Entropy In an experiment, 200 g of aluminum (with a specific...

Entropy In an experiment, 200 g of aluminum (with a specific heat of 900 $\mathrm{J} / \mathrm{kg} \cdot \mathrm{K} )$ at $100^{\circ} \mathrm{C}$ is mixed with 50.0 $\mathrm{g}$ of water at $20.0^{\circ} \mathrm{C},$ with the mixture thermally isolated. (a) What is the equilibrium temperature? What are the entropy changes of (b) the aluminum, (c) the water, and (d) the aluminum- water system?

Entropy
In an experiment, 200 g of aluminum (with a specific heat of 900 $\mathrm{J} / \mathrm{kg} \cdot \mathrm{K} )$ at $100^{\circ} \mathrm{C}$ is mixed with 50.0 $\mathrm{g}$ of water at $20.0^{\circ} \mathrm{C},$ with the mixture thermally isolated. (a) What is the equilibrium temperature? What are the entropy changes of (b) the aluminum, (c) the water, and (d) the aluminum- water system?

Key Concepts

Specific Heat Capacity

The specific heat capacity is a property that indicates how much heat energy is required to change a substance's temperature by a certain amount. In thermal calculations, it connects the amount of heat exchanged with the corresponding temperature change. Different substances have different specific heat capacities, which affects how much they heat up or cool down when energy is added or removed.

Entropy Increase in Irreversible Processes

In an isolated system undergoing an irreversible process, such as the spontaneous mixing of substances with different temperatures, the total entropy of the system increases. While energy is conserved in the system, the increase in entropy quantifies the irreversibility of the process, reflecting the natural tendency of systems to move toward states of greater disorder.

Entropy Change Calculation

Entropy is a measure of the disorder or randomness in a system, and its change during a process is given by integrating the reversible heat transfer over temperature, typically expressed as ?S = ?(?Q_rev/T). When a substance with constant specific heat capacity changes temperature, this integration simplifies to ?S = m·c·ln(T_f/T_i), where m is the mass, c is the specific heat capacity, T_i is the initial temperature, and T_f is the final temperature.

Isolated Systems and Energy Conservation

In a thermally isolated system, there is no heat exchange with the surroundings, meaning that the total energy of the system remains constant. When different substances with different initial temperatures are brought into contact, the heat lost by the warmer substance is exactly equal to the heat gained by the cooler substance. This principle is used to determine the final equilibrium temperature of the system by setting the sum of all energy transfers to zero.

Thermal Equilibrium

Thermal equilibrium is reached when all parts of a system have the same temperature, resulting in no net heat flow between them. In the context of mixing substances, the final equilibrium temperature is the temperature at which the opposing heat exchanges between the components have balanced out. This state is determined using the concept of energy conservation within an isolated system.

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