Boltzmann entropy equation; what is it, how to use s = klnW?

The formula s=klnW is known as the Boltzmann-Planck entropy equation. It relates the entropy of a system to the number of microstates available. The greater the number of microstates, the higher the entropy value.

s= klnW is important to understand the connection between macroscopic thermodynamic properties and the statistical microstates an ideal gas can occupy.

Through this article, you will learn all about s= klnW, especially how to use this equation to find solutions to a diversity of numerical problems.

What are the components of s=klnW?

In s=klnW:

• s= Statistical entropy of the system (Units: J/K)

Entropy is a form of internal energy possessed by a system.

It measures the degree or spread out of disorder or randomness present in a system. The more disorganized a system is, the greater the spread out of energy possessed by the particles of that system, which implies a higher entropy value.

• k= Boltzmann’s constant per particle (Units: J/K)

The value of k stays fixed at 1.3805 x 10^-23 J/K

• W = no of microstates, also called thermodynamic probability (No units)

What are macrostates and microstates in s= klnW?

To understand this concept better, take the example of a gaseous container comprising numerous gas molecules.

Each molecule is made up of a specific number of atoms. Each atom is subdivided into atomic particles.

External conditions such as concentration (n), temperature (T), pressure (P) and volume (V) influencing the gaseous system at a larger scale are known as macrostates.

Similarly, entropy (s) in s = klnW is a macroscopic property.

In contrast, the specific configurations that the constituent atoms can adopt within the system are referred to as microstates.

Let us see, for an example, the four particles that can be arranged into the following two different boxes (A and B) in 16 different configurations.

As per the law of statistics, for N particles, the different number of possible arrangements (permutations) are:

W = 𝑵!/𝒏𝟏! 𝒏𝟐! ,,,,, 

Where

• N= total number of particles
• n = number of items in each group (in this case, the number of particles in each box)
• ! represents a factorial

∴ W = 24/4 = 6

∴ Total number of possibilities for arranging four particles in two different boxes W = 1 + 1 + 4 + 4 + 6 = 16.

Therefore, the possible number of microstates in the above example = 16

In terms of chemistry, the total number of microstates possible is known as the thermodynamic probability of the system.

As per the equation s=klnW, entropy (s) is directly proportional to the possible number of microstates/thermodynamic probability (W) while k is a constant.

The greater the possibilities for particles to get arranged in a system, the higher the chances of disorder, i.e., entropy.

For a spontaneous change in a system such that;

• W₁ = initial number of microstates
• W₂ = final number of microstates

The entropy of the system changes from s₁ to s₂

Applying the integral on both sides of s= klnW gives us another formula, i.e.

Where and how to use s= klnW? – Examples

The formula s=klnW can be applied to find either s or W based on which of the two variables is unknown.

To find W using s=klnW:

As per mathematical laws, the relation between natural log (ln) and common log (log) is:

log_e x = ln x

Therefore, ln can be converted into log as follows:

ln x = 2.303 log x

ln W = 2.303 log W

Substituting the above value into the Boltzmann entropy formula:

s = k (2.303 log W)

s = 2.303 k log W

Making W the subject of the formula by taking antilog:

s/2.303 k = log W

∴ W = 10^{-s/2.303 k}

Now let us apply the above concepts to the examples given below.

For example, The thermodynamic probability (W) for a gas at a specific temperature and pressure is 10²⁸ J/K. Use the formula s = klnW to calculate the entropy of the gas under these conditions.

As per the question statement;

W = 10²⁸ J/K

We already know that;

Boltzmann constant (k) = 1.3805 x 10^-23 J/K

Substituting the above data into the Boltzmann entropy formula gives us:

s = (1.3805 x 10^-23) ln (10²⁸)

Calculating the value of ln 10²⁸ with the help of a scientific calculator:

s = (1.3805 x 10^-23) (64.47)

∴ s = 8.90 x 10^-22

Result: The entropy of the gas in this example is 8.90 x 10^-22 J/K.

Another example is- Find the thermodynamic probability (W) if the entropy of a system is 318 J/K.

 As per the question statement;

Entropy (s) = 318 J/K

We already know that;

Boltzmann constant (k) = 1.3805 x 10^-23 J/K

Substituting the above data into the Boltzmann entropy formula gives us:

⇒ s = k ln W

318 = (1.3805 x 10^-23) x 2.303 log W

318 = (3.18 x 10^-23) log W

log W = 1 x 10²⁵

Taking the antilog of 1 x 10²⁵ finally gives us:

W = 10^{-(1 x 10^25)}

∴ W = 10^{10^25}

Result: The thermodynamic probability of the system is 10^{10^25}, which hints at a huge number of possible microstates.

FAQ

What is represented by the formula s = klnW?

The formula s = klnW represents the Boltzmann entropy equation. It is used to find the disorder (entropy) of a system if the number of microstates (W) is given. In s = klnW: s = entropy of the system (in J/K) k = Boltzmann constant (k = 1.3805 x 10-23 J/K) W = no of microstates, more commonly known as thermodynamic probability (unitless).

What is the relationship between s and W, as per s = klnW?

As per s= k ln W, the entropy (s) of a system is directly related to its thermodynamic probability (W), while k is kept constant. The greater the possible number of microstates, the higher the spread of energy/disorder in the system.

From where is the equation s = klnW derived?

The Boltzmann entropy equation (s=klnW) is derived from a combination of the 1st and 2nd law of thermodynamics and the Clausius entropy equation. 1st law of thermodynamics: The energy of a system stays conserved. dU = dQ + dW where, dU = change in internal energy of the system dQ = heat entering or leaving the system dW = work done on or by the system 2nd law of thermodynamics: The entropy of a system always tends to increase. dS = dQ/T (Clausius entropy equation) where dS = change in entropy and T= temperature of the system