Statistical Interpretation of Entropy: Videos & Practice Problems

In thermodynamics, a macrostate is defined by measurable properties like pressure, volume, and temperature, while a microstate refers to the specific arrangements of particles within that macrostate. The number of microstates corresponding to a macrostate is termed multiplicity (Ω). The entropy (S) of a system is calculated using the equation $S=kln\left(\Omega \right)$, where k is the Boltzmann constant. As disorder increases, so does entropy, aligning with the second law of thermodynamics.

Microstates and Macrostates of a System

The concepts of macrostates and microstates are fundamental in understanding the entropy of a system. A macrostate refers to the macroscopic properties of a system, such as pressure, volume, temperature, internal energy, and entropy itself. These properties define the overall state of the system and can be measured directly. For example, in the case of ideal gases, the relationship between pressure, volume, and temperature is described by the ideal gas law, which helps define the macrostate.

Importantly, a specific macrostate is not unique; multiple arrangements of particles can lead to the same macrostate. These arrangements are known as microstates. For instance, two samples of gas can have the same temperature while having different configurations of gas particles. Each macrostate must have at least one corresponding microstate, as it would be meaningless to consider a macrostate that cannot be realized by any arrangement of particles.

The number of microstates corresponding to a particular macrostate is referred to as its multiplicity, denoted by the Greek letter omega (Ω). Different macrostates can have varying multiplicities. For example, if we consider a system of four coins, a macrostate could be defined as having two coins heads up. The different arrangements of the coins that still result in this macrostate represent the microstates. In this case, there are six distinct microstates for the macrostate of two heads up, leading to a multiplicity of 6.

Entropy, a key concept in thermodynamics, is mathematically defined in relation to the multiplicity of microstates. The formula for entropy (S) is given by:

$$ S = k \cdot \ln(\Omega) $$

where \( k \) is the Boltzmann constant, approximately \( 1.38 \times 10^{-23} \, \text{J/K} \), and \( \Omega \) is the multiplicity. For the coin example, the entropy can be calculated as:

$$ S = 1.38 \times 10^{-23} \cdot \ln(6) \approx 2.47 \times 10^{-23} \, \text{J/K} $$

It is essential to note that entropy is always a positive value, as there must be at least one microstate for any macrostate. As the disorder of a system increases, the number of available microstates also increases, leading to a rise in entropy. This relationship explains why systems naturally progress towards greater disorder, aligning with the second law of thermodynamics, which states that the total entropy of an isolated system can never decrease over time.

The macrostate of a set of coins is given by the number of coins that are heads-up. If you have 100 coins, initially with 20 heads-up, what is Δ? when the system is changed to have 50 heads-up? Note that the multiplicity of k coins which are heads-up, out of N total coins, is $\Omega=\frac{N!}{k!\left(N-k\right)!}$. Does this change in macrostate satisfy the second law of thermodynamics?

• A. - 2.63 × 10^-22 J/K
• B. 7.24 × 10^-25 J/K
• C. 2.63 × 10^-22 J/K
• D. 1.58 × 10^-21 J/K

Answer: (C)

Here’s what students ask on this topic:

What is the difference between a macrostate and a microstate in thermodynamics?

In thermodynamics, a macrostate is defined by measurable properties such as pressure, volume, and temperature. These are macroscopic properties that describe the overall state of a system. A microstate, on the other hand, refers to the specific arrangements of particles within that macrostate. While a macrostate can be described by a few variables, a microstate involves the detailed positions and velocities of all particles in the system. Multiple microstates can correspond to the same macrostate, and the number of these microstates is termed the multiplicity (Ω).

How is entropy related to the number of microstates in a system?

Entropy (S) is a measure of the disorder or randomness in a system and is directly related to the number of microstates (Ω) corresponding to a macrostate. The relationship is given by the equation:

$S=k\ln \left(\Omega \right)$

where k is the Boltzmann constant (1.38 × 10⁻²³ J/K). As the number of microstates increases, the entropy also increases. This aligns with the second law of thermodynamics, which states that systems tend to move towards more disorder, thereby increasing entropy.

What is multiplicity in the context of thermodynamics?

Multiplicity (Ω) in thermodynamics refers to the number of microstates corresponding to a particular macrostate. It quantifies the different ways particles in a system can be arranged while still resulting in the same macroscopic properties like pressure, volume, and temperature. For example, if you have a system of four coins, a macrostate of two heads up can have multiple microstates, such as heads-heads-tails-tails or heads-tails-heads-tails. The multiplicity is crucial for calculating entropy, as it is used in the equation:

$S=k\ln \left(\Omega \right)$

How do you calculate the entropy of a system using the Boltzmann constant?

To calculate the entropy (S) of a system using the Boltzmann constant (k), you use the equation:

$S=k\ln \left(\Omega \right)$

where Ω is the multiplicity, or the number of microstates corresponding to a particular macrostate. The Boltzmann constant (k) is 1.38 × 10⁻²³ J/K. For example, if a system has a multiplicity of 6, the entropy would be:

$S=1.3810^-23\ln \left(6\right)$

Plugging in the values, you get S ≈ 2.47 × 10⁻²³ J/K.

Why does entropy always increase in a system according to the second law of thermodynamics?

According to the second law of thermodynamics, entropy in an isolated system always increases because systems naturally evolve towards more disorder. As the number of available microstates (Ω) increases, the entropy (S) also increases, given by the equation:

$S=k\ln \left(\Omega \right)$

where k is the Boltzmann constant. This tendency towards higher entropy is because there are more ways for a system to be disordered than ordered. For example, a gas will spread out to fill its container because there are more microstates corresponding to a dispersed gas than a concentrated one, leading to higher entropy.