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The calculation of a reversible, isothermal expansion of a gas is explained on pages 265-266 of Chapter 8 in the textbook. The green shaded box shows the derivation of the formula using integrals, which is what Dr. Lavelle did in class today.
The derivations done by Dr. Lavelle in class today can be found in chapter 8. For further explanation, the equation is used to calculate reversible gas expansions. If you recall in last Friday's class, Dr. Lavelle derived w=-P∆V. In a similar manner, in today's class we derived the equation w=-nRT x ln (V2/V1). The difference between these two equations is that pressure is not constant in w=-nRT x ln (V2/V1) and is constant in w=-P∆V. Therefore, when pressure is not constant we will use the ideal gas equation PV=nRT to substitute nRT/V with pressure. Once you do that, the constant values nRT can be taken out of the integral, and you can solve for ∫1/V dV from V1 to V2. This is similar to ∫1/X dx which you might have seen before. This solution is ln(X) so the solution for ∫1/V dV would be lnV2-lnV1. Using logarithmic rules we know that is equivalent to ln (V2/V1) , therefore giving us the full equation w=-nRT x ln (V2/V1).
For isothermal reversible expansion, w=-P∆V doesn't work because reversible expansion means P isn't constant, but instead changes very gradually. Instead, we use PV=nRT to substitute nRT/V for P and then integrate it in terms of V, which results in the equation w=-nRTln(V2/V1).
Adriana Rangel 1A wrote:Why are these equations derived?
I derive them as the analysis is fundamental to understanding what the equations mean, under what conditions they apply, and how to use them.
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