Question

Calculate the quantity of heating required to convert the water in four ice cubes \((60.1 \mathrm{~g}\) each \()\) from \(\mathrm{H}_{2} \mathrm{O}(\mathrm{s})\) at \(0{ }^{\circ} \mathrm{C}\) to \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\) at \(100 .{ }^{\circ} \mathrm{C}\). The enthalpy of fusion of ice is \(333 \mathrm{~J} / \mathrm{g}\) and the enthalpy of vaporization of liquid water is \(2260 \mathrm{~J} / \mathrm{g}\).

Short answer

The total heating required is 723952.4 J.

Step-by-step solution

Calculate Total Mass of Ice Cubes

First, determine the total mass of the ice cubes. Since each ice cube is 60.1 g and there are 4 of them, the total mass is calculated as follows:\[\text{Total Mass} = 4 \times 60.1 \text{ g} = 240.4 \text{ g}\]

Calculate Heat for Melting Ice

Next, calculate the energy required to melt the ice at 0°C to water at 0°C. Use the enthalpy of fusion:\[\text{Heat for fusion} = \text{mass} \times \text{enthalpy of fusion} = 240.4 \text{ g} \times 333 \text{ J/g} = 80053.2 \text{ J}\]

Calculate Heat for Heating Water

Calculate the heat required to raise the temperature of the water from 0°C to 100°C using the specific heat capacity of water, which is approximately 4.18 J/g°C:\[\text{Heat for warming water} = \text{mass} \times \text{specific heat} \times \Delta T\]Here, \( \Delta T = 100°\text{C} \):\[= 240.4 \text{ g} \times 4.18 \text{ J/g°C} \times 100\text{°C} = 100595.2 \text{ J}\]

Calculate Heat for Vaporizing Water

Finally, calculate the heat required to vaporize the water at 100°C:\[\text{Heat for vaporization} = \text{mass} \times \text{enthalpy of vaporization} = 240.4 \text{ g} \times 2260 \text{ J/g} = 543304 \text{ J}\]

Total Heat Required

Add all the calculated heats to get the total heat required to convert the ice at 0°C to steam at 100°C:\[\text{Total Heat} = 80053.2 \text{ J} + 100595.2 \text{ J} + 543304 \text{ J} = 723952.4 \text{ J}\]

Key concepts

Enthalpy of Fusion

The enthalpy of fusion is a term used to describe the energy required to change a substance from solid to liquid at its melting point. This is an important concept in phase changes, particularly when melting ice into water. For example, if you have ice at 0°C, which is exactly its melting point, to turn this ice into liquid water, you need to add energy. This energy doesn't increase the temperature; instead, it changes the state of the matter from solid to liquid.

In the exercise, the enthalpy of fusion for ice is given as 333 J/g. This means that for every gram of ice, 333 joules of energy are needed to transform it into water at the same temperature. Calculating this is as simple as multiplying the mass of the ice by the enthalpy of fusion. So, for 240.4 grams of ice, you need 333 joules per gram, resulting in a total of 80,053.2 J.

Enthalpy of Vaporization

The enthalpy of vaporization refers to the heat required to convert a liquid into a gas at its boiling point. This is critical when you’re converting water to steam—the form we commonly call vapor. A key detail here is that the temperature stays constant during the phase change, even though energy is being added.

In the exercise scenario, the given enthalpy of vaporization for water is 2260 J/g. This tells us that each gram of liquid water requires 2260 joules to become vapor at 100°C. Once you have water at this temperature, all the energy goes into breaking the molecules apart to turn from liquid into gas. For the total mass of water (240.4 grams), this means a whopping 543,304 J is needed for the transformation.

Specific Heat Capacity

Specific heat capacity is a measure of how much energy it takes to raise the temperature of a certain mass of a substance by 1°C. It's a critical concept when heating a substance without changing its phase.

Water has a specific heat capacity of 4.18 J/g°C. This means that every gram of water needs 4.18 joules of heat to get 1°C warmer. When you need to raise the temperature of the water from 0°C to 100°C, you're not changing its phase or chemical makeup, just moving its molecules faster. Applying this to 240.4 grams of water for a temperature increase of 100°C, the calculation is straightforward—100,595.2 J in total will be used for this heating process.

Phase Change

Phase change is when a substance transitions from one state of matter to another, such as from solid to liquid, or liquid to gas. During a phase change, the temperature of the substance does not change, even though energy is inputted or released. This is because the energy is used to overcome intermolecular forces rather than increasing molecular motion.

This principle is crucial when working with the enthalpy of fusion and vaporization. For instance, when ice melts to water, the temperature remains at 0°C until all the ice has melted. Similarly, water stays at 100°C while it turns to steam, despite continuous heating. Understanding phase changes helps explain why so much energy can be inputted without actually changing the temperature of the substance.