Calculating the Required Heat: Heating Water in an Aluminum Pan

The pan and the water are always at the same temperature. When you put the pan on the stove, the temperature of the water and the pan is increased by the same amount. We use the equation for the heat transfer for the given temperature change and mass of water and aluminum. The specific heat values for water and aluminum are given in Table 14.1.

Because water is in thermal contact with the aluminum, the pan and the water are at the same temperature.

1. Calculate the temperature difference:
   $$\text{Δ}T=T_{\text{f}}-T_{\text{i}}=\text{60.0ºC.}$$

   14.3
2. Calculate the mass of water. Because the density of water is $\text{1000}\mathrm{<mspace\; width="0.25em"><msup><mtext>kg/m</mtext><mrow><mn>3</mn></mrow></msup></mspace>}$, one liter of water has a mass of 1 kg, and the mass of 0.250 liters of water is $m_{\mathrm{w}}=0\text{.}\text{250}\mathrm{<mspace\; width="0.25em"><mtext>kg</mtext></mspace>}$.
3. Calculate the heat transferred to the water. Use the specific heat of water in Table 14.1:
   $$Q_{\mathrm{w}}=m_{\mathrm{w}}{c}_{\mathrm{w}}\text{Δ}T=\left(0\text{.}\text{250}\mathrm{<mspace\; width="0.25em"><mtext>kg</mtext></mspace>}\right)\left(\text{4186}\text{J/kg}\text{ºC}\right)\left(\text{60.0}\text{ºC}\right)=\text{62}\mathrm{.8\; kJ.}$$

   14.4
4. Calculate the heat transferred to the aluminum. Use the specific heat for aluminum in Table 14.1:
   $$Q_{\text{Al}}=m_{\text{Al}}{c}_{\text{Al}}\text{Δ}T=\left(\text{0.500 kg}\right)\left(\text{900 J/kgºC}\right)\left(\text{60.0ºC}\right){\text{= 27.0 × 10}}^4\text{J = 27.0 kJ.}$$

   14.5
5. Compare the percentage of heat going into the pan versus that going into the water. First, find the total transferred heat:
   14.6

Thus, the amount of heat going into heating the pan is

and the amount going into heating the water is

In this example, the heat transferred to the container is a significant fraction of the total transferred heat. Although the mass of the pan is twice that of the water, the specific heat of water is over four times greater than that of aluminum. Therefore, it takes a bit more than twice the heat to achieve the given temperature change for the water as compared to the aluminum pan.