Question

One end of a metal rod is dipped in boiling water and the other is dipped in melting ice.

(a) All parts of the rod are in thermal equilibrium with each other.
(b) We can assign a temperature to the rod.
(c) We can assign a temperature to the rod after steady state is reached
(d) The state of the rod does not change after steady state is reached

Answer 1

(d) The state of the rod does not change after steady state is reached

The heat transfer will take place from the hot end to the cold end of the rod via conduction. So, with time, the temperature of the rod will increase from the end dipped in boiling water to the end dipped in melting, until it comes in equilibrium with its surroundings. In steady state, the temperature of the rod is nonuniform and constant, maximum at the end dipped in boiling water and minimum at the end dipped in melting ice.
Equilibrium means that the system is stable. So, all the macroscopic variables describing the system will not change with time. Hence, the temperature of the rod will become constant once equilibrium is reached, but its value is different at different positions of the rod.