1 kg of water is warmed from 20^∘ C to 100^∘ C (a) by placing it in contact with a reservoir at

step 1 We'd like to calculate the entropy changes for each particular case. So, in this case for our wa

1 kg of water is warmed from 20^∘ C to 100^∘ C (a) by...

$1 \mathrm{~kg}$ of water is warmed from $20^{\circ} \mathrm{C}$ to $100^{\circ} \mathrm{C}$ (a) by placing it in contact with a reservoir at $100^{\circ} \mathrm{C}$, (b) by placing it first in contact with a reservoir at $50^{\circ} \mathrm{C}$ until it reaches that temperature, and then in contact with the reservoir at $100^{\circ} \mathrm{C}$, and (c) by operating a reversible heat engine between it and the reservoir at $100^{\circ} \mathrm{C}$. In each case, what are the entropy changes of (i) the water, (ii) the reservoirs, and (iii) the universe? (Assume the heat capacity of water is independent of temperature.)

Key Concepts

Specific Heat Capacity

Specific heat capacity is the amount of energy required to raise the unit mass of a substance by one degree in temperature. When considering processes like the heating of water, a constant specific heat capacity allows the entropy change of the system to be determined by integrating the heat capacity over the temperature change. This parameter provides a direct link between energy input and temperature change, which is fundamental in calculating both energy transfers and associated entropy changes.

Heat Reservoirs

A heat reservoir is an idealized body with a large thermal capacity that can supply or absorb finite amounts of heat without undergoing any significant change in temperature. In thermodynamic problems, heat reservoirs allow the simplification of analysis by treating the reservoir’s temperature as constant during heat exchange. This concept is crucial when calculating the entropy change of the surroundings in processes such as the direct heating of a system.

Reversible Heat Engine

A reversible heat engine is an idealized engine that operates between two heat reservoirs and completes a cycle in which the total entropy change of the universe is zero. It represents the maximum possible efficiency achievable between two temperatures. The concept is essential in thermodynamics because it sets the benchmark for the performance of real engines and illustrates the practical limits imposed by the second law.

Entropy

Entropy is a state function that quantifies the degree of disorder or randomness in a system. It changes when a system absorbs or releases heat in a process. The entropy change for a system undergoing a reversible process can be calculated with the formula ?S = ?(dQ/T), where dQ is the differential heat transfer and T is the absolute temperature. This concept is fundamental in understanding how energy disperses and is used to determine the feasibility and directionality of thermodynamic processes.

Reversible and Irreversible Processes

Reversible processes are idealized thermodynamic changes that occur infinitely slowly such that the system remains in equilibrium at all stages. They have the distinctive feature that the total entropy change of the universe is zero. In contrast, irreversible processes occur at a finite rate, leading to additional entropy production in the universe. The analysis of reversible versus irreversible processes is critical in thermodynamics as it relates to the most efficient transformations between states and the intrinsic limitations imposed by the second law.

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