Problem 95 - Chemistry: The Central Science

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 \(1000 \mathrm{~kg}\).)

Step-by-Step Solution

Answer

The total energy required to melt a moderately large iceberg with a mass of \(1.25\) million metric tons is calculated as follows: 1. Convert mass to grams: \(1.25 \times 10^6\) metric tons = \(1.25 \times 10^9\) kg = \(1.25 \times 10^{12}\) g 2. Calculate moles of ice: \(\frac{1.25 \times 10^{12}\text{ g}}{18\text{ g/mol}} = 6.94 \times 10^{10}\) moles 3. Enthalpy change for melting ice = \(6.01 \mathrm{~kJ/mol}\) 4. Total energy required = \((6.94 \times 10^{10}\text{ moles}) \times (6.01 \mathrm{~kJ/mol}) = 4.17 \times 10^{14}\) kJ

Step 1: Convert the mass of the iceberg to grams

: First, we need to convert the mass of the iceberg from metric tons to grams. We know that 1 metric ton is 1000 kg, and 1 kg is 1000 grams. Mass of iceberg (grams) = mass of iceberg (metric tons) × 1000 kg/metric ton × 1000 g/kg

Step 2: Calculate the number of moles of ice in the iceberg

: Now that we have the mass of the iceberg in grams, we can determine the number of moles of ice. The molar mass of ice (H2O) is approximately 18 grams/mole (1 mole of hydrogen = 1 g/mole and 1 mole of oxygen = 16 g/mole). Number of moles = mass of iceberg (grams) / molar mass of ice (g/mole)

Step 3: Calculate the energy required to melt one mole of ice

: The enthalpy change for melting ice is given as 6.01 kJ/mol. This means that 6.01 kJ of energy is required to melt 1 mole of ice at 0°C and constant atmospheric pressure.

Step 4: Calculate the total energy required to melt the entire iceberg

: Now, we just need to multiply the energy required to melt one mole of ice by the total number of moles in the iceberg to get the total energy required: Total energy required (kJ) = (Number of moles) × (Enthalpy change) Plug in the values from steps 1, 2, and 3 to find the total energy required.

Key Concepts

Understanding Molar Mass in Melting Ice

Molar mass is a critical concept when calculating the energy needed to melt substances like ice. It refers to the mass of one mole of a substance.
In the case of water (H₂O), the molar mass is the combined mass of two hydrogen atoms and one oxygen atom. Each hydrogen atom has a molar mass of 1 g/mol, and oxygen has 16 g/mol.
Therefore, the molar mass of ice is calculated as:

• Molar mass of H₂O = 2(1 g/mol) + 16 g/mol = 18 g/mol

By understanding molar mass, you can easily find out how many moles of ice exist in a given mass. Knowing the moles is essential for calculating the energy required for melting.

The Process of Melting Ice

Melting ice involves a phase change from solid to liquid, requiring energy input. This energy is needed to break the hydrogen bonds between water molecules.
When ice melts, it's a physical change, meaning the water molecules change state but not their intrinsic properties. The melting occurs at a constant temperature of 0°C under standard atmospheric pressure.

• The process is highly predictable and repeatable, making calculations straightforward once the variables are known.

Understanding this process helps in determining how much energy is necessary to convert ice into water, as is needed in our iceberg scenario.

Calculating Energy Required for Melting

Energy calculation is at the heart of melting ice. To determine the total energy required, follow these steps:
First, convert the mass of the iceberg from metric tons to grams:

• 1 metric ton = 1,000 kg and 1 kg = 1,000 grams.

Next, calculate the total number of moles in the iceberg by using the molar mass of ice. Simply divide the mass of the iceberg in grams by the molar mass:

• Number of moles = mass (g) / 18 g/mol

Finally, using the known enthalpy of fusion, multiply the number of moles by the enthalpy change to find the total energy:

• Total energy (kJ) = Number of moles × 6.01 kJ/mol

This calculation reveals the energy needed to melt a specific mass of ice at 0°C.

Enthalpy of Fusion and Its Role

Enthalpy of fusion refers to the amount of energy required to change a substance from a solid to a liquid at its melting point. For ice, the enthalpy of fusion is 6.01 kJ/mol.
This value is crucial when calculating how much energy is needed for a phase transition, such as melting an iceberg.

• The 6.01 kJ/mol quantifies the energy to overcome molecular bonds in one mole of ice without changing temperature.

By understanding this concept, you can appreciate the thermodynamic principles at play. The enthalpy of fusion helps you accurately determine the total energy required to perform a complete melt of an ice mass, as seen in the iceberg example.