Problem 107 - Chemistry The Molecular Science

Question

The specific heat capacity of copper is \(0.385 \mathrm{~J} \mathrm{~g}^{-1}{\underline{\phantom{xx}}}^{\circ} \mathrm{C}^{-1}\), whereas it is \(0.128 \mathrm{~J} \mathrm{~g}^{-1}{\underline{\phantom{xx}}}^{\circ} \mathrm{C}^{-1}\) for gold. Assume you place \(100 . \mathrm{g}\) of each metal, originally at \(25^{\circ} \mathrm{C},\) in a boiling water bath at \(100^{\circ} \mathrm{C}\). If energy is transferred to each metal at the same rate, determine which piece of metal will reach \(100^{\circ} \mathrm{C}\) first.

Step-by-Step Solution

Answer

Gold reaches 100°C first.

Step 1: Understand the Problem

We need to determine which metal, copper or gold, heats up to 100°C faster when energy is transferred at the same rate. This requires considering specific heat and mass of each metal.

Step 2: Recall the Formula for Heat Transfer

The heat (\( q \)) required to change the temperature of a substance is calculated using the formula \( q = mc\Delta T \), where \( m \) is mass, \( c \) is specific heat capacity, and \( \Delta T \) is the change in temperature.

Step 3: Calculate the Heat Required for Copper

For copper, substitute the known values into the formula: mass \( m = 100 \text{ g} \), \( c = 0.385 \text{ J g}^{-1}\text{ °C}^{-1} \), and \( \Delta T = 100\, \text{°C} - 25\, \text{°C} = 75\, \text{°C} \). Thus, \( q = 100 \times 0.385 \times 75 = 2887.5 \text{ J} \).

Step 4: Calculate the Heat Required for Gold

For gold, use the values: \( m = 100 \text{ g} \), \( c = 0.128 \text{ J g}^{-1}\text{ °C}^{-1} \), and \( \Delta T = 75\, \text{°C} \). So, \( q = 100 \times 0.128 \times 75 = 960 \text{ J} \).

Step 5: Compare the Heat Required

Copper requires 2887.5 J to reach 100°C, whereas gold requires only 960 J. Since energy is transferred at the same rate to both metals, the metal that requires less energy will reach the target temperature first.

Step 6: Conclude Which Metal Heats Faster

Gold requires less heat energy to reach 100°C compared to copper. Thus, gold will reach 100°C first if both are heated at the same energy transfer rate.

Key Concepts

Heat Transfer Formula

Understanding how heat is transferred in materials can help us answer many practical questions, such as which material will heat up faster. The fundamental equation for calculating the amount of heat energy (\( q \) ) needed to change the temperature of a substance is known as the heat transfer formula. This formula is expressed as \( q = mc\Delta T \), where:

• \( q \) is the heat energy in joules (J).
• \( m \) is the mass of the substance in grams (g).
• \( c \) is the specific heat capacity, which indicates how much energy is needed to raise the temperature of 1 gram of a substance by 1°C, measured in \( \text{J g}^{-1}\text{ °C}^{-1} \).
• \( \Delta T \) is the temperature change, calculated as the final temperature minus the initial temperature.

Basing on this formula, you can calculate how much energy is necessary to heat a specific material to a desired temperature. This concept is crucial in comparing how different substances react to the same energy input.

Metal Heating Comparison

When comparing the heating of different metals, their specific heat capacities play a vital role. Specific heat capacity is an intrinsic property that indicates how much heat energy is required to change the temperature of a given mass by 1°C. In our exercise, copper has a specific heat capacity of \(0.385 \text{ J g}^{-1} \text{ °C}^{-1} \), while for gold, this value is \(0.128 \text{ J g}^{-1} \text{ °C}^{-1} \). Let's assume both metals have a mass of 100 g and are initially at the same temperature, 25°C. Their response to heating will differ due to these specific heat capacities:

• Copper: Higher specific heat capacity means it requires more energy to increase its temperature by the same amount compared to metals with lower specific heat capacities.
• Gold: Lower specific heat capacity means less energy is needed to achieve the same temperature change, making it easier and faster to heat.

Understanding these characteristics allows us to predict that, given the same energy input, gold will reach a specific temperature faster than copper.

Temperature Change Calculation

Calculating the amount of heat required to increase a substance's temperature involves determining the temperature change, \( \Delta T \), from the initial to the final temperature. In the example, both copper and gold start at an initial temperature of 25°C and need to be heated to 100°C.Therefore, the change in temperature for each metal is:

• \( \Delta T = 100°C - 25°C = 75°C \)

This change is the same for both metals, but the energy required differs due to their specific heat capacities. By substituting the values into the formula for each metal:

• Copper: \( q = 100 \times 0.385 \times 75 = 2887.5 \text{ J} \).
• Gold: \( q = 100 \times 0.128 \times 75 = 960 \text{ J} \).

From these calculations, it is evident that gold requires significantly less energy to achieve the same temperature change compared to copper. This highlights the importance of specific heat capacity and demonstrates why gold heats up faster in this scenario.