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Chapter 5: Problem 7

Which will release more heat as it cools from \(50^{\circ} \mathrm{C}\) to \(25^{\circ} \mathrm{C}, 1 \mathrm{~kg}\) of water or \(1 \mathrm{~kg}\) of aluminum? How do you know? [Section 5.5]

Short Answer

Expert verified
1 kg of water will release more heat as it cools from 50掳C to 25掳C compared to 1 kg of aluminum. This is because water has a higher specific heat capacity (\(4.18 \times 10^3 \mathrm{~J/(kg \cdot ^\circ C)}\)) than aluminum (903 \(\mathrm{~J/(kg \cdot ^\circ C)}\)), meaning it takes more energy to change the temperature of water than aluminum. The amount of heat released by water is \(-1.045 \times 10^5 \mathrm{~J}\), while for aluminum, it is \(-22,575 \mathrm{~J}\).

Step by step solution

01

Find the specific heat capacity values

We need the specific heat capacity of water and aluminum. For water, c = \(4.18 \times 10^3 \mathrm{~J/(kg \cdot ^\circ C)}\) For aluminum, c = 903 \(\mathrm{J/(kg \cdot ^\circ C)}\)
02

Determine the mass and temperature change for both substances

Both substances have a mass of 1 kg and undergo a cooling process from 50掳C to 25掳C, resulting in a temperature change of \(螖T = 25掳C - 50掳C = -25掳C\).
03

Calculate the amount of heat released for each substance

Use the heat energy formula Q = mc螖T to calculate the heat energy released for water and aluminum. For water: \(Q_w = m \cdot c_w \cdot 螖T_w = (1 \mathrm{~kg})(4.18 \times 10^3 \mathrm{~J/(kg \cdot ^\circ C)})(-25^\circ C) = -1.045 \times 10^5 \mathrm{~J}\) For aluminum: \(Q_{Al} = m \cdot c_{Al} \cdot 螖T_{Al} = (1 \mathrm{~kg})(903 \mathrm{~J/(kg \cdot ^\circ C)})(-25^\circ C) = -22,575 \mathrm{~J}\) The negative signs indicate the substances are losing heat energy (since they are cooling down).
04

Compare the heat released by water and aluminum

Comparing the heat energy released by water \((-1.045 \times 10^5 \mathrm{~J})\) and aluminum \((-22,575 \mathrm{~J})\), we find that water releases more heat as it cools.
05

State the conclusion

As it cools from 50掳C to 25掳C, 1 kg of water releases more heat compared to 1 kg of aluminum. This is due to the higher specific heat capacity of water, which means it takes more energy to change its temperature compared to aluminum.

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