Question

A \(225-\mathrm{g}\) sample of aluminum was heated to \(125.5^{\circ} \mathrm{C}\), then placed into \(500.0 \mathrm{~g}\) water at \(22.5^{\circ} \mathrm{C}\). (The specific heat of aluminum is \(0.900 \mathrm{~J} / \mathrm{g}^{\circ} \mathrm{C}\) ). Calculate the final temperature of the mixture. (Assume no heat is lost to the surroundings.)

Short answer

The final temperature of the mixture is the temperature \(T_f\) that satisfies the equation obtained by equating the heat lost by aluminum with the heat gained by the water.

Step-by-step solution

Understand the Concept

In a closed system, the heat lost by the hot object will be equal to the heat gained by the cold object. This principle is based on the law of conservation of energy. The formula for heat transfer is \(Q = mc\Delta T\), where \(Q\) is the heat transferred, \(m\) is the mass of the substance, \(c\) is the specific heat capacity, and \(\Delta T\) is the change in temperature.

Calculate Heat Lost by Aluminum

Use the specific heat formula for the aluminum to calculate the heat lost. Let the final temperature be \(T_f\).\[Q_{\text{al}} = m_{\text{al}}c_{\text{al}}(T_{\text{initial, al}} - T_f)\]Plug in the given values and solve for \(Q_{\text{al}}\).\[Q_{\text{al}} = 225\text{ g} \times 0.900 \frac{\text{J}}{\text{g}^{\text{o}}\text{C}}\times (125.5^{\text{o}}\text{C} - T_f)\]

Calculate Heat Gained by Water

Do the same for the water. The specific heat of water is typically \(4.184 \text{J/g}^{\text{o}}\text{C}\).\[Q_{\text{water}} = m_{\text{water}}c_{\text{water}}(T_f - T_{\text{initial, water}})\]Plug in the given values for the water and solve for \(Q_{\text{water}}\).\[Q_{\text{water}} = 500.0\text{ g} \times 4.184 \frac{\text{J}}{\text{g}^{\text{o}}\text{C}}\times (T_f - 22.5^{\text{o}}\text{C})\]

Set Heat Lost Equal to Heat Gained

No heat is lost to the surroundings, so the heat lost by aluminum \(Q_{\text{al}}\) is equal to the heat gained by the water \(Q_{\text{water}}\).\[Q_{\text{al}} = Q_{\text{water}}\]

Combine the Equations and Solve for Final Temperature

Combine the equations from steps 2 and 3 and solve for the final temperature \(T_f\).\[225 \times 0.900 \times (125.5 - T_f) = 500.0 \times 4.184 \times (T_f - 22.5)\]Expand and simplify the equation to solve for \(T_f\).

Calculate the Final Temperature

After simplifying the equation from the previous step, you should have a linear equation in terms of \(T_f\). Solve this equation to find the final temperature of the mixture.

Key concepts

Specific Heat Capacity

Have you ever wondered why some materials heat up quicker than others? This has to do with a property called specific heat capacity. It's a measure of how much energy is needed to raise the temperature of one gram of a substance by one degree Celsius. Different materials have different capacities to store heat, thus they require varying amounts of energy to change their temperature.

For instance, water has a specific heat capacity of about 4.184 J/g°C, which is relatively high, meaning it takes a lot of energy to heat water. In contrast, aluminum has a specific heat capacity of 0.900 J/g°C, which is lower, so it heats up faster. This concept is critical in thermochemistry calculations, as it affects how substances in a mixture will exchange heat until they reach thermal equilibrium - a fancy term for the final uniform temperature of the mixture.

Heat Transfer Equation

The key to understanding the exchange of heat between substances lies in the heat transfer equation. This equation, expressed as Q = mcΔT, allows you to calculate the amount of heat (Q) absorbed or released by a substance. Here, 'm' stands for mass, 'c' for specific heat capacity, and 'ΔT' for the change in temperature.

With this formula, you can find out how much heat energy is required to raise the temperature of a specific mass of substance, or conversely, how much energy is released when the substance cools down. It's effectively the mathematical translation of the specific heat capacity concept we discussed previously, turning it into a practical tool for solving real-world problems involving heat changes.

Law of Conservation of Energy

The principle that energy can neither be created nor destroyed, only converted from one form to another, is known in physics as the law of conservation of energy. This principle is a cornerstone of thermochemistry calculations, particularly when dealing with closed systems.

In the context of our exercise, it means that the total amount of heat energy before and after the aluminum is placed in the water must remain constant. In simpler terms, the heat lost by the hot aluminum as it cools will equal the heat gained by the cooler water until both materials reach the same final temperature. Understanding this law helps us set up our calculations correctly, knowing that we're essentially transferring energy from one body to another without any loss or gain.

Final Temperature Calculation

The final temperature calculation is the climax of our little thermochemistry drama where we find out the temperature at which our two actors, aluminum and water, coexist in thermal harmony. The process involves combining the heat transferred to and from each substance and setting these quantities equal to each other—because, thanks to the law of conservation of energy, they must be equal.

The equation looks something like this for our specific exercise: heat lost by aluminum equals heat gained by water. Solving for the final temperature (T_f) requires some algebra, but once you do, you've effectively determined the temperature at which the heat exchange stops, as both substances stabilize at the same temperature. It's a practical conclusion that has implications in everything from cooking to industrial processes.