heat equation,radiating at 1 end, find boundary conditions at that end

Consider the heat flow in a one-dimensional rod of length 4 with cross-section of 1 and constant thermal conductivity of 1. Assume that the rate at which the heat energy flows out of the rod at "**that**" end is twice the temperature difference between the temperature at the right end and the air temperature of 23 degrees celcius. Derive the boundary condition__s__ at the right end.

I am not sure what to make of " that " end for now, leaning toward the right end, but i'll do both ends. I just want to check if i am getting the right boundary conditions here.

Right end : Energy leaving right end is given by $\displaystyle -\lambda A \frac{\partial u}{\partial x}(4,t) = 2(u(4,t)-23)$ which becomes

$\displaystyle -\frac{\partial u}{\partial x}(4,t) = 2(u(4,t)-23)$ and then... not sure. I assume they want 2 boundary conditions at the right end because they ask for boundary condition__s__.

I would say the other boundary condition could be $\displaystyle u(4,t) = u(4,0)-t\frac{\partial u}{\partial x}(4,t)$

Am i on the right track here ? I don't have much experience with boundary conditions so i don't have much of a feel for what " fits ". Which boundary conditions would you put here ? Any help is much appreciated !