An insulated beaker with negligible mass contains 0.250 kg of water at 75.0°C. How many kilograms of ice at -20.0°C must be dropped into the water to make the final temperature of the system 40.0°C?

Verified step by step guidance

1. Calculate the heat required to raise the temperature of the ice from -20.0°C to 0°C using the specific heat capacity of ice. Use the formula: \(Q = m \cdot c_{\text{ice}} \cdot \Delta T\), where \(m\) is the mass of the ice, \(c_{\text{ice}}\) is the specific heat capacity of ice, and \(\Delta T\) is the temperature change.
2. Calculate the heat required to melt the ice at 0°C to water at 0°C using the latent heat of fusion of ice. Use the formula: \(Q = m \cdot L_f\), where \(L_f\) is the latent heat of fusion of ice.
3. Calculate the heat lost by the water as it cools from 75.0°C to 40.0°C using the specific heat capacity of water. Use the formula: \(Q = m \cdot c_{\text{water}} \cdot \Delta T\), where \(m\) is the mass of the water, \(c_{\text{water}}\) is the specific heat capacity of water, and \(\Delta T\) is the temperature change.
4. Set up an energy balance equation assuming no heat is lost to the surroundings. The heat gained by the ice and the resulting water must equal the heat lost by the initial water. The equation will be: \(Q_{\text{ice to 0°C}} + Q_{\text{melting}} + Q_{\text{melted ice to 40°C}} = Q_{\text{water cooling}}\).
5. Solve the energy balance equation for the mass of the ice, \(m\), to find out how many kilograms of ice are needed.

Key Concepts

Heat Transfer

Heat transfer is the process of thermal energy moving from a hotter object to a cooler one until thermal equilibrium is reached. In this scenario, heat will flow from the warm water to the cold ice, causing the water to cool down and the ice to warm up, eventually melting and reaching the final temperature.

Specific Heat Capacity

Specific heat capacity is the amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius. For water, this value is approximately 4.18 J/g°C, which means it takes a significant amount of energy to change its temperature. This concept is crucial for calculating the heat lost by the water and the heat gained by the ice.

Phase Change and Latent Heat

Phase change refers to the transition of a substance from one state of matter to another, such as ice melting into water. During this process, latent heat is absorbed or released without changing the temperature. For ice, the latent heat of fusion is the energy required to convert ice at 0°C to water at 0°C, which is essential for determining how much ice is needed to achieve the desired final temperature.