Question

A 25.5 -g aluminum block is warmed to \(65.4^{\circ} \mathrm{C}\) and plunged into an insulated beaker containing 55.2 \(\mathrm{g}\) water initially at \(22.2^{\circ} \mathrm{C}\) . The aluminum and the water are allowed to come to thermal equilibrium. Assuming that no heat is lost, what is the final temperature of the water and aluminum?

Short answer

The final temperature of the water and aluminum block when they reach thermal equilibrium.

Step-by-step solution

Write down the specific heat capacities

The specific heat capacity of water is approximately 4.18 J/g°C. For aluminum, it is approximately 0.897 J/g°C.

Set up the heat transfer equation

Since no heat is lost, and because of the conservation of energy, the heat gained by the water must be equal to the heat lost by the aluminum block. Use the formula: heat gained or lost = mass * specific heat capacity * change in temperature (q = m * c * ∆T). Set up the equation so that the heat lost by aluminum equals the heat gained by water.

Define change in temperature for each substance

The change in temperature for each substance can be defined as the final temperature minus its initial temperature: ∆T_aluminum = T_final - 65.4°C and ∆T_water = T_final - 22.2°C. Since we are looking for T_final, let's use these expressions in our next calculations.

Express the heat transfer equation with the given values

For aluminum: q_aluminum = 25.5 g * 0.897 J/g°C * (T_final - 65.4°C). For water: q_water = 55.2 g * 4.18 J/g°C * (T_final - 22.2°C). We now equate these to set up the equation 25.5 g * 0.897 J/g°C * (T_final - 65.4°C) = 55.2 g * 4.18 J/g°C * (T_final - 22.2°C).

Solve the equation for T_final

Expand and simplify the above equation to a standard form and solve for T_final. This will require distributing the multiplication across the parentheses and then collecting like terms to isolate T_final.

Extract the final temperature from the equation

After solving the equation, we will obtain the numeric value for T_final which is the final temperature in °C.

Key concepts

Specific Heat Capacity

The concept of specific heat capacity is crucial when dealing with thermal processes. It is defined as the amount of heat energy required to raise the temperature of one gram of a substance by one degree Celsius (or one Kelvin). Different materials have different capacities for storing thermal energy; for instance, water has a relatively high specific heat capacity, which means it can absorb a lot of heat before it starts to get hot.

Specific heat capacity can be thought of like the 'thermal inertia' of a substance; just as a heavyweight is harder to push to a higher speed, a substance with a high specific heat capacity is harder to push to a higher temperature. This property explains why land near the ocean has milder climates, as the water can absorb or release large amounts of heat without significant changes in temperature. Understanding this concept is vital when predicting how heat will transfer between substances.

Heat Transfer Equation

The heat transfer equation is a mathematical representation of how thermal energy is exchanged between substances. It is given by the formula \( q = m \cdot c \cdot \Delta T \), where \( q \) is the heat gained or lost, \( m \) is the mass of the substance, \( c \) is the specific heat capacity, and \( \Delta T \) is the change in temperature.

When a hot object is placed in contact with a cooler one, heat will transfer from the hotter to the cooler object until they reach thermal equilibrium - that is, they'll eventually settle at the same temperature. By using the heat transfer equation, we can calculate how much heat is exchanged during this process. This equation is a powerful tool in not only theoretical physics but also in practical applications like engineering, environmental science, and everyday cooking.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In terms of thermal equilibrium, it means that the total heat energy in an isolated system remains constant if no heat is lost to the surroundings.

In the thermal equilibrium calculations for our aluminum block and water scenario, we assume no heat is lost to the surrounding environment. This means that the heat lost by the aluminum block will be equal to the heat gained by the water until they reach the same temperature. By setting the two heat transfer equations equal to one another, we honor the conservation of energy and allow us to solve for the final shared temperature. This fundamental principle guides everything from the smallest atoms to the largest cosmic systems, making it a cornerstone of understanding not only physics but all of natural science.