Will All the Ice Melt? A Calorimetry Check
In an insulated cup, a student places 200 g of water at 30.0°C. She adds 80.0 g of ice at -10.0°C and stirs until no further temperature change occurs. Use c_water = 4.18 J g⁻¹ °C⁻¹, c_ice = 2.10 J g⁻¹ °C⁻¹, and Lf = 334 J g⁻¹; neglect the cup's heat capacity and heat loss. She then repeats the trial with 100 g of ice at -10.0°C and the same 200 g of water at 30.0°C. She expects the final temperature in the second trial to be lower than in the first in simple proportion to the extra ice. The instructor asks her to check whether all the ice can melt before she uses a thermometer reading to calculate a final temperature. The cup is open only briefly, the pressure is 1 atm, and the mixture is stirred until equilibrium.
1. For the 80 g trial, determine whether all the ice melts and find the final state and temperature.
Read the worked answer
Heat released by the water cooling to 0°C: [Q_w=200\times4.18\times30.0=25{,}080\ \text{J}.] Heat needed to warm the ice to 0°C: [Q_1=80\times2.10\times10=1{,}680\ \text{J}.] Heat needed to melt all the ice: [Q_2=80\times334=26{,}720\ \text{J}.] Total heat needed to warm and melt all ice: [Q_{\text{total}}=1{,}680+26{,}720=28{,}400\ \text{J}.] This is greater than 25,080 J, so not all the ice can melt. The final temperature is 0°C, with ice and water coexisting.
Mass melted: [m_{\text{melted}}=\frac{25{,}080-1{,}680}{334}=70.1\ \text{g}.] Remaining ice: [80-70.1=9.9\ \text{g}.]
2. For the 100 g trial, find the final state and compare it with the student's expectation.
Read the worked answer
Warm 100 g of ice to 0°C: [100\times2.10\times10=2{,}100\ \text{J}.] Melt all 100 g: [100\times334=33{,}400\ \text{J}.] Total heat needed: [2{,}100+33{,}400=35{,}500\ \text{J},>\ 25{,}080\ \text{J}.] Again not all the ice melts, and the final temperature is 0°C. The melted mass is [m_{\text{melted}}=\frac{25{,}080-2{,}100}{334}=68.8\ \text{g},] so 31.2 g of ice remains.
The student expected a lower final temperature, but the temperature cannot fall below 0°C while ice remains. The extra ice changes the amount of ice left, not the final temperature.
3. What is the maximum mass of ice at -10.0°C that the 200 g of water at 30.0°C can melt completely? What happens for slightly more ice?
Read the worked answer
Set the heat released by the water cooling to 0°C equal to the heat needed to warm and completely melt an ice mass (m): [m(2.10\times10+334)=25{,}080.] [m(355)=25{,}080,\qquad m=70.6\ \text{g}.] For (m\le70.6\ \text{g}), all the ice melts and the final temperature is above 0°C; any excess heat warms the resulting water. For (m>70.6\ \text{g}), the final state is 0°C with some ice remaining, as in both trials. A thermometer reading of 0°C therefore does not by itself prove that all the ice has melted; the heat balance must be checked.