Question

After spontaneous heat transfer between systems, the overall multiplicity is: a. lower than it was before the transfer of heat b. zero c. maximized d. minimized

Short answer

c. maximized

Step-by-step solution

Understanding Multiplicity

Multiplicity, in the context of thermodynamics, refers to the number of microstates corresponding to a particular macrostate of a system. It is related to the concept of entropy.

Connection Between Entropy and Multiplicity

Entropy is a measure of the number of possible ways a system can be arranged, often referred to as randomness or disorder. Multiplicity, denoted by \( \Omega \), directly influences entropy \( S \) via the relationship \( S = k \ln \Omega \), where \( k \) is the Boltzmann constant.

Heat Transfer and Entropy

When spontaneous heat transfer occurs, the entropy of an isolated system tends to increase, reflecting an increase in multiplicity, as more microstates become accessible.

Conclusion on Multiplicity After Heat Transfer

Following spontaneous heat transfer, the overall number of microstates (multiplicity) of the systems combined is maximized, consistent with the second law of thermodynamics.

Key concepts

Multiplicity in Thermodynamics

Multiplicity is a core concept in thermodynamics that helps describe the number of ways a system's particles can be arranged while maintaining the same energy, volume, and number of particles. This arrangement is known as a macrostate. Think of multiplicity as the different microstates that lead to the same observable properties in the system.

Each microstate is a specific arrangement of the system’s internal components. Whereas, a macrostate is characterized by bulk properties like temperature and pressure that we can measure. The greater the number of microstates, the higher the multiplicity.

In everyday language, multiplicity can be imagined as the different paths leading to the same destination. In thermodynamics, this is crucial because it underlies the concept of entropy. Systems with higher multiplicity tend to be more probable since there are more possible configurations, which leads us to our next topic, entropy.

Entropy and Microstates

Entropy is a fundamental concept in thermodynamics that quantifies the disorder or randomness of a system. In a more technical sense, it is a measure of the number of microstates (or ways) that a specific macrostate can be achieved. The relationship between entropy (\(S\)) and multiplicity (\(\Omega\)) is given by the equation: \[ S = k \ln \Omega \]where \(k\) is the Boltzmann constant.

This equation indicates that entropy is directly proportional to the logarithm of multiplicity. As the multiplicity of a system increases, so does its entropy. This is because a greater number of microstates translates to a higher level of uncertainty or disorder in the system. It’s like having more pieces to rearrange in a puzzle, leading to more combinations.

In practical terms, an increase in entropy typically indicates a change in the condition of a system, such as when heat is transferred, resulting in new microstates that become accessible. Thus, entropy can be viewed as a measure of energy dispersal at the microscopic level within a system.

Second Law of Thermodynamics

The second law of thermodynamics is a fundamental principle that dictates the direction of spontaneous processes and the general way energy is distributed in a system. It states that in an isolated system, the total entropy can never decrease over time. Instead, it tends to increase or remain constant.

This law is essential because it establishes that natural processes lead to an increase in entropy, explaining why isolated systems evolve toward equilibrium.

• When two systems interact, energy will spontaneously transfer from the hotter to the cooler system, increasing the entropy.
• As a result, multiplicity maximizes because more microstates become possible.
• The net effect of this energy transfer is a more disordered, higher entropy state across the combined systems.

Understanding this law helps explain phenomena such as why hot coffee cools down, why ice melts, or why energy cannot be perfectly recycled. Essentially, the second law of thermodynamics explains the arrow of time in terms of energetic changes, ensuring that once energy spreads out, it cannot spontaneously concentrate without external work.