Issue with unit conversion

Question

I am getting an error converting energy unit Pa*m^3 (Pascal * Meters^3) to J (Joules).

pi = Quantity[2.00*10^5, "Pascals"];
Vi = Quantity[4.00*10^-6, ("Meters")^3];
Vf = Quantity[8.00*10^-6, ("Meters")^3];
gamma = 1.4;
W = pi*Vi^(gamma)*Integrate[V^(-gamma), {V, Vi, Vf}]

Answer: 0.484283 m^3*Pa

When I try to convert to Joules:

UnitConvert[W, "Joules"]

I get an error message: Meters^3 Pascals and Joules are incompatible units. Also, I noticed the unit m^3 is actually m^2.999999.

Answer 1

The error message says:

(Meters)^3. Pascals and Joules are incompatible units, Notice the decimal point at the end of 3. Using Rationalize fixes it.

pi = Quantity[2.00*10^5, "Pascals"];
Vi = Quantity[4.00*10^-6, "Meters"^3];
Vf = Quantity[8.00*10^-6, ("Meters")^3];
gamma = 1.4;
W = pi*Vi^(gamma)*Integrate[V^(-gamma), {V, Vi, Vf}] // Rationalize

UnitConvert[W, "Joules"]

Quantity[0.484283, "Joules"]

Answer 2

Use an exact (rational) exponent γ:

pi = Quantity[2.00*10^5, "Pascals"];
Vi = Quantity[4.00*10^-6, ("Meters")^3];
Vf = Quantity[8.00*10^-6, ("Meters")^3];
gamma = 14/10;

W = pi*Vi^(gamma)*Integrate[V^(-gamma), {V, Vi, Vf}]
(*    Quantity[0.484283, ("Meters")^3 "Pascals"]    *)

UnitConvert[W, "Joules"]
(*    Quantity[0.484283, "Joules"]    *)