Kinetic Energy while melting I have a question and its really bothering me.
When a substance in its solid state is heated, then while it melts, the temperature does not increase. I used to assume that this is because the energy is being transferred to the chemical energy store of the substance in order to overcome the bonds rather than increase the average speed of the particles (internal energy store is made up of chemical and thermal energy) but my textbook says that the particles receive kinetic energy.
I am quite confused. Do both things happen at the same time? 
And if this is the case, then when energy is transferred to the kinetic energy stores of the particles, then why don't they melt immediately. Why does it take time for the solid to melt (while on the same temperature) completely?
 A: 
When a substance in its solid state is heated, then while it melts,
  the temperature does not increase.

That is generally the case when we consider only a physical change in state (solid to liquid). 

I used to assume that this is because the energy is being transferred
  to the chemical energy store of the substance in order to overcome the
  bonds rather than increase the average speed of the particles

Actually, for a purely physical change, the addition of heat called the latent heat of fusion increases the distances between molecules or atoms in opposition to the attractive intermolecular forces that are present (e.g., the Van Der Waals forces). It is the increased distance that accounts for the change in state.  Increasing the intermolecular separation increases the internal potential energy component of the internal energy of the substance, while the internal kinetic energy remains relatively constant (as evidenced by the temperature remaining constant).

but my textbook says that the particles receive kinetic energy. I am
  quite confused. Do both things happen at the same time? And if this is
  the case, then when energy is transferred to the kinetic energy stores
  of the particles, then why don't they melt immediately. Why does it
  take time for the solid to melt (while on the same temperature)
  completely?

Since I don't know the context of the statement made in your text book, I suspect it is referring to the complete process involved in causing a phase change. Achieving a phase change requires you to first supply heat to increase the temperature of the solid so that it reaches its melting point. For this part of the process the temperature (and thus the internal kinetic energy) does increase. For a solid, the heat required is the product the specific heat of the material times the increase in temperature needed to attain the melting temperature, or
$$Q=mC\Delta T$$
Where C is the specific heat ($\frac{J}{kg-K}$)
Once the melting temperature is reached, then the phase change from solid to liquid occurs at constant temperature (no change in internal kinetic energy), increasing the internal potential energy of the substance, according to
$$Q=mh$$
Where $h$ is the latent heat of fusion ($\frac {J}{kg}$)
Naturally, the entire process takes time based on the heat transfer rate for both parts of the process and the a amount of mass involved.
Hope this helps.
