Question

The specific heat of copper is \(0.385 \mathrm{~J} / \mathrm{g} \cdot{ }^{\circ} \mathrm{C}\). How much heat energy in kilojoules is required to raise the temperature of \(454 \mathrm{~g}\) of copper from \(40.0{ }^{\circ} \mathrm{C}\) to \(75.0^{\circ} \mathrm{C} ?\)

Short answer

The amount of heat energy required to raise the temperature of \(454 \mathrm{~g}\) of copper from \(40.0{ }^{\circ} \mathrm{C}\) to \(75.0^{\circ} \mathrm{C}\) is approximately \(6.198 \mathrm{~kJ}\).

Step-by-step solution

Identify the given values

In this exercise, we are given: - Specific heat capacity of copper (c) = 0.385 J/g°C - Mass of copper (m) = 454 g - Initial temperature (T1) = 40.0°C - Final temperature (T2) = 75.0°C

Calculate the temperature change (ΔT)

To find the change in temperature, we can use the formula ΔT = T2 - T1: ΔT = 75.0°C - 40.0°C = 35.0°C

Apply the heat transfer formula

We can now use the heat transfer formula, Q = mcΔT, to find the heat energy required to raise the temperature of the copper: Q = (454 g) × (0.385 J/g°C) × (35.0°C)

Calculate the heat energy

Multiply the values obtained in step 3: Q = 454 × 0.385 × 35 Q = 6197.9 J

Convert the heat energy to kilojoules

Since we want the result in kilojoules, we must convert the heat energy from joules to kilojoules: 1 Joule = 0.001 Kilojoules Thus, Q = 6197.9 J × 0.001 kJ/J = 6.1979 kJ The amount of heat energy required to raise the temperature of 454 g of copper from 40.0°C to 75.0°C is approximately 6.198 kJ.

Key concepts

Heat Transfer

Heat transfer is the process by which thermal energy moves from one material or substance to another. This can occur in three primary ways: conduction, when heat moves through a solid material; convection, which involves fluid or gas currents carrying heat; and radiation, where heat is transferred in the form of electromagnetic waves. In the context of our exercise, we're dealing with conduction, focusing on the transfer of heat through the metal, copper.

When a substance like copper is heated, its particles begin to vibrate more quickly and the energy is passed along from one particle to the next, resulting in an overall increase in temperature. It is important to realize that during this process, the amount of heat transferred depends on the specific heat capacity of the substance, its mass, and the temperature change. The specific heat capacity is a measure of how much energy is required to raise the temperature of one gram of a substance by one degree Celsius, often reflecting a material's ability to store heat.

Temperature Change Calculation

Understanding temperature change is vital when dealing with heat transfer problems. The temperature change, denoted as \( \Delta T \), is calculated by finding the difference between the final and initial temperatures. In our exercise, we've calculated \( \Delta T \) to be \( 75.0^{\circ}C \) minus \( 40.0^{\circ}C \) resulting in a \( 35.0^{\circ}C \) increase in temperature.

As \( \Delta T \) represents the net change, whether the temperature rises or falls, the sign is crucial in determining the direction of heat flow. In this scenario, because the temperature increases, the heat flow is positive, meaning energy is being absorbed by the copper to elevate its temperature.

Energy Conversion

Energy conversion, in thermodynamics, is the change of energy from one form to another. In our example, electrical energy provided by a heat source, such as a stove, is converted into thermal energy within the copper. This thermal energy is absorbed by the copper's particles, causing an increase in kinetic energy and, hence, an increase in temperature.

The formula \( Q = mc\Delta T \) that we use in the exercise assists in quantifying this energy conversion, where \( Q \) denotes the amount of heat energy in joules, \( m \) is the mass in grams, \( c \) is the specific heat capacity, and \( \Delta T \) is the temperature change in degrees Celsius. By calculating the energy required to achieve a certain temperature change, we are effectively bridging the gap between the abstract energy conversion and a tangible number, allowing us to understand and predict the outcomes of heating or cooling substances.