Four ice cubes at exactly 0 °C with a total mass of 53.5 g are combined with 115 g of water at 75 °C in an insulated container. If no heat is lost to the surroundings, what is the final temperature of the mixture?

Verified step by step guidance

1. Identify the heat transfer process: The heat lost by the warm water will be equal to the heat gained by the ice cubes as they melt and warm up to the final temperature.
2. Calculate the heat required to melt the ice: Use the formula \( q = m \times \Delta H_f \), where \( m \) is the mass of the ice and \( \Delta H_f \) is the heat of fusion for ice (334 J/g).
3. Calculate the heat required to raise the temperature of the melted ice to the final temperature: Use the formula \( q = m \times c \times \Delta T \), where \( c \) is the specific heat capacity of water (4.18 J/g°C) and \( \Delta T \) is the change in temperature.
4. Calculate the heat lost by the warm water as it cools to the final temperature: Use the formula \( q = m \times c \times \Delta T \), where \( m \) is the mass of the water, \( c \) is the specific heat capacity of water, and \( \Delta T \) is the change in temperature.
5. Set the heat gained by the ice equal to the heat lost by the water and solve for the final temperature.

Key Concepts

Heat Transfer and Conservation of Energy

In thermodynamics, the principle of conservation of energy states that energy cannot be created or destroyed, only transferred. In this scenario, heat will flow from the warmer water to the cooler ice cubes until thermal equilibrium is reached. The total heat lost by the warm water will equal the total heat gained by the ice cubes, allowing us to calculate 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. Different substances have different specific heat capacities, which affect how they respond to heat transfer. In this problem, the specific heat capacities of water and ice will be crucial for calculating the heat exchange and determining the final temperature.

Phase Change and Latent Heat

When ice at 0 °C is introduced to warmer water, it may undergo a phase change from solid to liquid, which requires energy known as latent heat. This energy does not change the temperature of the ice but is essential for melting it. Understanding the latent heat of fusion for ice is necessary to account for the energy required to convert the ice cubes into water before reaching thermal equilibrium with the warmer water.