Question

A \(2.0 \cdot 10^{2}\) g piece of copper at a temperature of \(450 \mathrm{~K}\) and a \(1.0 \cdot 10^{2} \mathrm{~g}\) piece of aluminum at a temperature of \(2.0 \cdot 10^{2} \mathrm{~K}\) are dropped into an insulated bucket containing \(5.0 \cdot 10^{2} \mathrm{~g}\) of water at \(280 \mathrm{~K}\). What is the equilibrium temperature of the mixture?

Short answer

Answer: The equilibrium temperature of the mixture is approximately 271.3 K.

Step-by-step solution

Remember the formula for heat exchange and conservation of energy.

The heat lost by a substance equals the heat gained by another substance in an isolated system (no heat lost to the surroundings). Mathematically, this can be expressed as: $$Q_{lost} = Q_{gained}$$ or $$mc \Delta T = mc \Delta T$$ where, m: mass of the substance c: specific heat capacity of the substance ∆T: change in temperature of the substance

Determine specific heat capacities for each substance.

We will need specific heat capacities for copper (c_c), aluminum (c_a), and water (c_w) to solve for the equilibrium temperature. These values are: c_c = 386 J/kg·K (for copper) c_a = 897 J/kg·K (for aluminum) c_w = 4186 J/kg·K (for water)

Convert the masses from grams to kilograms.

Converting mass from grams to kilograms (m_c for copper, m_a for aluminum, and m_w for water) makes it easier to calculate heat exchanges. m_c = 2.0 * 10^2 g = 0.2 kg m_a = 1.0 * 10^2 g = 0.1 kg m_w = 5.0 * 10^2 g = 0.5 kg

Set up the heat exchange equation.

Following conservation of energy, we can set up the following equation: $$m_c c_c (T_{final} - T_{c}) + m_a c_a (T_{final} - T_{a}) = m_w c_w (T_{w} - T_{final})$$ where, T_c = initial temperature of copper T_a = initial temperature of aluminum T_w = initial temperature of water T_final = equilibrium temperature

Substitute values and solve for the final temperature.

The given initial temperatures for copper, aluminum, and water are: T_c = 450 K T_a = 2.0 * 10^2 K = 200 K T_w = 280 K Substituting these values into the heat exchange equation we derived earlier: $$0.2 \cdot 386 (T_{final} - 450) + 0.1 \cdot 897 (T_{final} - 200) = 0.5 \cdot 4186 (280 - T_{final})$$ Solve this equation to find the equilibrium temperature (T_final): $$-77.2 T_{final} + 34740 + 89.7 T_{final} - 17940 = -2093 T_{final} + 585620$$ Combine the like terms and simplify the equation: $$2005.5 T_{final} = 544420$$ Finally, divide by 2005.5 to find the equilibrium temperature: $$T_{final} = \frac{544420}{2005.5} \approx 271.3 K$$ The equilibrium temperature of the mixture is approximately 271.3 K.

Key concepts

Specific Heat Capacity

Understanding specific heat capacity is essential in heat transfer calculations. Specific heat capacity (\(c\)) is the amount of heat required to raise the temperature of 1 kilogram of a substance by 1 degree Kelvin (K). Different substances have different specific heat capacities, meaning they require different amounts of heat to change their temperatures, even if they are of the same mass.

• Copper, for instance, has a specific heat capacity of 386 J/(kg·K),
• while Aluminum's is 897 J/(kg·K),
• and water's is particularly high at 4186 J/(kg·K).

These differences are crucial when considering how substances respond to the addition or loss of heat. A material with a high specific heat capacity, like water, will experience a slower temperature change upon gaining or losing heat compared to a material with a lower specific heat capacity.In our problem, the specific heat capacity values were used to understand how much heat each substance would gain or lose to reach thermal equilibrium.

Conservation of Energy

The principle of conservation of energy is a fundamental concept in physics, especially when dealing with heat transfer. This principle states that energy cannot be created or destroyed in an isolated system – it can only change forms.In practical terms, it means that any heat lost by a higher temperature object must be equal to the heat gained by a lower temperature object within an isolated system. This relationship is expressed mathematically through the equation \(Q_{lost} = Q_{gained}\)When dealing with calorimetry – the study of heat transfer – this principle helps us set up equations to solve for unknown variables, like the final equilibrium temperature of a mixture. By knowing that the heat lost by copper and aluminum equals the heat gained by water, we maintain the balance dictated by the conservation of energy.Thus, by ensuring the sum of energy exchanges equals zero, we can solve for the desired temperature.

Thermal Equilibrium

Thermal equilibrium occurs when two or more substances reach the same temperature and stop exchanging heat. This is an essential goal in many heat transfer calculations because it indicates that energy has been balanced among interacting bodies. In our example, thermal equilibrium is reached when the copper, aluminum, and water have all arrived at a common temperature, eliminating any temperature disparities among them.

• This state implies no further heat transfer within the system.
• It also confirms the application of the conservation of energy principle to achieve this balance.

Calculating the equilibrium temperature considers all factors including the masses, initial temperatures, and specific heat capacities, allowing for precise understanding of how the temperature will change until all materials reach the same thermal state.

Temperature Change Calculations

Calculating temperature changes involves understanding how heat transfer affects a substance's temperature. This process requires being familiar with the main formula:\( Q = mc\Delta T \)This equation expresses the relationship where

• \(Q\) is the heat absorbed or released,
• \(m\) is mass,
• \(c\) is specific heat capacity,
• and \(\Delta T\) is the change in temperature.

For our problem, the calculations become more complex because several substances with different properties are interacting.The initial temperatures of the copper (\(450\, K\)), aluminum (\(200\, K\)), and water (\(280\, K\)) define the starting points for each material. The equilibrium temperature (\(271.3\, K\)) was calculated by ensuring the net heat transfer within the system equals zero, in alignment with the formula for heat exchange. This involved calculating how much each substance's temperature would change before they all reach the same temperature.