Problem 126 - Introductory Chemistry Atoms First

Question

How many joules of heat energy would it take to raise the temperature of \(2.00\) pounds of iron from \(30.0^{\circ} \mathrm{C}\) to \(90.0^{\circ} \mathrm{C} ?[453.6 \mathrm{~g}=1\) pound \(]\)

Step-by-Step Solution

Answer

The heat energy required to raise the temperature of $2.00$ pounds of iron from $30.0^{\circ} \mathrm{C}$ to $90.0^{\circ} \mathrm{C}$ is \(24,393.6 \, \mathrm{J}\).

Step 1: Convert the mass from pounds to kilograms

First, we need to convert the mass of the iron from pounds to kilograms. We are given the conversion factor: 1 pound is equal to 453.6 grams, and 1 kilogram is equal to 1000 grams. So, \(2.00 \, lb \times \frac{453.6 \, g}{1 \, lb} \times \frac{1 \, kg}{1000 \, g} = 0.9072 \, kg\) The mass of the iron is 0.9072 kg.

Step 2: Calculate the change in temperature

The change in temperature (\(\Delta T\)) can be found by subtracting the initial temperature from the final temperature: \(\Delta T = T_{final} - T_{initial} = 90.0^\circ C - 30.0^\circ C = 60.0^\circ C\) The change in temperature is 60.0 °C.

Step 3: Calculate the heat energy

Now, we can use the heat energy formula, along with the mass, specific heat capacity, and change in temperature: \(Q = mc\Delta T\) \(Q = (0.9072 \, kg) (450 \, \frac{J}{kg \cdot^\circ C}) (60.0 \, ^\circ C)\) \(Q = 24393.6 \, J\) The heat energy required to raise the temperature of the iron is 24,393.6 joules.

Key Concepts

Specific Heat Capacity

Specific heat capacity is a crucial factor when dealing with thermal calculations. It's a property of a substance that indicates the amount of heat required to change the temperature of a unit mass by one degree Celsius (or Kelvin). For instance, the specific heat capacity of iron is given as \(450 \, \frac{J}{kg \cdot ^\circ C}\). This tells us that for every kilogram of iron, 450 joules are needed to increase the temperature by one degree Celsius.
Knowing the specific heat capacity allows us to calculate the energy required to change the temperature of a given mass of the material. This concept allows engineers and scientists to understand how materials react to heat and are essential for designing thermal systems.

• Substances with a high specific heat capacity require more energy to change their temperature.
• Water, for example, has a particularly high specific heat capacity, which is why it's effective for regulating temperature in environments.
• The formula to calculate heat using specific heat capacity is \(Q = mc\Delta T\), where \(Q\) is the heat energy.

Temperature Change

Temperature change refers to the difference between an initial and a final temperature, and it is a vital component in the heat calculation formula. In this exercise, the initial temperature of iron is \(30.0^\circ \mathrm{C}\) and the final temperature is \(90.0^\circ \mathrm{C}\).
To determine the temperature change, we subtract the initial temperature from the final temperature. The result is the temperature change (\(\Delta T\)):
\[\Delta T = T_{\text{final}} - T_{\text{initial}} = 90.0^\circ C - 30.0^\circ C = 60.0^\circ C\]
This calculation indicates that the iron's temperature needs to increase by \(60.0^\circ C\).

• Temperature change impacts the total energy needed: the greater the change, the more energy required.
• Knowing how much the temperature must change allows for precise energy calculations in various applications.

Conversion of Units

Unit conversion is a fundamental skill in physics and other sciences, especially when working with different measurement systems. In this exercise, we are shifting the mass measurement from pounds to kilograms, which is the standard SI unit used in physics.
The conversion process begins with the given information that \(1\) pound equals \(453.6\) grams. To convert pounds to kilograms, follow these steps:

• Start by converting pounds to grams: \(2.00 \, \text{lb} \times \frac{453.6 \, \text{g}}{1 \, \text{lb}} = 907.2\, \text{g}\).
• Next, convert grams to kilograms since there are \(1000\) grams in a kilogram: \(\text{mass in kg} = \frac{907.2 \, \text{g}}{1000} = 0.9072\, \text{kg}\).

These conversions allow for accurate calculations using the specific heat capacity, which is often given in \(\text{J/kg}\cdot^\circ \text{C}\).
Understanding unit conversion is essential for avoiding errors and ensuring that calculations are consistent across different systems of measurement.

Iron

Iron is one of the most commonly used metals in the world, known for its utility in construction, manufacturing, and other industries. Its wide application is due to its abundance and favorable properties such as strength, ductility, and its magnetic nature. However, when it comes to thermal properties, understanding iron's specific heat capacity is essential for calculating energy usage.
In terms of its thermal attributes, iron, having a specific heat capacity of \(450 \, \frac{J}{kg \cdot ^\circ C}\), requires a moderate amount of energy to increase its temperature. This makes it suitable for applications where controlled heating is necessary, like in cookware and various heating systems.

• Its strength also allows it to withstand high temperatures without deforming easily.
• This property is essential in scenarios where thermal stability is required over long periods.

Overall, understanding iron’s thermal properties helps us make informed decisions in both everyday and industrial applications.