Question

A "cold" object, \(T_{1}=300 \mathrm{~K}\), is briefly put in contact with a "hot" object, \(T_{2} \simeq 400 \mathrm{~K}\), and \(60 \mathrm{~J}\) of heat flows from the hot object to the cold one. The objects are then separated, their temperatures having changed negligibly due to their large sizes. (a) What are the changes in entropy of each object and the system as a whole? (b) Knowing only that these objects are in contact and at the given temperatures, what is the ratio of the probabilities of their being found in the second (final) state W that of their being found in the first (initial) state? What does chis result suggest?

Short answer

The change in entropy of the hot and cold objects are -0.15 J/K and +0.20 J/K respectively, making total change in entropy of the system to be +0.05 J/K. The final state of the objects being in contact is massively more probable than the initial state.

Step-by-step solution

Calculation of change in entropy for each object

The formula for change in entropy \(\Delta S\) when heat flows at a constant temperature is given by \(\Delta S = Q / T\), where Q is the heat transferred and T is the temperature. For the hot object, the heat Q is flowing out, so it is -60 J. The initial temperature \(T_{2}\) is 400 K. Hence, the change in entropy for the hot object \(\Delta S_{2} = -60 / 400 = -0.15 \ J/K\). For the cold object, the heat Q is flowing in, so it is +60 J. The initial temperature \(T_{1}\) is 300 K. Hence, the change in entropy for the cold object \(\Delta S_{1} = +60 / 300 = +0.20 \ J/K\). Remember, the entropy change is positive for heat gain and negative for heat loss at constant temperature.

Calculation of change in entropy for the whole system

The total entropy change of the system is the sum of the entropy changes of the hot and cold objects. Thus, \(\Delta S_{total} = \Delta S_{1} + \Delta S_{2} = +0.20 - 0.15 = +0.05 \ J/K\). As expected, the entropy of a system increases with time, due to the second law of thermodynamics that predict the arrow of time.

Calculation of the ratio of probabilities of the states of the objects

According to the principle of equal a priori probabilities, the probability of a certain state is proportional to the number of ways, W, that this state can be realized. According to Boltzmann's entropy formula, \(S = k \ln(W)\), where k is Boltzmann's constant, and W the number of microstates. The ratio of probabilities follows from this by exponentiating: \(W_{final} / W_{initial} = e^{(\Delta S / k)}\), with \(\Delta S = S_{final} - S_{initial} = +0.05\ J/K\), and k is the Boltzmann constant \(k = 1.4 \times 10^{-23}\ J/K\). Hence, \(W_{final} / W_{initial} = e^{(+0.05 / 1.4 \times 10^{-23})}\). The resulting ratio is a huge number, indicating that the final state is overwhelmingly more probable than the initial state, as expected due to the increase in entropy.

Key concepts

Understanding the Second Law of Thermodynamics

The Second Law of Thermodynamics is a fundamental principle which states that the entropy of an isolated system always increases over time, or remains constant in ideal cases of reversible processes. In simple terms, entropy can be thought of as a measure of disorder or randomness: the greater the disorder, the higher the entropy.

The implication of this law is profound: it introduces the concept of a 'direction' in time, often referred to as the 'arrow of time'. In the context of heat exchange, like in our exercise, this law implies that heat will naturally flow from a hotter object to a cooler one, increasing the overall entropy of the system.

In practical terms, this manifests as an inability to fully convert heat energy into work without some loss of energy to the surroundings, hence why perfectly efficient engines are impossible according to the second law. Its importance cannot be overstated as it governs the principles of heat transfer, energy dissipation, and has implications for the ultimate fate of the universe.

The Basics of Heat Transfer

Heat transfer is a fundamental concept in thermodynamics that describes the physical act of thermal energy being exchanged between different bodies or a body and its environment. In the exercise problem, heat is transferred from the hot object at 400 K to the cold one at 300 K. This conforms with our expected behavior from the second law of thermodynamics.

There are three modes of heat transfer: conduction, which occurs through solid materials; convection, which occurs through fluids (liquids or gases) often involving fluid motion; and radiation, which involves emitting or absorbing thermal radiation. In our scenario, it's likely that conduction would be the mode of heat transfer given that the objects are in direct contact.

The current problem simplifies the situation by assuming that the heat transfer does not alter the temperatures of the objects significantly due to their large size. This means that the calculation of entropy change can be done using the heat transfer and the initial temperatures, without needing to consider the complexities of changing temperature during the process.

Boltzmann's Constant and its Role in Entropy

Boltzmann's constant, denoted as \(k\), plays a crucial role in statistical mechanics and thermodynamics, relating the micro-level dynamics to macro-level observations. It is a bridge between the molecular world and the macroscopic properties we can measure, such as temperature and entropy.

With a value of approximately \(1.38 \times 10^{-23} \ J/K\), this constant is a scaling factor that translates temperature into kinetic energy and provides a measure of the thermal energy per particle per degree of freedom. In the context of entropy, using Boltzmann's relation \(S = k \ln(W)\), Boltzmann's constant provides a way to quantify the multiplicities of microstates (W) that correspond to a macrostate characterized by a certain entropy.

In our exercise, we use Boltzmann's constant when calculating the ratio of probabilities of different states based on the change in entropy. This is a statistical method that underscores the probabilistic nature of thermodynamics, particularly in systems with a large number of particles.

Thermodynamic Probability and Microstates

Thermodynamic probability often referred to as the number of accessible microstates \(W\), is a concept that's interwoven with the fabric of statistical mechanics. It represents the number of different ways in which a system can be arranged at the microscopic level, while still appearing the same at the macroscopic level.

Understanding thermodynamic probability is key to grasping why certain events occur with a higher likelihood than others. The exercise shows a scenario where the system evolves from an initial state to a final state, with the final state having a greater number of accessible microstates — this indicates a higher thermodynamic probability, and therefore, a higher entropy.

Using Boltzmann's formula, where entropy is proportional to the logarithm of the number of microstates, we relate entropy change to thermodynamic probability. This allows us to calculate the extremely large probability ratio of finding the system in the final state compared to the initial state. This ratio portrays the quintessential tendency of isolated systems to evolve towards configurations with the greatest number of microstates, thus respecting the second law of thermodynamics.