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 m^{3} for the volume, remembering that 1 L = 1 dm^{3}. That means the units on the numerator are:

(m^{3}) (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 / m^{2}) (m^{3})

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) / s^{2}

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

[(kg m) / (s^{2} m^{2}] (m^{3})

The m and m^{3} are both in the numerator to give m^{4} and the m^{2} cancels with it to give m^{2} (meter squared) in the numerator, learving this as a final answer:

(kg m^{2}) / s^{2}

which is the unit (from mv^{2}) known to be the unit for Joules.
Q.E.D.