Boltzmann Entropy Calculator
Calculate the entropy of a system using Boltzmann's entropy formula S = k ln W.
Explore the connection between microstates, probability, and thermodynamic entropy.
Boltzmann’s Entropy Formula S = k_B x ln(W). This equation, carved on Ludwig Boltzmann’s tombstone in Vienna, Austria, connects the microscopic world (atoms and molecules) to macroscopic thermodynamics. It was formulated around 1877. S is the entropy (in Joules per Kelvin). k_B is Boltzmann’s constant (1.381 x 10^-23 J/K). W is the number of microstates — the number of different ways the microscopic components can be arranged while producing the same macroscopic state.
What Are Microstates? A microstate is one specific arrangement of all particles in a system. For example, if you have 4 gas molecules in a box divided into two halves, there are 2^4 = 16 possible arrangements. Only 1 arrangement has all 4 molecules on the left side (low entropy), but 6 arrangements have exactly 2 on each side (high entropy). Systems naturally evolve toward states with more microstates — this is the Second Law of Thermodynamics.
Why Entropy Increases A system with more microstates is more probable. If you start with all molecules on one side, there are vastly more ways for them to spread out than to stay concentrated. With 10^23 molecules (a typical gas sample), the probability of spontaneous concentration is so astronomically small it effectively never happens. This is why heat flows from hot to cold, gases expand to fill containers, and ice melts at room temperature.
Entropy and Information Claude Shannon, working at Bell Labs in 1948 in the United States, discovered that information entropy has the same mathematical form as Boltzmann’s entropy. Information entropy measures uncertainty or surprise in a message. The connection between thermodynamic entropy and information theory is one of the deepest results in physics.
Practical Values Standard molar entropy of a perfect crystal at 0 K: 0 J/(molK) (Third Law of Thermodynamics). Water at 25 degrees C: 69.9 J/(molK). Nitrogen gas at 25 degrees C: 191.6 J/(mol*K). Entropy always increases in an isolated system — this defines the direction of time itself.