# Is there a limit to adding energy to a two-state system?

Suppose you have a two energy level system: $E_0=0$ and $E_1=\epsilon$ and $N$ molecules. We then have the number of molecules in the excited state to be:

$\displaystyle N_{\epsilon}(T)=\frac{N}{e^{\epsilon / k_bT}+1}$ Also: $\lim_{T\to \infty}N_\epsilon=N/2$.

Does this mean that we cannot add anymore energy to this system when $N_\epsilon=N/2$? Is there something which prohibits me from exciting another molecule to $\epsilon$?

You can add energy to a state of N two-level molecules with infinite positive temperature. Then you will have states with negative temperature, which have higher energy. However, the total energy is still limited by $N\epsilon$.
• "Then you will have states with negative energy." I do not have any $E_n<0$. A molecule either has energy $\epsilon$ or $0$. You mention negative temperatures. Negative temperatures are not associated with negative energies. Moreover in order to have negative temperature, $N_\epsilon>N/2$. Which is what I am asking: can you put more energy in the system even though $N_\epsilon=N/2$ Feb 28, 2017 at 18:30