Question

The heat energy required to melt \(1.00 \mathrm{g}\) of ice at \(0^{\circ} \mathrm{C}\) is 333 J. If one ice cube has a mass of \(62.0 \mathrm{g},\) and a tray contains 16 ice cubes, what quantity of energy is required to melt a tray of ice cubes to form liquid water at \(0^{\circ} \mathrm{C} ?\)

Short answer

330,336 J is required to melt the tray of ice cubes.

Step-by-step solution

Determine Total Mass of Ice

First, calculate the total mass of the ice cubes in the tray. Since each ice cube has a mass of 62.0 g and there are 16 ice cubes in total, you multiply the mass of one ice cube by the number of cubes: \[ \text{Total mass} = 62.0 \, \text{g/cube} \times 16 \, \text{cubes} = 992 \, \text{g}.\]

Calculate Total Energy to Melt the Ice

Next, compute the total amount of energy required to melt the entire tray. Since it takes 333 J to melt 1 g of ice, the energy needed for 992 g of ice is calculated by multiplying the energy per gram by the total mass: \[ \text{Total energy} = 333 \, \text{J/g} \times 992 \, \text{g} = 330,336 \, \text{J}.\]

Key concepts

Phase Change

A phase change occurs when a substance transitions from one state of matter to another. In our exercise, we are focusing on the phase change from solid (ice) to liquid (water). This is an example of melting, a common phase change.

During melting, ice absorbs heat energy without a change in temperature, which is referred to as the heat of fusion. This absorbed energy breaks down the rigid structure of ice, allowing molecules to move freely as liquid water. Importantly, the temperature stays at 0°C during the entire phase change process until all the ice has melted.

• Melting Point: The fixed temperature at which a solid becomes a liquid.
• Latent Heat: The energy absorbed or released during a phase change, which does not change the temperature. For ice, it is 333 J per gram.

Understanding phase changes helps in energy calculations during the melting processes and is fundamental to thermodynamics.

Thermodynamics

Thermodynamics is the study of heat, energy, and work. It's critical in understanding the energy changes occurring during a phase change. When ice changes to water, energy is absorbed as heat, which is a part of thermodynamic calculations.

In the context of this problem, we consider the first law of thermodynamics, which states that energy cannot be created or destroyed in an isolated system. Instead, it is transformed. Here, heat energy is absorbed by ice for the phase change.

• First Law of Thermodynamics: Energy in a system is constant, it can change forms but never disappears.
• System and Surroundings: The tray of ice is the system. Everything else is the surroundings.
• Energy Transfer: Heat is transferred from the surroundings into the ice.

Whenever heat energy is involved, thermodynamics provides the principles that explain how it flows and transforms during processes like melting.

Energy Calculation

Performing energy calculations is crucial for predicting the amount of heat required for phase changes. In this problem, we calculate how much energy is needed to melt the ice in the tray.

First, calculate the total mass of the ice, which includes the number of ice cubes and the mass of each. For example:\[\text{Total mass} = 62.0 \, \text{g/cube} \times 16 \, \text{cubes} = 992 \, \text{g}\]Once the total mass is known, use the heat of fusion to find the total energy. The energy required is calculated by multiplying the energy needed per gram of ice by the total mass:\[\text{Total energy} = 333 \, \text{J/g} \times 992 \, \text{g} = 330,336 \, \text{J}\]

• Energy per Gram: This is the specific latent heat per gram needed, in our case, 333 J/g.
• Total Energy: Sum of energy for every gram of ice.

By understanding these energy calculations, you can systematically determine how much energy is required in similar thermodynamic processes.