Problem 31 - Chemistry The Molecular Science

Question

A piece of iron ( \(400 . \mathrm{g}\) ) is heated in a flame and then plunged into a beaker containing \(1.00 \mathrm{~kg}\) water. The original temperature of the water was \(20.0^{\circ} \mathrm{C}\), but it is \(32.8^{\circ} \mathrm{C}\) after the iron bar is put in and thermal equilibrium is reached. Calculate the original temperature of the hot iron bar.

Step-by-Step Solution

Answer

The initial temperature of the iron is approximately \(330^{\circ} \mathrm{C}\).

Step 1: Identify Known Variables

First, let's identify the provided information in the problem. The mass of the iron is \(400 \text{ g}\) (or \(0.4 \text{ kg}\)), the mass of the water is \(1.00 \text{ kg}\), the specific heat capacity of water is \(4.18 \text{ J/g} \cdot { }^{\circ} \mathrm{C}\), and the specific heat capacity of iron is approximately \(0.45 \text{ J/g} \cdot { }^{\circ} \mathrm{C}\). The initial temperature of the water \(T_{\text{water initial}}\) is \(20.0^{\circ} \mathrm{C}\), and its final temperature \(T_{\text{final}}\) is \(32.8^{\circ} \mathrm{C}\). The final temperature of the iron will be the same as that of the water once equilibrium is reached.

Step 2: Set Up Heat Transfer Equation

When the system reaches thermal equilibrium, the heat lost by the hot iron will equal the heat gained by the water. This can be expressed as: \(m_{\text{iron}}c_{\text{iron}}(T_{\text{iron initial}} - T_{\text{final}}) = m_{\text{water}}c_{\text{water}}(T_{\text{final}} - T_{\text{water initial}})\)

Step 3: Substitute Known Values

Substitute the known values into the heat transfer equation:\(0.4 \times 0.45 \times (T_{\text{iron initial}} - 32.8) = 1.00 \times 4.18 \times (32.8 - 20.0)\)Calculate the respective sides to simplify the equation.

Step 4: Calculate Energy for Water

Calculate the energy that the water has absorbed:\[1.00 \times 4.18 \times (32.8 - 20.0) = 1.00 \times 4.18 \times 12.8 = 53.504 \text{ J}\]

Step 5: Calculate Energy Change for Iron

Calculate the iron's equivalent equation from the previous step:\[ 0.4 \cdot 0.45 \cdot (T_{\text{iron initial}} - 32.8) = 53.504 \]

Step 6: Solve for Iron's Initial Temperature

Solve the equation for the iron's initial temperature:1. \( 0.18(T_{\text{iron initial}} - 32.8) = 53.504 \)2. Distribute the 0.18: \( 0.18 \cdot T_{\text{iron initial}} - 5.904 = 53.504 \)3. Add 5.904 to both sides: \( 0.18 \cdot T_{\text{iron initial}} = 59.408 \)4. Divide by 0.18: \( T_{\text{iron initial}} = \frac{59.408}{0.18} = 330.04 \approx 330^{\circ} \mathrm{C} \)The initial temperature of the iron is approximately \(330^{\circ} \mathrm{C}\).

Key Concepts

Understanding Heat Transfer

Heat transfer is the movement of thermal energy from one object to another. In our exercise, heat transfers from the hot iron to the cooler water until both reach the same temperature. This process is driven by the temperature difference. The heat lost by the iron equals the heat gained by the water. This principle helps us calculate unknown temperatures. In equation form, it's expressed as:

• \( Q_{ ext{lost}} = Q_{ ext{gained}} \)
• Energy conservation is key in these calculations.

Understanding this helps us figure out how heat moves and equilibrates in various situations.

Exploring Specific Heat Capacity

Specific heat capacity is the amount of heat needed to raise the temperature of 1 gram of a substance by 1 °C. Different materials have different specific heat capacities. In our example, water has a specific heat capacity of 4.18 J/g°C, while iron's is 0.45 J/g°C. This means water requires more energy to change its temperature than iron does. Calculations involving specific heat help us understand how much heat is involved when substances heat up or cool down. The formula is:

• \( Q = mc\Delta T \)
• This indicates the relation between mass, specific heat, and temperature change.

Knowing the specific heat capacities allows for precise temperature predictions in mixed systems.

Achieving Thermal Equilibrium

Thermal equilibrium occurs when two objects reach the same temperature and heat transfer stops. In our situation, the iron and water achieve this balance at 32.8 °C. At this point, the heat lost by the iron exactly equals the heat gained by the water. This condition allows us to set up equations to find unknown initial temperatures. When objects in contact no longer transfer heat, thermal equilibrium is achieved. It's a foundational concept for understanding how energy distributes in physical systems.

Introduction to Calorimetry

Calorimetry is the science of measuring heat changes in physical and chemical processes. It uses the concept of heat transfer to determine unknown quantities like temperatures and heat capacities. In our problem, we essentially performed a calorimetry experiment to find the initial temperature of iron. By knowing the heat gained by the water, we calculated the heat lost by the iron. Calorimetry involves:

• Using equations and specific parameters to solve for unknowns.
• Applying the principles of energy conservation.

It's a practical application of thermochemistry concepts in real-world scenarios.