Question

What can be deduced from energy and entropy balances about a system undergoing a thermodynamic cycle while receiving energy by heat transfer at temperature \(T_{\mathrm{C}}\) and discharging energy by heat transfer at a higher temperature \(T_{\mathrm{H}}\), if these are the only energy transfers the system experiences?

Short answer

From energy and entropy balances, for a system discharging heat at \(T_{\mathrm{H}}\) and receiving heat at \(T_{\mathrm{C}}\), the net work done is: \[ W = Q_{\mathrm{H}} \left( \frac{T_{\mathrm{C}} - T_{\mathrm{H}}}{T_{\mathrm{H}}} \right) \].

Step-by-step solution

Identify the Heat Transfer Temperatures

Note that the system receives energy by heat transfer at temperature \(T_{\mathrm{C}}\) and discharges energy by heat transfer at a higher temperature \(T_{\mathrm{H}}\).

Apply the First Law of Thermodynamics

By the First Law of Thermodynamics (conservation of energy), the net work done by the system over one complete cycle is the difference between the heat received and the heat discharged. Let the heat received be \(Q_{\mathrm{C}}\) and the heat discharged be \(Q_{\mathrm{H}}\). The work done, \(W\), is given by: \[ W = Q_{\mathrm{C}} - Q_{\mathrm{H}} \]

Consider the Second Law of Thermodynamics

Using the Second Law of Thermodynamics, the net entropy change of the system over one complete cycle is zero since entropy is a state function. For the heat received and discharged, the entropy change can be represented as: \[ \Delta S = \frac{Q_{\mathrm{C}}}{T_{\mathrm{C}}} - \frac{Q_{\mathrm{H}}}{T_{\mathrm{H}}} \]Since the system undergoes a cyclic process, \( \Delta S = 0 \). Thus: \[ \frac{Q_{\mathrm{C}}}{T_{\mathrm{C}}} = \frac{Q_{\mathrm{H}}}{T_{\mathrm{H}}} \]

Solve for the Relationship Between Heats

From the entropy balance, solve for the relationship between the heat transfers: \[ Q_{\mathrm{C}} = Q_{\mathrm{H}} \frac{T_{\mathrm{C}}}{T_{\mathrm{H}}} \]This shows that the heat received by the system is scaled by the ratio of the temperatures.

Combine Results from Both Laws

Using the work done and the relationship between heat transfers, \[ W = Q_{\mathrm{C}} - Q_{\mathrm{H}} \]Substituting the value of \(Q_{\mathrm{C}}\) from the above entropy balance equation: \[ W = Q_{\mathrm{H}} \left( \frac{T_{\mathrm{C}}}{T_{\mathrm{H}}} \right) - Q_{\mathrm{H}} = Q_{\mathrm{H}} \left( \frac{T_{\mathrm{C}} - T_{\mathrm{H}}}{T_{\mathrm{H}}} \right) \]This relationship indicates the maximum work obtainable from the system.

Key concepts

First Law of Thermodynamics

The First Law of Thermodynamics is often referred to as the law of energy conservation. It's one of the most fundamental principles of physics. It states that energy cannot be created or destroyed, only transformed from one form to another. In the context of our thermodynamic cycle, this law implies that the net work done by the system is the difference between the energy (in the form of heat) that the system receives and the energy that it discharges. Let's break this down. If we denote the heat received by the system as \(Q_C\) and the heat discharged as \(Q_H\), then the work done \(W\) by the system over one complete cycle can be expressed as:
\[ W = Q_C - Q_H \] This equation tells us that the work output is fundamentally tied to the difference between heat input and heat output. No matter what processes the system undergoes, energy conservation must always hold true.

Second Law of Thermodynamics

The Second Law of Thermodynamics introduces the concept of entropy, a measure of the system's energy dispersal or disorder. Entropy always increases or remains stable for processes in an isolated system. In a thermodynamic cycle, the system reaches a state where the net entropy change over one complete cycle is zero. This is because entropy is a state function - it depends only on the state of the system, not on how it got there.
For our system, where heat is received at temperature \(T_C\) and discharged at a higher temperature \(T_H\), the entropy change can be shown as: \[ \frac{Q_C}{T_C} - \frac{Q_H}{T_H} \] Since the system undergoes a complete cycle, the net entropy change \(\Delta S = 0\). This gives us:
\[ \frac{Q_C}{T_C} = \frac{Q_H}{T_H} \] This relation is crucial because it ties together the amounts of heat transfer with the temperatures at which these transfers occur.

Entropy Balance

Entropy balance is a valuable tool for analyzing thermodynamic processes. As mentioned earlier, for a complete cycle in a thermodynamic system, the net entropy change must be zero. This is practically useful because it sets a relationship between the heat exchanges and the temperatures involved. Let's further explore this:

• From the Second Law, we already have \[ \frac{Q_C}{T_C} = \frac{Q_H}{T_H} \]
• Simplifying this, we can rearrange to find \[ Q_C = Q_H \frac{T_C}{T_H} \]

This implies that the heat received by the system (\(Q_C\)) is directly proportional to the temperature ratio \(T_C/T_H\). We can use this relationship in conjunction with the First Law to find the work output. Substituting \(Q_C\) in our work equation:

• \[ W = Q_C - Q_H \]
• \[ W = Q_H \frac{T_C}{T_H} - Q_H \]
• \[ W = Q_H \frac{T_C - T_H}{T_H} \]

This result shows the maximum work obtainable from the system, tying together all the principles of energy and entropy balance.