Question 9.4: A glass contains 250.0 g of warm water at 78.0°C. A piece of......
A glass contains 250.0 g of warm water at 78.0°C. A piece of gold at 2.30°C is placed in the water. The final temperature reached by this system is 76.9°C. What was the mass of gold? The specific heat of water is 4.184 J/g °C, and that of gold is 0.129 J/g °C.
Strategy We will need to assume that heat flows only between the gold and the water, with no heat lost to or gained from the glass or the surroundings. Then the heats must balance; the heat gained by the gold was lost by the water. Each of those can be calculated using Equation 9.3, and the two must be equal and opposite in sign.
\Delta E=E_{\mathrm{final}}-E_{\mathrm{initial}} (9.3)
q_{\mathrm{gold}}=-q_{\mathrm{water}}
m_{\mathrm{gold}}\times c_{\mathrm{gold}}\times\Delta T_{\mathrm{gold}}=-\mathrm{m_{water}}\times c_{\mathrm{water}}\times\Delta T_{\mathrm{water}}
m_{\mathrm{gold}}\times{\frac{0.129\ J}{g^{\circ}{\mathrm{C}}}}\times74.6^{\circ}{\mathrm{C}}=-250.0\ {\mathrm{g}}\times{\frac{4.184\,{\mathrm{J}}}{{\mathrm{g}}^{\circ}{\mathrm{C}}}}\times-1.1^{\circ}{\mathrm{C}}
Rearranging gives us
m_{\mathrm{gold}}=\frac{-250.0\ {\mathrm{g}}\times{\frac{4.184\,{\mathrm{J}}}{{\mathrm{g}}^{\circ}{\mathrm{C}}}}\times-1.1^{\circ}{\mathrm{C}}}{{\frac{0.129~\mathrm{J}}{g^{\circ}{\mathrm{C}}}}\times74.6^{\circ}{\mathrm{C}}}=120\ g
Analyze Your Answer From the original problem statement, we can see that the temperature of the hot water doesn’t change very much. This small ΔT for the hot water implies that the gold sample must be fairly small. Next, consider that the heat capacity for water is roughly 40 times larger than that for gold, and the temperature change for gold is almost 80 times greater than for water. These two observations suggest that the gold sample should be close to half the size of the water sample, and our calculated result confirms this.
Check Your Understanding A 125-g sample of cold water and a 283-g sample of hot water are mixed in an insulated thermos bottle and allowed to equilibrate. If the initial temperature of the cold water is 3.0°C, and the initial temperature of the hot water is 91.0°C, what will be the final temperature?