Question

Consider two bodies of identical mass \(m\) and specific heat \(c\) used as thermal reservoirs (source and sink) for a heat engine. The first body is initially at an absolute temperature \(T_{1}\) while the second one is at a lower absolute temperature \(T_{2}\). Heat is transferred from the first body to the heat engine, which rejects the waste heat to the second body. The process continues until the final temperatures of the two bodies \(T_{f}\) become equal. Show that \(T_{f}=\sqrt{T_{1} T_{2}}\) when the heat engine produces the maximum possible work.

Short answer

Answer: The final temperature when the heat engine produces the maximum possible work is T_f = √(T1*T2).

Step-by-step solution

Expressing Conservation of Energy

Since energy is conserved, the sum of the heat absorbed by the first body and the heat absorbed by the second body must equal the work done by the heat engine. Mathematically, we can write this as: \[Q_1 + Q_2 = W\] However, we know that the heat absorbed by the first body \(Q_1\) must be equal to the heat absorbed by the second body \(Q_2\) and work done by the heat engine \(W\). So, we can write: \[Q_1 = W - Q_2\]

Representing Heat Absorbed in Terms of Temperatures

Let \(\Delta T_1 = T_1 - T_f\) be the temperature difference of the first body, and \(\Delta T_2 = T_f - T_2\) be the temperature difference of the second body. The heat absorbed by the first body is given by \(Q_1 = mc \Delta T_{1}\), and the heat absorbed by the second body is given by \(Q_2 = mc \Delta T_{2}\). Substituting these values in the previous equation, we obtain: \[mc(\Delta T_1) = W - mc(\Delta T_2)\]

Expressing The Efficiency of The Heat Engine

The efficiency \(\eta\) of a heat engine is given by the ratio of the work done \(W\) to the heat absorbed from the first body, which is \(Q_1\). Thus, \[\eta = \frac{W}{Q_1}\] Considering that the heat engine produces the maximum possible work, its efficiency will be given by the Carnot efficiency, which is: \[\eta_C = 1 - \frac{T_2}{T_1}\] To maximize the work done, we equate \(\eta_C\) to \(\eta\). Thus: \[\eta_C = \frac{W}{mc(\Delta T_1)}\]

Solving for the Final Temperature \(T_f\)

We substitute the value of \(W\) from Step 2 into the equation from Step 3 and solve for \(T_f\). \[1 - \frac{T_2}{T_1} = \frac{mc(\Delta T_1) - mc(\Delta T_2)}{mc(\Delta T_1)}\] Simplifying and isolating \(\Delta T_1\) and \(\Delta T_2\), we get: \[\Delta T_1 - \Delta T_2 = T_1 - T_2\] Substituting the expressions for \(\Delta T_1\) and \(\Delta T_2\) in terms of \(T_f\), we have: \[(T_1 - T_f) - (T_f - T_2) = T_1 - T_2\] Solving for \(T_f\), we get: \[T_f = \sqrt{T_1 T_2}\] Thus, the final temperature \(T_f\) when the heat engine produces the maximum possible work is \(T_f = \sqrt{T_1 T_2}\).

Key concepts

Heat Engine

A heat engine is a system that converts heat or thermal energy to mechanical work. It operates by taking in heat from a high-temperature source, converting some of this energy into work, and then expelling the remaining energy to a low-temperature sink.

For students understanding the heat engine concept, it's essential to visualize it as a process with three main parts: a heat source, a working substance (like a gas inside a piston), and a heat sink. During the engine's cycle, the working substance expands when heated, doing work on the surroundings, and contracts when cooled without transferring the energy back to the heat source.

The performance of a heat engine is crucially determined by the temperatures of the source and the sink, represented by absolute temperatures which use units like kelvin (K). These temperatures are not arbitrary; they depict the thermodynamic limits of the engine's performance. The greater the difference in temperature between the source and the sink, the higher the potential efficiency of the engine.

Conservation of Energy

The principle of the conservation of energy states that energy cannot be created or destroyed, but it can change from one form to another. Within the context of a heat engine, this principle translates to the idea that the total energy input (the heat absorbed from the high-temperature source) will be equal to the combined energy of the output (the work done by the engine) and the wasted energy (the heat dumped into the low-temperature sink).

This conservation principle underpins the mathematical relationships presented in the heat engine problem. When solving for the final temperature \(T_f\) where the two bodies reach equilibrium, we account for this by equating the heat lost by the hot reservoir to the heat gained by the cold reservoir plus the work extracted by the engine. Failing to consider energy conservation would result in an incorrect understanding of the system's behavior. Therefore, it's vital to recognize energy conservation as a fundamental concept that assists in deriving the relationship between the temperatures and work in heat engine processes.

Specific Heat

Specific heat, denoted as \(c\) in physical equations, is the amount of heat per unit mass required to raise the temperature of a substance by one degree Celsius (or one kelvin). It measures a material's ability to store thermal energy.

In the context of our heat engine problem, specific heat tells us how much energy needs to be transferred to or from our thermal reservoirs (the two bodies) to change their temperatures. When they reach equilibrium, we've used the concept of specific heat to relate the amount of heat exchanged, \(Q\), with the mass of the bodies and their change in temperature (\(\Delta T\)).

For students grappling with the role of specific heat in heat transfer, it's helpful to think of it as a 'thermal capacitor'. Just as a capacitor in electronics stores electrical energy, specific heat describes how a body stores thermal energy. Different substances will have different abilities to store and transfer heat due to their varying specific heats, directly impacting the calculations for the final equilibrium temperature in our example.

Absolute Temperature

Absolute temperature is the temperature of an object as measured on a scale where zero is taken as absolute zero. Absolute zero is theoretically the lowest temperature possible, where a system's particles have minimal thermal energy and cannot be cooled further. On the Kelvin scale, which is commonly used for these measurements, water freezes at about 273 K and boils at about 373 K.

In thermodynamics, absolute temperature is vital because it provides a standardized reference for temperature measurements, ensuring that equations and comparisons of thermal energy are consistent. Absolute temperature is used in the Carnot efficiency formula, which reflects the maximum possible efficiency of a heat engine with a given high and low-temperature reservoir. Through the relationship \(T_f = \sqrt{T_1 T_2}\), the exercise demonstrates how the final temperature of two bodies when the maximum work is done by a heat engine is the geometric mean of their initial absolute temperatures.

This use of absolute temperature avoids the complexities that arise when using Celsius or Fahrenheit scales, as such temperature scales can have negative values which do not correctly portray the thermodynamic properties of a system.