The idea in this method is to keep track of energy going in $ \int_0^t Q\left( t' \right) dt' $ and out $ \int_0^t L\left(t' \right) dt' $ of a system to infer the energy change.

$ \Delta E = \int_0^t Q\left( t' \right) dt' - \int_0^t L\left(t' \right) dt' $

If the energy is conserved then the left hand side vanishes.

Adiabatic Expansion Edit

If a system is expanding adiabatically, it experiences non radiative energy losses. If it expands at a constant velocity, this loss can be accounted for, by making a slight change to the above equation

$ \int_0^t Q \left( t' \right) t' dt' = \int_0^t H \left( t' \right) t' dt' $

The reason this works is that the energy of a spherical radiation dominated gas is reciprocal in the radius

$ E_2 = E_1 \left(\frac{V_1}{V_2} \right)^{\gamma -1 } = E_1 \left( \left(\frac{v t_1}{v t_2}\right)^3 \right)^{1/3} = E_1 \frac{t_1}{t_2} $

Where we assumed that the displacement of each fluid element is much greater than its initial radius.