Month Of Equations: What Does The Boltzmann’s Equation Really Mean?

                             October 23: The Boltzmann’s Equation

The Boltzmann's Equation
The Boltzmann’s Equation

Meaning of the Equation:

The Boltzmann’s law states that the entropy of a system is the measure of the number of possible microscopic states of the system corresponding to a particular macrostate.

Known as the foundation of the statistical mechanics, Boltzmann’s principle gives an expression for the entropy of a system in thermodynamic equilibrium. Boltzmann’s equation applies to macrostates of the universe as a whole while considering that each possible microstate of the universe equally probable. According to this principle, the entropy of a system of an ideal gas is equal to the possible number of microstates of particles in that system. In order to understand this principle, we need to understand in detail about the various terms involved in its expression. Let us begin with some definitions :

  1. Entropy: The measure of disorder in a system is termed as the entropy of the system and a system having a maximum disorder has minimum energy. So, entropy is also considered as the unavailability of energy in a system for doing any work. For example, the stretching of rubber puts its atoms into order and hence, a state of minimum energy and zero entropy. However, when it was unstretched, the disorder and hence, the entropy was maximum and the work could be done for stretching the rubber.
  2. Microstate: A microstate is a particular microscopic configuration of the particles comprising a thermodynamic system such as an ideal gas in a container.
  3. Macrostate: A macrostate is characterized by the macroscopic variables of the system such as temperature, pressure, volume etc.

If these definitions are still mind-boggling to you, then suppose you put a drop of blue ink in a glass of water. Within a few seconds, the ink dissolves in water and the water attains a uniform blue color. At this point, we can depict the state of the water by just defining the color of it so this becomes a macrostate of the system. However, the color is actually a result of the diffusion of particles of blue ink within the water and attaining particular positions. Hence, the configuration of individual particles within the system indicates the microstate of that water-ink system.

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Now, we all know that there are a large number of particles even in a very small volume of the gas. Hence, if we have to define the positions of individual molecules of the system, it will become very difficult to understand and handle, hence, we use the macroscopic quantities such as temperature and pressure to define the state of a system easily. It is noteworthy that there may be a number of microstates corresponding to a specific macrostate of the system.

Let me now again take you back to the Boltzmann’s law of entropy: Boltzmann’s Equation signifies that the entropy of a thermodynamic system in equilibrium is defined as the probability of a macrostate for some probability distribution of possible microstates. For a system in thermodynamic equilibrium, the probability of each microstate is equal. Therefore, it becomes equal to the total number of possible microstates of the system. The term depicted by capital omega in the equation is obtained by the permutations of particles within for a given macrostate.

It is interesting that the entropy ‘S’ is itself a thermodynamic property of the system and hence, is a bridge between the macroscopic and the microscopic properties if the system. Thus this equation connects two important branches of physics: statistical mechanics and thermodynamics and hence is an important equation in Physics.

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