Question

The initial temperature of a 344 -g sample of iron is \(18.2^{\circ} \mathrm{C}\). If the sample absorbs \(2.25 \mathrm{kJ}\) of energy as heat, what is its final temperature?

Short answer

The final temperature of the iron sample is approximately 32.84°C.

Step-by-step solution

Understand the Given Values

We are given that the mass of the iron sample is 344 g, the initial temperature is \(18.2^{\circ} \mathrm{C}\), and the energy absorbed is 2.25 kJ. We will need these values for calculation.

Convert Energy to Joules

Since energy is commonly measured in joules rather than kilojoules in thermal calculations, convert 2.25 kJ to joules. Thus, \(2.25 \mathrm{kJ} = 2250 \mathrm{J}\).

Use the Specific Heat Formula

The formula to find the change in temperature using specific heat is \(q = m \cdot c \cdot \Delta T\), where \(q\) is the energy absorbed (2250 J), \(m\) is the mass (344 g), \(c\) is the specific heat capacity of iron (0.449 J/g°C), and \(\Delta T\) is the change in temperature.

Solve for Change in Temperature \(\Delta T\)

Rearrange the formula to solve for \(\Delta T\): \[\Delta T = \frac{q}{m \cdot c} = \frac{2250 \text{ J}}{344 \text{ g} \times 0.449 \text{ J/g°C}}\].Calculate the value to find \(\Delta T\).

Calculate Final Temperature

\(\Delta T\) calculated earlier was approximately 14.64°C. Now add this change to the initial temperature: \(T_{final} = 18.2^{\circ} \mathrm{C} + 14.64^{\circ} \mathrm{C} = 32.84^{\circ} \mathrm{C}\).

Key concepts

Understanding Thermal Energy Conversion

Thermal energy conversion refers to the process of changing heat energy from one form to another. In many scientific and real-world applications, converting heat energy is vital for processes like warming objects or changing their physical state.
In this context, the heat energy absorbed by the iron is converted into an increase in temperature. It's crucial to understand that when a substance absorbs thermal energy, its molecular activity increases, causing its temperature to rise.
For calculations, energy is often measured in joules (J), as seen in the problem. Here, the iron sample absorbs energy measured initially in kilojoules (kJ), which then needs conversion to joules to aid in further calculations.

• 1 kilojoule (kJ) = 1000 joules (J), so the conversion results in 2250 J.
• This energy conversion is essential for calculations involving specific heat capacity, which will facilitate determining temperature change.

Calculating Temperature Change Using Specific Heat Capacity

Specific heat capacity is a property of a substance that describes how much heat is needed to change the temperature of a given mass by one degree Celsius. It varies among materials, and for iron, it is approximately 0.449 J/g°C.
In this exercise, we use the specific heat capacity to calculate the temperature change of the iron sample.
The formula for finding the temperature change, \( q = m \cdot c \cdot \Delta T, \) where:

• \( q \) is the heat absorbed (2250 J),
• \( m \) is the mass (344 g),
• \( c \) is the specific heat capacity of iron (0.449 J/g°C),
• \( \Delta T \) is the change in temperature.

To find the temperature change \( \Delta T \), rearrange the equation to solve for \( \Delta T \):\[ \Delta T = \frac{q}{m \cdot c}. \] Substitute the given values to find \( \Delta T \) = \( \frac{2250}{344 \times 0.449} \). This results in a temperature change of approximately 14.64°C.

Iron's Heat Absorption Effect

The heat absorption of iron depends largely on how much thermal energy it can store which directly influences the temperature change. Different materials have different specific heat capacities, meaning they require varying amounts of energy to achieve the same temperature change.
Iron, with its specific heat capacity of 0.449 J/g°C, shows that it relatively quickly absorbs energy resulting in a notable temperature increase.
In practical terms, after energy absorption and calculating the temperature change, understanding iron's behavior in thermodynamics helps determine its practical applications, from constructing vehicles to basic cookware.
In our exercise, after calculating the temperature increase (\(\Delta T\)) to be 14.64°C, it is then added to the initial temperature (18.2°C) to find the final temperature of the iron, which results in approximately 32.84°C. This illustrates iron's efficiency in heat absorption and reflects its effectiveness in various thermal applications where rapid heating is necessary.