Error Propagation for Bound Variables

Question

Say I'm trying to calculate the energy term Pressure*Volume based on measurement of P and V over many different trials. Given a constant temperature, pressure and volume are bound variables as PV=nRT assuming near-ideal situation; how would you calculate the error term for this?

I know that the formula for error propagation of two multiplied variables ab=x is Sx/x=sqrt((Sa/a)^2+(Sb/b)^2)), but the derivation of this formula assumes that the two variables a and b are independent making dadb=0. How would you calculate the error propagation for two non-independent variables?

Answer 1

The assumption of independency is about errors of independently measured quantities (as there is almost always some function $f$ which connects all the measured quantities), i.e. if you measured P and V independently (for example using different measuring devices) - their errors are independent of all the other measurements, and thus you can use it.

Just to make it clear, the general formula for calculating error of $ f(x_1,x_2,...,x_n)$ (where $x_i$ are independently measured quantities) is $$ \delta f = \sqrt{\sum_{i=1}^{n}(\frac{\partial f}{\partial x_i}\delta x_i)^2}$$

$\delta x_i$ are uncertainties in $x_i$ measurement. All the derivatives are calculated at the point $(x_1,x_2,...,x_n)$ - measured values.