(III) A general theorem states that the amount of energy...

(III) A general theorem states that the amount of energy that becomes unavailable to do useful work in any process is equal to $T_{\mathrm{L}} \Delta S,$ where $T_{\mathrm{L}}$ is the lowest temperature available and $\Delta S$ is the total change in entropy during the process. Show that this is valid in the specific cases of $(a)$ a falling rock that comes to rest when it hits the ground; $(b)$ the free adiabatic expansion of an ideal gas; and $(c)$ the conduction of heat, $Q$ from a high-temperature $\left(T_{\mathrm{H}}\right)$ reservoir to a low-temperature $\left(T_{\mathrm{L}}\right)$ reservoir. [Hint: In part $(c)$ compare to a Carnot engine.]

(III) A general theorem states that the amount of energy that
becomes unavailable to do useful work in any process is equal
to $T_{\mathrm{L}} \Delta S,$ where $T_{\mathrm{L}}$ is the lowest temperature available and
$\Delta S$ is the total change in entropy during the process. Show
that this is valid in the specific cases of $(a)$ a falling rock that
comes to rest when it hits the ground; $(b)$ the free adiabatic
expansion of an ideal gas; and $(c)$ the conduction of heat, $Q$
from a high-temperature $\left(T_{\mathrm{H}}\right)$ reservoir to a low-temperature
$\left(T_{\mathrm{L}}\right)$ reservoir. [Hint: In part $(c)$ compare to a Carnot engine.]

Key Concepts

Entropy

Entropy is a fundamental thermodynamic quantity that measures the degree of disorder or randomness in a system. In the context of energy conversion, an increase in entropy implies that a portion of energy has become unavailable for doing useful work because it has been irreversibly dispersed. This concept is central to understanding why real processes cannot convert all available energy into work.

Exergy (Availability)

Exergy, or availability, refers to the maximum work that can be obtained from a system as it comes into equilibrium with its environment. It represents the quality or usefulness of energy and emphasizes that not all energy is convertible to work because of inherent system limitations imposed by the second law of thermodynamics.

Irreversible Processes

Irreversible processes are those in which dissipation occurs due to factors like friction, uncontrolled expansion, or heat conduction. These processes generate entropy, causing some energy to become unavailable for work. Recognizing the irreversible nature of real-world processes is key to applying the theorem that relates unavailable energy to the product of the lowest temperature and the total change in entropy.

Carnot Engine Efficiency

The Carnot engine establishes the upper limit on the efficiency of converting heat into work when operating between two temperature reservoirs. It serves as an ideal benchmark, illustrating that the loss of work potential is directly linked to the entropy change and the temperatures involved. This concept underpins the analysis of energy degradation in processes where heat flows from a higher to a lower temperature source.

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