Question

Suppose that the same amount of heat energy-say, \(260 \mathrm{~J}-\) is transferred by conduction from a heat reservoir at a temperature of \(400 \mathrm{~K}\) to another reservoir, the temperature of which is \((a) 100 \mathrm{~K}\), (b) \(200 \mathrm{~K},(c) 300 \mathrm{~K}\), and \((d) 360 \mathrm{~K}\). Calculate the changes in entropy and discuss the trend.

Short answer

The change in entropy when transferring 260J of heat from a reservoir at 400 K to another reservoir decreases as the temperature of the colder reservoir increases. It's 2.6 J/K for 100K, 1.3 J/K for 200K, 0.87 J/K for 300K, and 0.72 J/K for 360K. This shows an inverse relationship between temperature and entropy change.

Step-by-step solution

Calculate initial entropy change

In the first scenario, we need to calculate the change in entropy when the heat is transferred from a reservoir at 400 K to another reservoir at 100 K. We use the formula \(\Delta S = Q/T\), where Q=260 J and T=100 K. Therefore, \(\Delta S_1 = 260/100 = 2.6 J/K\).

Calculate second entropy change

In the second scenario, the heat is transferred to a reservoir at 200 K. By using the same formula, with Q=260 J and T=200 K, we calculate \(\Delta S_2 = 260/200 = 1.3 J/K\).

Calculate third entropy change

For the third scenario, the colder reservoir is at 300 K. Inserting Q=260 J and T=300 K into the formula, we get \(\Delta S_3 = 260/300 = 0.87 J/K\).

Calculate final entropy change

In the final scenario, the colder reservoir is at 360 K. Therefore, we calculate as \(\Delta S_4 = 260/360 = 0.72 J/K\).

Observing the trend

If we observe the entropy changes \(\Delta S_1, \Delta S_2, \Delta S_3\), and \(\Delta S_4\), we see that the entropy change decreases as the temperature of the cold reservoir increases.

Key concepts

Heat Transfer

Heat transfer is a fundamental concept in thermodynamics and involves the transfer of heat energy from one object or system to another. This typically occurs through three main mechanisms: conduction, convection, and radiation. In our original exercise, we are dealing with conduction, which is the process where heat is directly transmitted through a substance when there is a difference in temperature between adjoining regions, without movement of the material.
This form of heat transfer occurs on the molecular level, where faster-moving molecules in a hot area transfer their kinetic energy to slower-moving molecules in a cooler area.

• In conduction, the rate of heat transfer depends on the material's thermal conductivity, the surface area through which heat passes, and the temperature difference between the hot and cold regions.
• Materials like metals are excellent conductors of heat, while materials like wood and plastic are poor conductors.

Understanding heat transfer is essential for calculating entropy changes in thermodynamic processes, as shown in the exercise when heat energy is transferred between reservoirs with different temperatures.

Second Law of Thermodynamics

The Second Law of Thermodynamics is one of the central principles in the study of thermodynamics. It essentially states that the total entropy of an isolated system can never decrease over time. Entropy, often thought of as disorder or randomness, tends to increase, making this law key in determining the direction of thermodynamic processes.
In our exercise, where heat energy is transferred from one reservoir to another, the second law helps us understand why changes in entropy occur. The law implies that when heat moves from a hotter to a colder body, the entropy of the overall system increases.

• This is because the heat increase in the colder reservoir raises its entropy more than the heat decrease in the hotter reservoir lowers its entropy.
• The second law also tells us that achieving the maximum possible efficiency in these transfers is impossible without increasing entropy elsewhere in the system.

By observing the changes in entropy given in the exercise, we can see how it aligns with the Second Law of Thermodynamics, reinforcing the idea that the process naturally proceeds towards states of higher entropy.

Thermodynamic Processes

Thermodynamic processes refer to the various ways in which a system changes from one equilibrium state to another. These processes are defined based on how energy and matter move within the system and can influence the calculation of system properties such as temperature, pressure, volume, and, of course, entropy.
Four main types of thermodynamic processes are:

• Isothermal: Occurs at a constant temperature.
• Adiabatic: Occurs without heat transfer into or out of the system.
• Isobaric: Occurs at a constant pressure.
• Isochoric: Occurs at a constant volume.

In the exercise, the transfer of heat can be seen as an isothermal process for each reservoir because the temperature of the reservoirs remains constant during each individual transfer.
Understanding how these processes work helps in predicting the behavior of a system and allows for the calculation of important metrics such as the work done by or on the system and the changes in entropy. The uniformity of entropy change calculations across different temperature scenarios highlights how vital these processes are in real-world applications and scientific understanding.