(II) A 150- g insulated aluminum cup at 15^∘ C is filled with 215 g of water at 100^∘ C. Determ

step 1 to find the change in entropy. Again, we're going to use, here we're going to use the integral f

(II) A 150- g insulated aluminum cup at 15^∘ C is filled...

(II) A $150-\mathrm{~g}$ insulated aluminum cup at $15^{\circ} \mathrm{C}$ is filled with $215 \mathrm{~g}$ of water at $100^{\circ} \mathrm{C}$. Determine $(a)$ the final temperature of the mixture, and (b) the total change in entropy as a result of the mixing process (use $\left.\Delta S=\int d Q / T\right) .$

Key Concepts

Integration of dS = dQ/T

This concept involves integrating the small increments of heat transfer divided by the temperature over the process of heating or cooling. When calculating the entropy change for each substance at varying temperatures, the integration accounts for the continuous change in temperature rather than assuming a constant temperature throughout the process.

Entropy Change

Entropy is a measure of the disorder or randomness in a system, and its change is an important aspect of the second law of thermodynamics. The problem requires calculating the total change in entropy as a result of mixing, which involves evaluating the expression ?S = ?(dQ/T). It provides insights into the irreversibility and the natural tendency of the system towards increased disorder.

Thermal Equilibrium

Thermal equilibrium is reached when all components in a system attain the same temperature, resulting in no net heat flow between them. In this problem, finding the final equilibrium temperature involves setting the heat exchange between the cup and the water equal, using the conservation of energy principle.

Specific Heat Capacity

Specific heat capacity is the amount of heat required to change the temperature of a unit mass of a substance by one degree. It is crucial in calculating the heat gained or lost by both the water and the aluminum cup, using the formula q = m × c × ?T. This concept allows us to quantify the energy transfer between substances based on their masses and specific heat values.

Conservation of Energy

This concept dictates that in an isolated system, the total energy remains constant. In the context of the problem, this means that the heat lost by the hot water must equal the heat gained by the cold aluminum cup. The energy balance is the key to determining the final equilibrium temperature after the two substances are mixed.

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