You drop 428 g of solid ice at −35°C into a bucket of water (1,200 g) at 45°C. Eventually, the ice all melts, and the entire system comes into thermal equilibrium. Find the final temperature of the system.

To solve this problem, we need to use the principle of conservation of energy. This principle states that the total energy in a closed system will remain constant. In this case, the energy gained by the ice must be equal to the energy lost by the water.

Here are the steps to solve this problem:

1. Calculate the energy needed to raise the temperature of the ice from -35°C to 0°C. The specific heat capacity of ice is 2.09 J/g°C. So, the energy (Q1) can be calculated using the formula Q = mcΔT, where m is the mass, c is the specific heat capacity, and ΔT is the change in temperature. Q1 = 428g * 2.09 J/g°C * 35°C = 31,511 J

2. Calculate the energy needed to melt the ice at 0°C to water at 0°C. The heat of fusion of ice is 334 J/g. So, the energy (Q2) can be calculated using the formula Q = mL, where m is the mass and L is the heat of fusion. Q2 = 428g * 334 J/g = 142,952 J

3. Calculate the total energy gained by the ice (Q_total_ice = Q1 + Q2). Q_total_ice = 31,511 J + 142,952 J = 174,463 J

4. Calculate the maximum possible energy lost by the water when it cools from 45°C to 0°C. The specific heat capacity of water is 4.18 J/g°C. So, the energy (Q3) can be calculated using the formula Q = mcΔT. Q3 = 1200g * 4.18 J/g°C * 45°C = 225,720 J

5. If Q_total_ice is less than Q3, the final temperature of the system will be 0°C, and the remaining energy will be used to freeze some of the water. If Q_total_ice is more than Q3, all the water will cool to 0°C, and the remaining energy will be used to raise the temperature of the water and ice mixture.

In this case, Q_total_ice (174,463 J) is less than Q3 (225,720 J), so the final temperature of the system will be 0°C.