A flexible balloon contains a sample of nitrogen gas that initially has density \(\rho_0\), absolute temperature \(T_0\), and pressure \(P_0\). The balloon is moved to a low-pressure chamber where the internal gas pressure reaches a new equilibrium value of \(\dfrac{1}{3}P_0\). During this process, energy is transferred to the gas such that the root-mean-square speed of the nitrogen molecules doubles. Assuming the nitrogen gas behaves ideally, what is the final density \(\rho_f\) of the nitrogen gas in terms of \(\rho_0\)?