Heat of fusion how much thermal energy is needed change 50.0 g of ice at -20.0°c to water at 10.0°c?

We should divide the problem into 3 separate processes.

1) Bring the temperature of the ice from  to its melting point (): the amount of heat needed in this process is

where
is the mass of the ice
is the specific heat capacity of ice
is the increase of temperature

Plugging numbers into the equation, we find


2) Fusion of ice
When the ice is at melting point, we need to add a certain amount of heat in order to melt it, and this amount of it is given by:

where 
is the mass of ice
is the latent heat of fusion of ice

Plugging numbers into the equation, we find

During this phase transition, the temperature of the ice/water does not change.

3) Bring the temperature of the water from  to 

The amount of heat needed for this process is

where
is the mass of water
is the specific heat capacity of water
is the increase of temperature

Plugging numbers into the equation, we find


--> therefore, the total energy needed for the whole process is: