Why is a Liter-Atmosphere a Unit of Energy?
Isn't That What a Joule is?

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That's right and they are equal to each other. The proof depends on using all the proper SI definitions. From the study of gases, we know this to be true:

PV = nRT

Rearranging:

R = (PV) / (nT)

Depending on the units chosen R can have different values. Two of the most popular are L-atm / mol-K and J / mol-K. So what I propose to do is an analysis of the units in the numerator and change them over from liter-atmospheres to Joules.

The first thing is to use Pascals as the pressure unit rather than atmospheres. Those of you who don't think I can do this, please consider that 101,325 Pa = 1 atm. When I substitute Pa for atm., all I change is the numerical value of R. I also substituted m3 for the volume, remembering that 1 L = 1 dm3. That means the units on the numerator are:

(m3) (Pa)

Now, Pascals are a pressure unit and keep in mind that pressure equals force per unit area. Since a Pascal equals a Newton per square meter, we have this now:

(N / m2) (m3)

That Newton per square meter stuff is coming to you out of the clear blue sky, I know that. You can study up on it later. For the moment, trust your friendly ChemTeam!!

Next, we need the definition of a Newton. It is the SI unit for force and it is:

the amount of net force that gives an acceleration of one meter per second squared
to a body with a mass of one kilogram.

and the units would be:

(kg m) / s2

I'll put the units for a Newton in place of the symbol N to get:

[(kg m) / (s2 m2] (m3)

The m and m3 are both in the numerator to give m4 and the m2 cancels with it to give m2 (meter squared) in the numerator, learving this as a final answer:

(kg m2) / s2

which is the unit (from mv2) known to be the unit for Joules. Q.E.D.

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