Question

Residual molar entropy is given by the expression (a) \(S=R \ln W\) (b) \(S=N \ln W\) (c) \(S=\frac{R}{N} \ln W\) (d) \(S=n R \ln W\)

Short answer

The correct expression for the residual molar entropy is \(S = nR \ln W\), so option (d) is the answer.

Step-by-step solution

Recall Boltzmann's Entropy Formula

The correct formula for entropy according to Boltzmann's entropy formula is given by \(S = k \ln W\), where \(k\) is the Boltzmann constant and \(W\) is the number of microstates.

Relate Boltzmann Constant to Gas Constant

The Boltzmann constant \(k\) can be written in terms of the gas constant \(R\) and Avogadro's number \(N_A\), as \(k = R/N_A\). Here, \(N_A\) is Avogadro's number which is the number of atoms per mole. The entropy equation becomes \(S = \frac{R}{N_A}\ln W\).

Express Entropy in Terms of Moles

In molar terms, \(N_A\) is replaced by the number of moles \(n\), leading to \(S = nR \ln W\).

Key concepts

Boltzmann's Entropy Formula

Entropy is a fundamental concept in thermodynamics that measures disorder or randomness. Ludwig Boltzmann, a pioneering physicist, developed a formula to quantify this disorder. This formula is known as Boltzmann's Entropy Formula, and it's crucial in understanding entropy at a molecular level.
Boltzmann's formula is expressed as: \[S = k \ln W\]where:

• S represents the entropy.
• k is the Boltzmann constant.
• W denotes the number of microstates.

The Boltzmann constant \(k\) is a fundamental physical constant that relates the average kinetic energy of particles in a gas with the temperature of the gas. It bridges macroscopic and microscopic physics by linking entropy with probability distributions. In essence, this formula tells us that the greater the number of microstates \(W\), the higher the entropy \(S\). This relationship underscores the connection between disorder and the number of ways a system can be configured.

Gas Constant

The gas constant, often symbolized by \(R\), is an essential factor in equations of state for gases. It's particularly important when connecting microscopic and macroscopic thermodynamic properties. Let's delve a bit deeper into this concept.
In Boltzmann's context, the gas constant helps translate his formula from molecular to macroscopic scales. The gas constant is related to another critical constant:

• Boltzmann Constant \(k\): This constant applies at the molecular scale.

The relationship between \(R\) and \(k\) is given by:\[k = \frac{R}{N_A}\]where \(N_A\) is Avogadro's number, representing the number of molecules in one mole of a substance. This formulation helps in expressing entropy (\(S\)) in larger, tangible terms that encompass the mole, a standard unit in chemistry for expressing amounts of a chemical substance. Therefore, \(R\) serves as a bridge, helping to switch from individual molecular behavior to the behavior of an entire mole of substance.

Microstates

In thermodynamics, microstates are specific configurations that a system can achieve at a given energy level. Each microstate represents a unique way the components of the system can be arranged while achieving the same macroscopic state, such as temperature or pressure.
The number of microstates \(W\) plays a vital role in Boltzmann's Entropy Formula. If we consider a system in distinct energy states, the microstates become a crucial aspect in determining the entropy \(S\). With these states, entropy can be seen as a measure of uncertainty or disorder:

• Higher number of microstates \(\rightarrow\) Higher entropy
• Fewer number of microstates \(\rightarrow\) Lower entropy

The idea here is that the more microstates available to a system, the greater the disorder, and hence, the higher the entropy. Entropy reaches its maximum when the distribution of energy among available microstates is most probable, meaning it is the state of maximum uncertainty.Understanding microstates is fundamental in explaining macroscopic phenomena through probability and statistics, forming a foundation for statistical mechanics.