Question

A completely reversible heat engine operates with a source at \(1500 \mathrm{R}\) and a sink at \(500 \mathrm{R}\). If the entropy of the sink increases by \(10 \mathrm{Btu} / \mathrm{R}\), how much will the entropy of the source decrease? How much heat, in \(\mathrm{Btu}\), is transferred from the source?

Short answer

Answer: The decrease in entropy of the source is 10 Btu/R, and the amount of heat transferred from the source is 15000 Btu.

Step-by-step solution

Determine the total entropy change of the universe

Since this is a completely reversible heat engine, the total entropy change of the universe (system plus surroundings) would be equal to zero. Therefore, the decrease in entropy of the source must be equal to the increase in entropy of the sink.

Calculate the decrease in entropy of the source

The increase in entropy of the sink is given as 10 Btu/R. Since the total entropy change should be zero, the decrease in entropy of the source will also be 10 Btu/R.

Calculate the heat transferred from the source

To determine the heat transferred from the source, we will use the equation for entropy change and relate it to the heat transferred. The equation for entropy change is: $$\Delta S = \frac{Q}{T}$$ Where \(\Delta S\) is the entropy change, \(Q\) is the heat transferred, and \(T\) is the absolute temperature. Since we know the decrease in entropy of the source (\(\Delta S = -10 \mathrm{Btu} / \mathrm{R}\)) and the source temperature (\(T= 1500 \mathrm{R}\)), we can solve for the heat transferred (\(Q\)) as follows: $$Q = \Delta S \times T$$ Substituting the known values, we get: $$Q = -10 \mathrm{Btu} / \mathrm{R} \times 1500 \mathrm{R}$$

Calculate the heat transferred from the source

Multiplying the values, we obtain the heat transferred from the source: $$Q = -15000 \mathrm{Btu}$$ The negative sign indicates that heat is being transferred from the source to the engine. So, the decrease in entropy of the source is 10 Btu/R, and the amount of heat transferred from the source is 15000 Btu.

Key concepts

Entropy Change

Understanding entropy change is fundamental in thermodynamics, particularly when dealing with heat engines. At its core, entropy is a measure of a system's disorder or randomness. An increase in entropy reflects a system's tendency to move towards greater disorder, while a decrease indicates a more ordered state.

In the context of a heat engine, like the reversible heat engine mentioned in the exercise, we examine the entropy change of both the heat source and the sink. A key principle here is that for a completely reversible process, the total entropy change of the universe remains constant — it neither increases nor decreases. This concept is crucial because it sets the framework for which the calculations in our exercise are based.

When the sink's entropy increases by a certain amount, say \(10 \mathrm{Btu} / \mathrm{R}\), the principle of conservation of entropy dictates that the source's entropy must decrease by the same amount. This principle underpins the first and second laws of thermodynamics and supports the premise that energy (and entropy) can neither be created nor destroyed; it can only be transferred or transform from one form to another.

Heat Transfer

The term heat transfer refers to the movement of thermal energy from one thing to another as a result of a temperature difference.

In thermodynamics, and especially in the study of heat engines, heat transfer is a pivotal concept. It is through the transfer of heat energy from the high-temperature source to the cooler sink that the engine operates. This concept is not only vital in understanding how engines function but also in calculating the efficiency and work output of such systems. The second step of our exercise, where we calculate heat transfer, utilizes the formula for entropy change \(\Delta S = \frac{Q}{T}\), which relates the heat transferred (\(Q\)) to entropy change (\(\Delta S\)) and temperature (\(T\)).

Here, the negative sign conventionally implies the direction of heat transfer — in this case, from the source to the environment. It's this precise understanding of heat transfer and its relationship to entropy change that enables engineers and scientists to design more efficient thermal systems and engines.

Thermodynamic Cycles

A thermodynamic cycle consists of a series of processes that a system undergoes, returning to its initial state by the end of the cycle. In the realm of heat engines, thermodynamic cycles are the backbone of operation.

Common cycles include the Carnot, Otto, and Rankine cycles, among others. Each cycle has a set of processes, including isothermal expansions and compressions, where temperature remains constant, and adiabatic processes, where no heat is transferred into or out of the system.

Reversible vs. Irreversible Cycles

One critical distinction to be made is between reversible and irreversible cycles. Our exercise specifically deals with a reversible cycle, which is idealized and assumes no energy loss due to friction, sound, or any form of dissipation. In the real world, all processes possess some degree of irreversibility which makes actual cycles less efficient than their reversible counterparts.

Studying thermodynamic cycles is crucial, as they are the fundamental principles upon which engines operate, allowing us to understand and optimize their performance. The exercise provided offers a glimpse into such cycles and the crucial role entropy and heat transfer play within them.