# The temperature of 100 g of water initially at 25 degrees Celsius in a calorimeter increases to...

## Question:

The temperature of 100 g of water initially at 25 degrees Celsius in a calorimeter increases to 30 degrees Celsius after adding a 20 g piece of hot aluminum. What was the temperature of the aluminum before adding it to water?

## Heat Transfer:

The heat transfer is quantified considering the mass, m, heat capacity, c, and the change in temperature, {eq}\displaystyle \Delta {/eq}T. The heat transferred is quantified through the equation, {eq}\displaystyle q = mc\Delta T {/eq}.

Determine the initial temperature of the Aluminum, {eq}\displaystyle T_{i_{Al}} {/eq} by equating the heat lost by the aluminnum, {eq}\displaystyle -q_{Al} {/eq}, to the heat gained by the water, {eq}\displaystyle q_{H_2O} {/eq}. We must take note that the heat transfer equation is given as {eq}\displaystyle q = mc(T_f - T_i) {/eq}. We are given the following values:

• {eq}\displaystyle m_{H_2O} = 100\ g {/eq}
• {eq}\displaystyle c_{H_2O} = 4.18\ J/g ^\circ C {/eq}
• {eq}\displaystyle T_f = 30 ^\circ C {/eq}
• {eq}\displaystyle T_{i_{H_2O}} = 25 ^\circ C {/eq}
• {eq}\displaystyle m_{Al} = 20\ g {/eq}
• {eq}\displaystyle c_{Al} = 0.900\ J/g ^\circ C {/eq}

We proceed with the solution.

{eq}\begin{align} \displaystyle -q_{Al} &= q_{H_2O}\\ -m_{Al}c_{Al}(T_f-T_{i_{Al}}) &= m_{H_2O}c_{H_2O}(T_f-T_{i_{H_2O}})\\ T_f-T_{i_{Al}}&= -\frac{m_{H_2O}c_{H_2O}(T_f-T_{i_{H_2O}})}{m_{Al}c_{Al}}\\ T_{i_{Al}}&= T_f +\frac{m_{H_2O}c_{H_2O}(T_f-T_{i_{H_2O}})}{m_{Al}c_{Al}}\\ &= 30 ^\circ C + \frac{100\ g\times 4.18\ J/g ^\circ C\times (30 ^\circ C - 25 ^\circ C)}{20\ g\times 0.900\ J/g ^\circ C}\\ &\approx\boxed{\rm 146\ ^\circ C} \end{align} {/eq} 