation can be extended to include multiple gases or non-ideal systems.

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