Question

A \(25.0-\mathrm{g}\) sample of pure iron at 85 ' \(\mathrm{C}\) is dropped into \(75 \mathrm{~g}\) of water at \(20 .\) ' \(\mathrm{C}\). What is the final temperature of the water-iron mixture?

Short answer

The final temperature of the water-iron mixture is approximately \(22.24 °C\).

Step-by-step solution

List the given information

The problem provides the following information: - Mass of iron (m₁): \(25.0 g\) - Initial temperature of iron (T₁i): \(85°C\) - Mass of water (m₂): \(75 g\) - Initial temperature of water (T₂i): \(20°C\) - Specific heat capacity of iron (c₁): \(0.45 J/(g °C)\) - Specific heat capacity of water (c₂): \(4.18 J/(g °C)\) Note that we have the specific heat capacity values for iron and water, which are constants.

Write the heat conservation formula

Since the heat lost by the iron will equal the heat gained by the water, we can write the equation: \(m_1c_1(T_f - T_{1i}) = m_2c_2(T_{2i} - T_f)\) where T_f is the final temperature of the water-iron mixture.

Solve for the final temperature (T_f)

Now we'll plug in the known values into the equation: \((25.0 g)(0.45 J/(g °C))(T_f - 85 °C) = (75 g)(4.18 J/(g °C))(20 °C - T_f)\) Next, we'll perform the multiplications and solve for T_f: \(11.25 (T_f - 85 °C) = 313.5(20°C - T_f)\) \(11.25T_f - 956.25 = 6270 - 313.5T_f\) Combine the terms with T_f: \(324.75T_f = 7226.25\) Finally, divide by 324.75 to find the final temperature: \(T_f = \dfrac{7226.25}{324.75} ≈ 22.24 °C\)

State the final temperature

The final temperature of the water-iron mixture is approximately \(22.24 °C\).

Key concepts

Understanding Specific Heat Capacity

Specific heat capacity is a property that measures how much heat energy is needed to raise the temperature of a substance by one degree Celsius (or one kilogram-Kelvin). This value is crucial for solving problems involving temperature changes and heat transfer because it helps us understand how different materials react to heat. It's often symbolized by 'c' and the units are typically expressed in joules per gram degree Celsius (\(J/(g°C)\)).

To illustrate, in the exercise, the specific heat capacity of water (\(c_2 = 4.18 J/(g °C)\) is much higher than that of iron (\(c_1 = 0.45 J/(g °C)\)). This means that water requires more energy to change its temperature compared to iron. When solving for the final temperature in a heat exchange problem, knowing the specific heat capacities of the substances involved is essential.

The Flow of Heat Transfer

Heat transfer is a fundamental concept in thermochemistry that refers to the movement of thermal energy between objects or substances of different temperatures. It occurs until thermal equilibrium is reached, meaning the objects involved attain the same temperature. There are three primary modes of heat transfer: conduction, convection, and radiation, but in the context of the given exercise, we deal with conduction, the direct transfer of heat through matter.

When a hot iron sample is introduced into cooler water, heat flows from the iron to the water. The speed and amount of energy transferred depend on the materials' specific heat capacities and masses. We use the heat transfer equation to quantify the energy exchanged, ensuring the heat lost by the hot substance equals the heat gained by the cooler substance, as shown in the solution steps.

Conservation of Energy in Thermal Processes

Conservation of energy is a fundamental principle in physics stating that energy cannot be created or destroyed, only transferred or converted from one form to another. In the context of our problem with iron and water, when no heat is lost to the surroundings, the amount of heat lost by the iron must be equal to the amount of heat gained by the water.

Through the lens of this principle, the exercise establishes an equation that balances the energy exchange, ensuring the conservation of energy. By doing this, we can find an unknown final temperature that conforms to the principle. Understanding that the total energy before and after the heat exchange must remain constant is critical in accurately determining the outcome of such thermochemical processes.