Why does the internal energy change in few cases when there is no change in temperature? Why does the internal energy  of a ice- water system increase when a ice slab melts at 273K at atmospheric pressure? I don't  understand  how the internal  energy tends to change when there is no change in the temperature  at the melting point. Can someone clear my doubt?
 A: 
I don't understand how the internal energy tends to change when there
is no change in the temperature at the melting point.

Internal energy of the ice-water system consists of the sum of both kinetic and potential energy at the molecular level. A change in temperature only reflects a change in the kinetic energy component of the system. When heat is added to melt ice it increases the separation of the molecules increasing the internal molecular potential energy, without increasing the motion (kinetic energy) of the molecules.
Hope this helps.
A: The magic word you're probably reaching for is the latent internal energy.
The process of melting is a first-order phase transition (form solid state to a liquid state) which needs energy, and this takes place at constant temperature (and also at constant pressure). This amount of energy is called the latent heat (also known as the latent internal energy) which is energy released or absorbed, by a body or a thermodynamic system, during a constant-temperature process. At the level of the molecules, the energy supplied is used only to separate the molecules and no part of it is used to increase the kinetic energy of the system's molecules. In your problem, as ice melts, the molecules change state (from a solid to a liquid).
The total amount of latent heat is given by
$$L = \frac{Q}{m},$$
where $Q$ is the energy released or absorbed during phase change and $m$ is the mass of the substance. $L$ is that energy (per mass) which your system, i.e. ice, acquires (specific latent heat of the substance), for phase transition to occur, and the temperature of the system will stay constant as heat is added, so there is no temperature change until a phase change is complete.
And to make it more clear, the heat $Q$ comes from the increase in entropy, i.e., $Q=T \Delta S$. During a first-order phase transition, the entropy of the system increases if the phase transition is towards higher internal energy (e.g., like ice melting to water) and decreases if the phase transition is towards lower internal energy (e.g., like water freezing to ice).
A: The bonds holding the water molecules together in the ice crystal have been broken.  The energy to break the bonds was added from outside as the latent heat of melting, and can be recovered by allowing the water to refreeze into ice. That added energy is therefore counted as part of the change of internal energy
$$
\Delta U= T \Delta S -P \Delta V
$$
in the system.  The latent heat is $Q=T\Delta S$, so a small part  of the bond-breaking  energy also came from the $-P\Delta V$ work done by the atmosphere as the ice reduced its volume while  melting.
