Problem 32 - Chemistry and Chemical Reactivity

Question

The specific heat capacity of silver is \(0.235 \mathrm{J} / \mathrm{g} \cdot \mathrm{K}\) Its melting point is \(962^{\circ} \mathrm{C},\) and its enthalpy of fusion is \(11.3 \mathrm{kJ} / \mathrm{mol} .\) What quantity of energy, in joules, is required to change \(5.00 \mathrm{g}\) of silver from a solid at \(25^{\circ} \mathrm{C}\) to a liquid at \(962^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

The energy required is approximately 1625.20 J.

Step 1: Calculate Energy Required to Heat the Silver to Melting Point

To find the energy required to heat the silver from 25°C to 962°C, we use the formula for heat energy: \[ q_1 = m imes c imes \Delta T \]where:- \( m = 5.00 \, \text{g} \) is the mass,- \( c = 0.235 \, \text{J/g}\cdot\text{K} \) is the specific heat capacity,- \( \Delta T = 962 - 25 = 937 \, \text{K} \) is the temperature change.Plugging these values into the formula:\[ q_1 = 5.00 \, \text{g} \times 0.235 \, \text{J/g}\cdot\text{K} \times 937 \, \text{K} = 1,100.875 \, \text{J} \]

Step 2: Calculate Energy Required for the Phase Change

To find the energy required for the phase change from solid to liquid at the melting point, we use the formula:\[ q_2 = n \times \Delta H_f \]where:- \( n \) is the number of moles of silver,- \( \Delta H_f = 11.3 \, \text{kJ/mol} \) is the enthalpy of fusion.First, calculate the number of moles \( n \):- Molar mass of silver \( = 107.87 \, \text{g/mol} \)- \( n = \frac{5.00 \, \text{g}}{107.87 \, \text{g/mol}} \approx 0.0464 \, \text{mol} \)Now calculate \( q_2 \):\[ q_2 = 0.0464 \, \text{mol} \times 11.3 \, \text{kJ/mol} \times 1000 \, \text{J/kJ} = 524.32 \, \text{J} \]

Step 3: Calculate Total Energy Required

The total energy required is the sum of the energy to heat the silver and the energy for the phase change:\[ q_{\text{total}} = q_1 + q_2 = 1,100.875 \, \text{J} + 524.32 \, \text{J} \]\[ q_{\text{total}} = 1,625.195 \, \text{J} \]

Step 4: Final Answer

The total energy required to change 5.00 g of silver from a solid at 25°C to a liquid at 962°C is 1,625.195 J.

Key Concepts

Enthalpy of Fusion

Enthalpy of fusion, also known as heat of fusion, is an important thermodynamic concept. It represents the amount of energy required to change a substance from a solid phase to a liquid phase at constant temperature. This energy is necessary to overcome the intermolecular forces that hold the solid in its rigid form.

For silver, the enthalpy of fusion is given as 11.3 kJ/mol. This means that 11.3 kilojoules of energy are needed to convert one mole of solid silver at its melting point into liquid silver, without changing its temperature.

When solving problems involving enthalpy of fusion, it is crucial to:

• Identify the substance and its enthalpy of fusion value.
• Ensure the amount of substance is measured in moles, as enthalpy of fusion is typically expressed per mole.
• Use the formula \( q = n \times \Delta H_f \), where \( n \) is the number of moles and \( \Delta H_f \) is the enthalpy of fusion.

This concept helps in quantifying the energy involved in phase changes.

Molar Mass Calculation

Molar mass is a measure that tells us the mass of one mole of a given substance, often expressed in grams per mole (g/mol). It plays a pivotal role in converting between mass and moles, which is essential for many stoichiometry problems.

To calculate the molar mass of silver, you take the atomic weight of silver, which is 107.87 g/mol. This means that one mole of silver atoms weighs 107.87 grams.

When working on problems like determining the energy required for phase changes, you'll often need to:

• Calculate the number of moles from a given mass of the substance using the formula \( n = \frac{m}{M} \), where \( m \) is the mass in grams, and \( M \) is the molar mass.
• Ensure precision in converting grams to moles for accurate energy calculations.

Understanding molar mass is crucial for linking the physical mass of a substance to its chemical properties.

Phase Change Energy Calculation

Phase change energy calculation involves determining the total energy required to change a substance from one phase to another. In this case, transforming silver from a solid at 25°C to a liquid at 962°C involves several energy calculations.

Initially, you need to calculate the energy to heat the solid silver up to its melting point using specific heat capacity. Specific heat capacity is the amount of heat per unit mass required to raise the temperature by one degree Celsius. The formula is \( q = m \times c \times \Delta T \), with \( m \) being the mass, \( c \) the specific heat, and \( \Delta T \) the temperature change.

Subsequently, the energy required for the actual phase transition at the melting point is calculated using the enthalpy of fusion. Summarizing, the total energy is the sum of the heating energy and the phase change energy:

• Calculate the heating energy for temperature rise.
• Add the energy for phase change using enthalpy of fusion.
• Total energy \( q_{\text{total}} = q_1 + q_2 \).

This comprehensive approach ensures the correct calculation of the total energy needed for the phase change.