An underlying theory suggests that in any given process, the quantity of energy that becomes inaccessible (or lost) for doing useful work equals $T_L\Delta S$, where $T_L$ signifies the lowest accessible temperature and $\Delta S$ denotes the total change in entropy throughout the process. Determine the expression for the lost energy E_lost for the free adiabatic expansion that occurs inside of an idealized gas.

$E_{\text{lost }}=T_L\left(nR\ln \left(\frac{T_2}{T_1}\right)\right)$

$E_{\text{lost }}=T_L\left(nR\ln \left(\frac{P_2}{P_1}\right)\right)$

$E_{\text{lost }}=T_L\left(nR\ln \left(\frac{V_2}{V_1}\right)\right)$

$E_{\text{lost }}=T_L\left(nR\ln \left(\frac{P_1}{P_2}\right)\right)$