Problem 51 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

A reversible heat engine absorbs \(40 \mathrm{~kJ}\) of heat at \(500 \mathrm{~K}\) and performs \(10 \mathrm{~kJ}\) of work rejecting the remaining amount to the sink at \(300 \mathrm{~K}\). The entropy change for the universe is (a) \(-80 \mathrm{~J} / \mathrm{K}\) (b) \(100 \mathrm{~J} / \mathrm{K}\) (c) \(20 \mathrm{~J} / \mathrm{K}\) (d) \(180 \mathrm{~J} / \mathrm{K}\)

Step-by-Step Solution

Answer

The entropy change for the universe is (d) 180 J/K.

Step 1: Calculate the amount of heat rejected

To find the amount of heat rejected, use the first law of thermodynamics which states that the net heat absorbed by the system (Q) is equal to the work done by the system (W) plus the heat rejected to the sink (Q_sink). This is represented as \(Q = W + Q_{sink}\). Rearrange the equation to solve for \(Q_{sink}\), which gives \(Q_{sink} = Q - W\). Plug in the values to get \(Q_{sink} = 40 - 10\) kJ.

Step 2: Calculate the entropy change of the source

The entropy change of the source (\(\Delta S_{source}\)) when heat is absorbed reversibly can be calculated using the equation \(\Delta S_{source} = \frac{Q}{T_{source}}\), where \(Q\) is the heat absorbed and \(T_{source}\) is the source temperature. Therefore, \(\Delta S_{source} = \frac{40}{500}\) kJ/K = \(0.08\) kJ/K or \(80\) J/K.

Step 3: Calculate the entropy change of the sink

Similarly, the entropy change of the sink (\(\Delta S_{sink}\)) when heat is rejected reversibly is given by \(\Delta S_{sink} = \frac{Q_{sink}}{T_{sink}}\), where \(Q_{sink}\) is the heat rejected to the sink and \(T_{sink}\) is the sink temperature. First, calculate \(Q_{sink}\) from Step 1 (which is \(30\) kJ), then use \(\Delta S_{sink} = \frac{30}{300}\) kJ/K = \(0.1\) kJ/K or \(100\) J/K.

Step 4: Calculate the total entropy change

The total entropy change for the universe (\(\Delta S_{universe}\)) is the sum of the entropy changes of the source and the sink. Therefore, \(\Delta S_{universe} = \Delta S_{source} + \Delta S_{sink}\). Using the values from Steps 2 and 3, \(\Delta S_{universe} = 80 \, J/K + 100 \, J/K = 180 \, J/K\).

Key Concepts

Understanding the First Law of Thermodynamics

The first law of thermodynamics is a fundamental concept in physics that applies to a wide variety of scenarios, including the operation of heat engines. This law, also known as the principle of energy conservation, states that energy cannot be created or destroyed, only transformed from one form to another. In the context of a heat engine, the first law can be expressed as:

equation: \(Q = W + Q_{\text{sink}}\)

In this equation, \(Q\) represents the heat absorbed by the engine, \(W\) is the work done by the engine, and \(Q_{\text{sink}}\) is the heat rejected to the sink (a cooler reservoir). The equation asserts that the net heat absorbed (\(Q\)) is the sum of the useful work output (\(W\)) and the waste heat (\(Q_{\text{sink}}\)).

To cement your understanding, it's helpful to visualize the heat engine as a system where energy enters in the form of heat, part of which is converted into work — the purpose of the engine — and the rest is discarded as less useful heat energy to the sink. The law emphasizes conservation in any energy exchange or transformation, laying the foundation for further analysis of thermodynamic processes.

Entropy Change Calculation in a Heat Engine

In thermodynamics, entropy is a measure of the disorder or randomness of a system. Calculating entropy change plays a crucial role in understanding how systems evolve over time, especially in processes involving heat transfer. The second law of thermodynamics introduces entropy as a quality that increases in a closed system as it moves towards equilibrium. The equation for calculating entropy change (\(equation: \Delta S\)) when heat is transferred is:

equation: \(\Delta S = \frac{Q}{T}\)

where \(Q\) is the heat transfer and \(T\) is the absolute temperature at which the transfer occurs. For a reversible process, we consider the entropy change at the source (where heat is absorbed) and the sink (where heat is rejected), and calculate each separately. The sign of the entropy change will depend on whether the system gains or loses heat. It’s worth noting that in any real process the total entropy of the universe increases, which is a reflection of the second law of thermodynamics.

To apply these concepts, we look at the heat engine example given to us. The entropy change for the source (\(\Delta S_{\text{source}}\)) is the heat added divided by the source's temperature. For the sink (\(\Delta S_{\text{sink}}\)), it's the heat rejected divided by the sink’s temperature. By adding these two values, considering their proper signs, we get the total change in entropy for the process.

Reversible Processes in Thermodynamics

A reversible process in thermodynamics is an idealized concept where a system changes states in such a manner that the system and environment can both return to their original conditions without any net changes. These processes are characterized by equilibrium at every stage and involve no dissipation of energy as waste heat or friction.

Real processes always contain some irreversibilities, such as friction, turbulence, and uncontrolled heat transfer making them irreversible. However, reversible processes are a valuable theoretical tool for analyzing the maximum efficiency that can be achieved in any given process.

in a reversible heat engine like the one in the exercise, entropy changes in the system during the processes of heat absorption and rejection can be exactly balanced out because the sum of the entropy changes in the system and the surroundings remains constant. This balance is why we can use the earlier entropy change calculations to assess the state of the system and its surroundings as a whole. The concept of reversibility is closely tied to the idea of maximum efficiency in thermodynamic cycles, such as the Carnot cycle, where the efficiency is determined by the temperature difference between the hot source and cold sink.