Thermodynamics: PV Work: compression-expansion work

derivation of PV work formulas


1. isobaric conditions: p = c


W = F * d

F = A * p


substituting,

W = A * p * d


but:

V = area * height

dV = A * d



W = p * dV  ---> PV work for constant pressure process



2. isothermal conditions: T = c


W = S(V1 V2) p * dV


but:

pV = RT


and

p = RT/V


substituting,


W = S(V1 V2) p * dV

W = S(V1 V2) RT/V * dV


because temperature T = constant,

W = RT * S(V1 V2) 1/V * dV


but the

Integral of 1/v * dv = ln V


putting the limits (V1 V2)

W = RT * (ln V2 - ln V1)


but from logarithmic properties

ln (u/v) = ln u - ln v

ln (uv) = ln u + ln v

ln u^n = n ln u



finally getting


W = RT * ln (V2/V1)  ---> PV work for constant temperature process



where:

W = Work done by compression or expansion(- if done on the surroundings)

F = Force

d = distance

p = absolute pressure

A = area

V = volume

dV = change in volume

T = absolute temperature

V2 = volume at point2

V1 = volume at point1

R = gas constant

S(V1 V2) = integral from V1 to V2