Deriving the Internal Energy of a gas using Work-Energy Theorem

Question

I’m trying to derive the internal energy (IE) of a gas using a method similar to how the electric potential energy is derived for a system of two charges.

Here’s what I’ve done so far:

Imagine a situation where gas molecules start at rest (zero kinetic energy) and are separated by infinite distances (so there is no interaction between them initially). As the molecules are brought together into a finite volume, work is done on the system, and the molecules gain kinetic energy.

To make things simple, I assume the system has only two gas molecules. Bringing the second molecule towards the first requires work done by an external agent. Here, it is not necessary to move the molecule quasi-statically (because even if the molecule is accelerated, it will not create electrodynamic effects, unlike the case of charges).

Here’s how I set up the problem mathematically:

The total work done is:

$W_{\text{net}} = W_{\text{ext}} + W_{\text{interaction}}$

[ $W_{ext}$ is the work done by the external agent and, $W_{interaction}$ is the work done due to the interaction between the molecules ]

The work-energy theorem says:

$W_{\text{net}} = \Delta KE$

$W_{\text{net}} = KE_f = W_{\text{ext}} + W_{\text{interaction}}$

Since $W_{\text{interaction}}$ is due to a conservative force, it can be written as:

$W_{\text{interaction}} = -\Delta U$

$\Delta U = U_f - U_i$

Thus, $W_{\text{interaction}} = -U_f$

Substituting this into the work equation: $KE_f = W_{\text{ext}} - U_f$

At this point, I’m stuck. I can’t figure out how to use the equation I got, to derive:

$IE = U_K + U_P$

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PS: The professor who taught me Work, Energy, and Power (the name of a chapter in my curriculum) explained that potential energy is defined as the negative of the work done by a conservative force and is introduced purely to simplify problem-solving. That’s why potential energy doesn’t directly appear in the work-energy theorem. While I’ve found this approach helpful in other problems, I can’t figure out how to proceed here using it.

I can shift the $-U_f$ to LHS and get the equation $KE_f + U_f=W_{\text{ext}}$, but that would not be meaningful in this situation.

Thank you for putting your time and reading this. ☺️

Answer 1

I’m trying to derive the internal energy (IE) of a gas using a method similar to how the electric potential energy is derived for a system of two charges.

I assume that your intention is to treat a simplified case, because with your current level of knowledge attemping an exhaustive treatment would be way above your means.

The obvious simplification is to treat the case of an idealized gas.

In the case of an idealized gas:
The objects are treated as perfect spheres
When the objects collide with each other the collision is perfectly elastic collision.

Implicitly: the atoms of an idealized gas are electrically neutral. That means that they neither attract each other nor repel each other; other than elastic force during collision they don't have an interaction.

The only way for an idealized gas to change internal energy is when an external force is doing work.

When a gas is enclosed in a cylinder with a piston, and the piston is moved to reduce the volume of the gas then the temperature of the gas increases, in proportion to the work done by the moving piston.

Repeating the key point that I want to get across:
At your current level of knowledge only the idealized case is tractable.
In the idealized case there is no interaction between the atoms of the gas (other than elastic force during collision).

[Later edit}

I am puzzled as to what your curriculum is asking you to produce, given that your current level of knowledge is limited.

You have stated twice that what you are supposed to demonstrate is the formula:

$$ EI = U_k + U_p \tag{1} $$

But you haven't specified what $U_k$ and $U_p$ stand for. Presumably they stand for kinetic energy and potential energy respectively. My recommendation to you is: when using notation like $U_k$ and $U_p$, always specify what it stands for.

It could be that the goal (of the curriculum) is to actually make the collisions of the atoms of the idealized gas as the phenomenon to be examined.

As a visualization of those collisions:
Youtube video titled: Bowling ball elastic collisions

The collisions of the atoms of an idealized gas are like that, ony the velocities are far, far larger.

When two objects collide:
As the objects become elastically deformed the objects decelerate each other. That is: kinetic energy is being transformed to elastic potential energy. And then there is the rebound during which elastic potential energy is transformed back to kinetic energy.

The work-energy theorem provides the motivation for the concepts of kinetic energy and potential energy.

For a derivation of the work-energy theorem I link to an answer that I posted in november of 2023: the work-energy theorem