# What is a time-temperature diagram

## Internal energy - heat capacity

a) The calculation takes place in three steps. First, the required energy is calculated and the period of time in which the heat source applies this energy with \ (P = 20 \ rm {\ frac {kJ} {\ min}} \): \ (\ Delta E = P \ cdot \ Delta t \ Leftrightarrow \ Delta t = \ frac {{\ Delta E}} {P} \); calculated

Step 1 Warming the ice from –20 ° C to 0 ° C:

\ [\ Delta E_1 = c _ {\ rm {ice cream}} \ cdot m \ cdot \ Delta \ vartheta \ Rightarrow \ Delta E_1 = 2 {,} 1 \, \ rm {\ frac {kJ} {kg \ cdot {} ^ {\ circ} \ rm {C}}} \ cdot 0 {,} 1 \, \ rm {kg} \ cdot 20 \, {} ^ {\ circ} \ rm {C} = 4 {,} 2 \ , \ rm {kJ} \] \ [\ Rightarrow \ Delta {t_1} = \ frac {{4 {,} 2} \, \ rm {kJ}} {{20 \, \ rm {\ frac {kJ} { min}}}} \ min = 0 {,} 21 \, \ rm {min} \]

step 2 Melting the ice:

\ [\ Delta E_2 = s _ {\ rm {ice cream}} \ cdot m \ Rightarrow \ Delta E_2 = 335 \, \ rm {\ frac {kJ} {kg}} \ cdot 0 {,} 1 \, \ rm { kg} = 34 \, \ rm {kJ} \] \ [\ Delta {t_2} = \ frac {{34 \, \ rm {kJ}}} {{20 \, \ rm {\ frac {kJ} {min }}}} \ min = 1 {,} 7 \, \ rm {min} \]

step 3 Evaporation of the water:

\ [\ Delta E_3 = c _ {\ rm {water}} \ cdot m \ cdot \ Delta \ vartheta \ Rightarrow \ Delta E_3 = 4 {,} 2 \, \ rm {\ frac {kJ} {kg \ cdot {} ^ {\ circ} \ rm {C}}} \ cdot 0 {,} 1 \, \ rm {kg} \ cdot 100 \, {} ^ {\ circ} \ rm {C} = 42 \, \ rm { kJ} \] \ [\ Rightarrow \ Delta {t_3} = \ frac {{42} \, \ rm {kJ}} {{20 \, \ rm {\ frac {kJ} {min}}}} \ min = 2 {,} 1 \, \ rm {min} \]

b) The same formula applies to the evaporation of the water as to the melting:

\ [\ Delta E_4 = r _ {\ rm {Water}} \ cdot m \ Rightarrow \ Delta E_2 = 2260 \, \ rm {\ frac {kJ} {kg}} \ cdot 0 {,} 1 \, \ rm { kg} = 226 \, \ rm {kJ} \] \ [\ Delta {t_4} = \ frac {{226 \, \ rm {kJ}}} {{20 \, \ rm {\ frac {kJ} {min }}}} \ min = 11 {,} 3 \, \ rm {min} \]