## Boltzmann’s entropy formula

**entropy**is a quantitative measure of disorder, or of the energy in a system to do work.

In statistical physics, entropy is a measure of the disorder of a system. What disorder refers to is really the **number of microscopic configurations**, **W**, that a thermodynamic system can have when in a state as specified by certain macroscopic variables (volume, energy, pressure, and temperature). By “microscopic states”, we mean the exact states of all the molecules making up the system.

Mathematically, the exact definition is:

**Entropy = (Boltzmann’s constant k) x logarithm of number of possible states**

**S = k _{B} logW**

This equation, known as the **Boltzmann’s entropy formula**, relates the microscopic details, or microstates, of the system (via *W*) to its macroscopic state (via the **entropy S**). It is the key idea of statistical mechanics. In a closed system, entropy never decreases, so in the Universe entropy is irreversibly increasing. In an open system (for example, a growing tree), entropy can decrease and order can increase, but only at the expense of an increase in entropy somewhere else (e.g. in the Sun).

Order is decreasing.

Entropy is increasing.

## Irreversibility of Natural Processes

**According to the second law of thermodynamics:**

*The entropy of any isolated system never decreases. In a natural thermodynamic process, the sum of the entropies of the interacting thermodynamic systems increases. *

This law indicates the **irreversibility** of **natural processes**. Reversible processes are a useful and convenient theoretical fiction, but do not occur in nature. From this law follows that it is impossible to construct a device that operates on a cycle and whose sole effect is the transfer of heat from a cooler body to a hotter body. It follows, perpetual motion machines of the second kind are impossible.

## Entropy at Absolute Zero

**According to third law of thermodynamics:**

*The entropy of a system approaches a constant value as the temperature approaches absolute zero.*

Based on empirical evidence, this law states that the **entropy of a pure crystalline substance is zero** at the **absolute zero of temperature**, 0 K and that it is impossible by means of any process, no matter how idealized, to reduce the temperature of a system to absolute zero in a finite number of steps. This allows us to define a zero point for the thermal energy of a body.

Absolute zero is the coldest theoretical temperature, at which the thermal motion of atoms and molecules reaches its minimum. This is a state at which the enthalpy and entropy of a cooled ideal gas reaches its minimum value, taken as 0. **Classically**, this would be a state of **motionlessness**, but **quantum** uncertainty dictates that the particles still possess a **finite zero-point energy**. **Absolute zero** is denoted as 0 K on the Kelvin scale, **−273.15 °C** on the Celsius scale, and **−459.67 °F** on the Fahrenheit scale.

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