Quick and Dirty Information on Calculus in Thermodymanics
Power Functions and P vs. V diagrams:
- The derivative of a power function y = x^n is y' = n
x^(n-1).
- To integrate a power function you turn the integral around
and find the antiderivative, so an antiderivative of f(x) = n x^(n-1) would be F(x) = x^n.
- In general an antiderivative of x^n is thereofore 1/n
x^(n-1).
- If V is the variable then the derivative of V^n would be n
V^(n-1).
- An antiderivative of P(V) = V^n would be 1/n V^(n-1).
- To integrate P(V) = V^n between volumes V1 and V2 you find
the antiderivative function then subtract its value at V1 from its value at V2.
- You get 1/n V2^(n-1) - 1/n V1^(n-1).
- This is the area under the curve, which is equivalent to
the work done in the expansion.
Application to Isothermal and Adiabatic Processes
- For an adiabatic process we have the Greek
gamma , equal to Cp / Cv (e.g., about 1.4 for most diatomic molecules), instead of n, but
that's no big deal. We also have a constant multiple of V^gamma, but the constant just
goes along for the ride.
- For an isothermal process P = nRT/V so n =
1 and we have nRT as the constant in P = const * (1 / V).
- So we have to integrate 1 / V.
- The derivative of ln(x) is 1/x so an
antiderivative of 1/x is ln(x), and similarly when integrating with respect to V an
antiderivative of 1/V is ln(V).
- Evaluating the antiderivative between V1 and V2 we
get ln(V2) - ln(V1), which by laws of logarithms is ln(V2/V1).