Question

The enthalpy change for melting ice at \(0^{\circ} \mathrm{C}\) and constant atmospheric pressure is \(6.01 \mathrm{~kJ} / \mathrm{mol}\). Calculate the quantity of energy required to melt a moderately large iceberg with a mass of \(1.25\) million metric tons. (A metric ton is \(1000 \mathrm{~kg}\).)

Short answer

The quantity of energy required to melt a moderately large iceberg with a mass of 1.25 million metric tons is approximately \(4.17 \times 10^{11} ~\text{kJ}\).

Step-by-step solution

Convert the mass of the iceberg to kilograms

Since a metric ton is equal to 1000 kg, we can convert the mass of the iceberg to kilograms by multiplying: \(1.25 \times 10^6 ~\text{metric tons} \times 1000 ~\text{kg/metric ton} = 1.25 \times 10^9 ~\text{kg}\)

Determine the number of moles of ice

To find the number of moles of ice, we can use the formula: \(\text{moles} = \frac{\text{mass}}{\text{molar mass}}\) The molar mass of water (H₂O) is approximately 18.015 g/mol. It's also important to convert this to kilograms per mol by dividing by 1000: Molar mass of water = \(18.015 ~\text{g/mol} \div 1000 = 0.018015 ~\text{kg/mol}\) Now, we can find the number of moles: \(\text{moles} = \frac{1.25 \times 10^9 ~\text{kg}}{0.018015 ~\text{kg/mol}} = 6.935 \times 10^{10} ~\text{moles}\)

Calculate the energy required to melt the iceberg

We are given the enthalpy change for melting ice at 0°C, which is 6.01 kJ/mol. Now, we need to multiply the number of moles by the enthalpy change to find the required energy: \(\text{Energy} = \text{moles} \times \text{enthalpy change}\) \(\text{Energy} = (6.935 \times 10^{10} ~\text{moles}) \times (6.01 ~\text{kJ/mol}) = 4.17 \times 10^{11} ~\text{kJ}\) So the quantity of energy required to melt a moderately large iceberg with a mass of 1.25 million metric tons is approximately \(4.17 \times 10^{11} ~\text{kJ}\).

Key concepts

Melting Ice

The process of melting ice involves changing solid ice into liquid water. This happens when the ice absorbs enough energy in the form of heat. At 0°C, ice begins to melt under normal atmospheric pressure. It's important to understand that melting is an endothermic process, meaning it requires energy.

When you melt ice, you're overcoming the forces holding the water molecules in a rigid structure. This energy absorbed by the ice causes the molecules to vibrate more freely, transforming the solid into a liquid state.

To analyze melting from a chemistry perspective, knowing how to calculate the energy needed for this phase transition is crucial. This involves understanding concepts like enthalpy and molar mass.

Enthalpy

Enthalpy refers to the total heat content of a system and is denoted by the symbol "H". It's an important concept in thermodynamics that helps us understand energy changes in physical processes.

Specifically, the enthalpy change ( riangle H ) during a reaction indicates whether it’s absorbing or releasing energy. For melting ice, the enthalpy change is positive, meaning energy is absorbed. In our context, the enthalpy change for melting ice at 0°C is given as 6.01 kJ/mol.

This value tells us the amount of energy needed to melt one mole of ice. By understanding this change, we can calculate the total energy required for larger masses of ice, such as an iceberg.

Molar Mass

Molar mass is the weight of one mole of a substance, expressed in grams per mole (g/mol). For water, the molar mass is approximately 18.015 g/mol. This is a key piece of information for converting mass into moles, which is essential for calculating energy requirements.

It's useful to convert this molar mass into kilograms per mole. By dividing by 1000, we find that the molar mass of water is 0.018015 kg/mol.

Knowing this value allows us to convert the mass of ice from kilograms into moles. This conversion is necessary because enthalpy is often expressed per mole, aligning with the way energy changes are calculated in chemistry.

Energy Calculation

Calculating the energy required to melt ice involves several steps, integrating many of the concepts we've explored.

Here’s how we can do it:

• First, convert the mass of the iceberg into kilograms. For our problem, that's 1.25 million metric tons, which equals 1.25 × 10^9 kg.
• Next, use the molar mass to find the number of moles. Divide the mass by the molar mass of water (0.018015 kg/mol), resulting in 6.935 × 10¹⁰ moles.
• Multiply the number of moles by the enthalpy change ( 6.01 kJ/mol) to find the total energy.

This gives us an approximate total energy requirement of 4.17 × 10¹¹ kJ. By following these steps, you can calculate the energy needed for various processes involving changes in state, not just melting ice.