Given 400.0 g of hot tea at 80.0 °C, what mass of ice at 0 °C must be added to obtain iced tea at 10.0 °C? The specific heat of the tea is 4.18 J>1g °C2 and ΔHfusion for ice is + 6.01 kJ>mol.

Verified step by step guidance

1. Calculate the amount of heat (Q) lost by the tea as it cools from 80.0 °C to 10.0 °C using the formula Q = mcΔT, where m is the mass of the tea, c is the specific heat of the tea, and ΔT is the change in temperature.
2. Convert the heat lost by the tea from joules to kilojoules (since the heat of fusion is given in kJ/mol).
3. Calculate the amount of heat required to melt the ice using the formula Q = nΔHfusion, where n is the number of moles of ice and ΔHfusion is the heat of fusion of ice.
4. Convert the mass of ice to moles using the molar mass of water (approximately 18.0 g/mol).
5. Set the heat lost by the tea equal to the heat gained by the ice (from melting) and solve for the mass of the ice.

Key Concepts

Heat Transfer and Specific Heat Capacity

Heat transfer involves the movement of thermal energy from a hotter object to a cooler one until thermal equilibrium is reached. The specific heat capacity is the amount of heat required to raise the temperature of one gram of a substance by one degree Celsius. In this problem, the specific heat of the tea (4.18 J/g°C) is crucial for calculating how much heat the tea will lose as it cools down to 10.0 °C.

Phase Change and Latent Heat

Phase change refers to the transition of a substance from one state of matter to another, such as from solid to liquid. During this process, energy is absorbed or released without changing the temperature of the substance. The latent heat of fusion for ice (+6.01 kJ/mol) indicates 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 absorb heat from the tea.

Energy Conservation in Thermodynamic Systems

The principle of energy conservation states that energy cannot be created or destroyed, only transformed from one form to another. In this scenario, the heat lost by the hot tea must equal the heat gained by the ice as it melts and warms up to the final temperature of 10.0 °C. This relationship allows us to set up an equation to solve for the mass of ice needed to achieve the desired temperature.