Question

A kilogram of aluminum metal and a kilogram of water are each warmed to \(75^{\circ} \mathrm{C}\) and placed in two identical insulated containers. One hour later, the two containers are opened, and the temperature of each substance is measured. The aluminum has cooled to \(35^{\circ} \mathrm{C},\) while the water has cooled only to \(66{ }^{\circ} \mathrm{C}\). Explain this difference.

Short answer

The difference in temperature after cooling is due to water's higher specific heat capacity compared to aluminum, meaning water retains its heat longer and cools down more slowly.

Step-by-step solution

Understand the Concept of Specific Heat Capacity

The difference in temperatures after one hour can be explained by the concept of specific heat capacity, which is the amount of heat required to raise the temperature of one kilogram of a substance by one degree Celsius. Water has a higher specific heat capacity than aluminum, meaning it requires more heat to change its temperature.

Heat Loss Comparison

Given that aluminum has a specific heat capacity of approximately 0.900 J/g°C and water's is approximately 4.184 J/g°C, and knowing that they were both at the same initial temperature, it's clear that aluminum would cool down faster since it requires less heat to be lost to decrease its temperature compared to water.

Analyze Heat Transfer

As the substances cool in the insulated containers, the heat is transferred to the surrounding environment. Due to the lower specific heat capacity of aluminum, it loses its heat more rapidly than water. This differential cooling rate results in the observed temperature difference after one hour.

Key concepts

Heat Transfer

The principle of heat transfer is pivotal to understanding why materials cool down or warm up at different rates. It refers to the movement of heat energy from a warmer object to a cooler one. This process continues until thermal equilibrium is reached, meaning that both objects are at the same temperature.

In our exercise, both aluminum and water begin at the same initial temperature and are placed in identical insulated containers to minimize external influences. Over time, heat energy is lost to the surroundings, cooling the substances down. Since the insulation is the same, the key difference in the cooling rates must arise from the substances themselves. This difference is due to variations in thermal properties, such as specific heat capacity, which significantly affect heat transfer rates.

Temperature Change

Fundamentally, temperature change in a substance occurs when it gains or loses heat energy. The magnitude of this change depends on various factors, including the amount of heat added or removed and the substance's specific heat capacity.

Throughout the exercise, we witness that aluminum and water, although starting at the same temperature, do not display the same final temperature after losing heat. Aluminum experiences a greater temperature change, falling from 75 to 35 degrees Celsius, while water's temperature only drops from 75 to 66 degrees Celsius. This disparity arises from each material's ability to resist temperature change when heat is transferred, which leads us back to their specific heat capacities.

Thermal Properties of Materials

The thermal properties of materials, specifically specific heat capacity, dictate how they will respond to the addition or removal of heat energy. Specific heat capacity is the quantity of heat required to change the temperature of a unit mass of a substance by one degree Celsius (or Kelvin).

As we delve into this exercise, the crux of the matter lies in the comparison between the specific heat capacities of aluminum and water. With aluminum's specific heat capacity at approximately 0.900 J/g°C and water's at a much higher 4.184 J/g°C, it becomes evident why their temperatures vary after cooling. Water, requiring more heat to alter its temperature, retains its warmth longer than aluminum. This property not only shapes the outcome of our exercise but also influences practical applications in everyday life and various industries.