We should divide the problem into 3 separate processes.
1) Bring the temperature of the ice from [tex]-20.0^{\circ}C[/tex] to its melting point ([tex]0^{\circ}C[/tex]): the amount of heat needed in this process is
[tex]Q_1=mC_s \Delta T[/tex]
where
[tex]m=50.0 g[/tex] is the mass of the ice
[tex]C_s = 2.108 J/g^{\circ}C[/tex] is the specific heat capacity of ice
[tex]\Delta T=0^{\circ}C-(-20^{\circ}C)=20^{\circ}C[/tex] is the increase of temperature
Plugging numbers into the equation, we find
[tex]Q_1 = (50.0 g)(2.108 J/g^{\circ}C)(20^{\circ}C)=2108 J[/tex]
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:
[tex]Q_2 = mL_f[/tex]
where
[tex]m=50.0 g[/tex] is the mass of ice
[tex]L_f = 334 J/g[/tex] is the latent heat of fusion of ice
Plugging numbers into the equation, we find
[tex]Q_2 = mL_f = (50.0g)(334 J/g)=16700 J[/tex]
During this phase transition, the temperature of the ice/water does not change.
3) Bring the temperature of the water from [tex]0^{\circ}C [/tex] to [tex]10^{\circ}C[/tex]
The amount of heat needed for this process is
[tex]Q_3 = mC_s \Delta T[/tex]
where
[tex]m=50.0 g[/tex] is the mass of water
[tex]C_s = 4.187 J/g^{\circ}C[/tex] is the specific heat capacity of water
[tex]\Delta T=10^{\circ}C-0^{\circ}C=10^{\circ}C[/tex] is the increase of temperature
Plugging numbers into the equation, we find
[tex]Q_3 = (50.0 g)(4.187 J/g^{\circ}C)(10.0^{\circ}C)=2094 J[/tex]
--> therefore, the total energy needed for the whole process is:
[tex]Q=Q_1+Q_2+Q_3=2108 J+16700 J+2094 J=20902 J=20.9 kJ[/tex]
