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 heat of fusion is \(11.3 \mathrm{kJ} / \mathrm{mol}\) What quantity of heat, 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

1624.73 J is needed to heat and melt the silver as described.

Step 1: Calculate Heat Required to Raise Temperature

Determine the amount of heat needed to raise the temperature of 5 g of silver from 25°C to its melting point at 962°C using the formula: \[ q_1 = m \cdot c \cdot \Delta T \]where:- \( m = 5.00 \text{ g} \)- \( c = 0.235 \text{ J/g} \cdot \text{K} \)- \( \Delta T = 962 - 25 = 937 \text{ °C (or K)}\)Thus,\[ q_1 = 5.00 \times 0.235 \times 937 = 1100.975 \text{ J} \]

Step 2: Convert Heat of Fusion to J/g

Convert the heat of fusion from \( \text{kJ/mol} \) to \( \text{J/g} \) using the molar mass of silver, which is 107.87 \( \text{g/mol} \):\[ \frac{11.3 \text{ kJ/mol}}{107.87 \text{ g/mol}} = \frac{11300 \text{ J/mol}}{107.87 \text{ g/mol}} = 104.75 \text{ J/g} \]

Step 3: Calculate Heat Required for Fusion

Calculate the heat required to melt 5.00 g of silver using the heat of fusion:\[ q_2 = m \cdot \text{(Heat of fusion per gram)} = 5.00 \times 104.75 = 523.75 \text{ J} \]

Step 4: Add Heat Quantities

Add the heat required to raise the temperature (from Step 1) and the heat required for fusion (from Step 3):\[ q_{\text{total}} = q_1 + q_2 = 1100.975 + 523.75 = 1624.725 \text{ J} \]

Step 5: Final Answer

The total quantity of heat required for the entire process is approximately:\[ 1624.73 \text{ J} \]

Key Concepts

Specific Heat

Specific heat is a property that describes how much heat energy a substance needs to increase its temperature by 1 degree Celsius (or 1 Kelvin) per unit mass. It is often represented by the symbol, \( c \), and measured in joules per gram per degree Celsius \( \mathrm{J}/\mathrm{g} \cdot \mathrm{K} \). For silver, the specific heat is \(0.235\, \mathrm{J}/\mathrm{g} \cdot \mathrm{K}\). This means that heating a single gram of silver by one degree requires only 0.235 joules of energy, reflecting its efficient heat conduction abilities.

• To calculate the total heat required to heat a substance, use the formula: \( q = m \cdot c \cdot \Delta T \), where \( q \) is the heat in joules, \( m \) is the mass in grams, \( c \) is the specific heat, and \( \Delta T \) is the change in temperature.
• Specific heat varies between different materials, indicating how quickly or slowly they heat up or cool down.

Heat of Fusion

The heat of fusion is the amount of energy required to change a substance from a solid to a liquid at its melting point without changing its temperature. For silver, the heat of fusion is \(11.3\, \mathrm{kJ/mol}\) or approximately \(104.75\, \mathrm{J/g}\) when converted using the molar mass of silver (107.87 g/mol). This value signifies the energy needed to overcome the solid structure of silver atoms as it transitions into the liquid phase.

• It is crucial because it helps us understand the energy expenditure associated with phase changes, where temperature remains constant.
• When calculating fusion heat, employ the equation: \( q = m \cdot \text{Heat of fusion per gram} \).

Understanding the heat of fusion concept is vital in processes like smelting or shaping materials, where thermal management is essential.

Temperature Change

Temperature change in physics refers to the difference between an initial and a final temperature of a given substance. In the exercise, the silver's temperature shifts from 25°C to 962°C, resulting in a change (\( \Delta T \)) of 937°C.

• Temperature change is important when calculating how much energy is required to bring a substance to a specific state.
• It significantly influences the heat calculation since it directly multiplies with specific heat and mass in the formula \( q = m \cdot c \cdot \Delta T \).

This significant increase in silver's temperature demonstrates the large amount of energy involved in heating it to its melting point.

Melting Point

The melting point is the specific temperature at which a solid becomes a liquid. For silver, this transition occurs at 962°C. It is crucial because it defines the conditions under which a substance shifts its state and determines thermal processing limits in industrial applications.

• Knowing the melting point is essential for engineers and scientists when designing systems that involve material heating, to prevent structural failures.
• In the context of this exercise, reaching the melting point marks the energy input needed to begin the phase change where additional energy is required for fusion.

The melting point can vary based on material quality and purity but remains a key characteristic for thermal calculations in both science and industry.