Problem 47 - General Chemistry: Principles and Modern Applications

Question

The normal melting point of copper is \(1357 \mathrm{K}\), and \(\Delta \mathrm{H}_{\text {fus }}\) of \(\mathrm{Cu}\) is \(13.05 \mathrm{kJ} \mathrm{mol}^{-1}\). (a) How much heat, in kilojoules, is evolved when a \(3.78 \mathrm{kg}\) sample of molten Cu freezes? (b) How much heat, in kilojoules, must be absorbed at 1357 K to melt a bar of copper that is \(75 \mathrm{cm} \times\) \(15 \mathrm{cm} \times 12 \mathrm{cm} ?\) (Assume \(d=8.92 \mathrm{g} / \mathrm{cm}^{3}\) for \(\mathrm{Cu}\).)

Step-by-Step Solution

Answer

The heat evolved in part (a) when a 3.78 kg sample of molten Cu freezes is -776.23 kJ and the heat absorbed in part (b) at 1357 K to melt a bar of copper that is 75 cm x 15 cm x 12 cm is 24735.89 kJ.

Step 1: Convert the mass of copper into moles for part (a)

To convert mass into moles, the formula used is Moles = mass ÷ molar mass. The mass of copper is given as 3.78 kg, which is 3780g (since 1kg = 1000g). The molar mass of copper (Cu) is 63.55g. Moles of Cu = 3780g ÷ 63.55g/mol = 59.5 mol.

Step 2: Calculate the heat evolved when the molten Cu freezes

We can find the heat evolved when copper freezes using the enthalpy of fusion formula i.e. q = nΔH_fus. Therefore, q = 59.5 mol × -13.05 kJ/mol= -776.23 kJ. The answer is negative because energy is being released (evolved) when Cu changes from liquid to solid.

Step 3: Calculate the volume of copper in part (b)

The volume of the copper bar can be calculated using the formula for the volume of a rectangular parallelepiped which is volume = length × breadth × height. Therefore, volume = 75 cm × 15 cm × 12 cm = 13500 cm^3.

Step 4: Calculate the mass of copper in part (b)

The mass of the copper can be calculated by multiplying its volume by the density. Therefore, mass = volume × density = 13500 cm^3 × 8.92 g/cm^3 = 120420 g.

Step 5: Convert the mass of copper into moles for part (b)

Converting mass into moles, we get Moles = mass ÷ molar mass = 120420g ÷ 63.55g/mol = 1895.39 mol.

Step 6: Calculate the heat absorbed when the Cu melts

Use the enthalpy of fusion formula i.e. q = nΔH_fus = 1895.39 mol × 13.05 kJ/mol = 24735.89 kJ. The answer is positive because energy is being absorbed when Cu changes from solid to liquid.

Key Concepts

Enthalpy of Fusion

The enthalpy of fusion, often symbolized as \( \Delta H_{\text{fus}} \), is a measure of the heat energy needed to convert a solid into a liquid at its melting point without changing its temperature. This concept is crucial in understanding how substances transition from solid to liquid. In the context of this problem, the enthalpy of fusion for copper (Cu) is provided as 13.05 kJ/mol. This value indicates the amount of energy required to completely melt one mole of copper at its normal melting point of 1357 K.
Understanding the enthalpy of fusion is essential for calculating the energy changes during phase transitions. When a substance melts, it absorbs heat, resulting in a positive enthalpy change. Conversely, when it freezes, it releases heat, causing a negative enthalpy change. In this exercise, we apply the formula \( q = n \Delta H_{\text{fus}} \), where \( q \) represents the heat absorbed or released, to determine these energy changes as copper transitions between solid and liquid forms.

Phase Changes

Phase changes involve the transition of matter from one state to another, such as solid to liquid or vice versa. These changes occur through processes like melting, freezing, vaporization, and condensation. In this exercise, we specifically focus on the melting and freezing of copper.
During the melting of copper, energy is absorbed to overcome the intermolecular forces holding the solid structure together, turning it into a liquid state. This requires the input of heat, as calculated using the enthalpy of fusion. On the contrary, when copper freezes, it transitions from a liquid to a solid state, releasing the absorbed heat back into the environment.
Here's a simplified breakdown of the two processes involved:

• Melting: Solid to liquid transition. Requires absorption of heat. Example: melting a 75 cm x 15 cm x 12 cm copper bar.
• Freezing: Liquid to solid transition. Releases heat. Example: freezing a 3.78 kg sample of molten copper.

Molar Mass Calculation

Calculating the molar mass is crucial for converting between grams and moles, a key step in thermodynamic calculations. The molar mass is defined as the mass of one mole of a substance, expressed in g/mol. For copper (Cu) in this exercise, the molar mass is given as 63.55 g/mol.
To determine the number of moles from a given mass, you use the formula:\[ \text{Moles} = \frac{\text{Mass}}{\text{Molar Mass}} \]In part (a) of our exercise, we converted 3.78 kg of copper into moles by dividing its mass in grams (3780 g) by the molar mass. In part (b), we employed the same method: first calculating the volume of the copper bar and then its mass, before converting it into moles using the molar mass.
This conversion is a fundamental step in chemistry that bridges the measurable physical mass in the laboratory with the abstract notion of substance amount in moles, allowing accurate calculation of energy changes and other quantitative relationships in chemical reactions.