Boltzmann Constant
The Boltzmann constant (k_B) is a fundamental physical constant that relates the average kinetic energy of particles in a gas to the absolute temperature of the gas, acting as the bridge between macroscopic thermodynamic quantities and microscopic statistical mechanics. It appears in Boltzmann's entropy formula S = k_B ln Ω, the ideal gas law in per-particle form, and the Maxwell-Boltzmann energy distribution, making it one of the most universal constants in physics. Since the 2019 SI redefinition, the Boltzmann constant has an exact defined value of 1.380649 × 10⁻²³ J/K.
Key Formula
k_B = 1.380649 × 10⁻²³ J/K
LaTeX: k_B = 1.380649 \times 10^{-23} \text{ J/K}
| Symbol | Meaning | Unit |
|---|---|---|
| k_B | Boltzmann constant | J/K |
| R | Universal gas constant (R = N_A × k_B = 8.314 J/mol·K) | J/(mol·K) |
| N_A | Avogadro's number (6.022 × 10²³ mol⁻¹) | mol⁻¹ |
Worked Example
Problem
Calculate the average kinetic energy of a single nitrogen molecule at room temperature (T = 300 K) using the equipartition theorem.
Solution
Step 1: For a monatomic ideal gas, average translational KE per particle = (3/2) k_B T. Step 2: k_B = 1.380649 × 10⁻²³ J/K, T = 300 K. Step 3: KE = (3/2) × 1.380649 × 10⁻²³ × 300 = (3/2) × 4.142 × 10⁻²¹ J. Step 4: KE = 6.21 × 10⁻²¹ J.
Answer
Average translational kinetic energy ≈ 6.21 × 10⁻²¹ J per molecule at 300 K
Key Equations Featuring the Boltzmann Constant
| Equation | Formula | Description | Field |
|---|---|---|---|
| Entropy | S = k_B ln Ω | Entropy from number of microstates | Statistical mechanics |
| Avg. kinetic energy | KE = (3/2) k_B T | Average translational KE per particle | Kinetic theory |
| Ideal gas (per particle) | pV = Nk_B T | Ideal gas law in microscopic form | Thermodynamics |
| Thermal voltage | V_T = k_B T / e | Thermal voltage in semiconductor physics | Electronics |
| Boltzmann factor | e^(−E/k_B T) | Probability of state with energy E at T | Statistical mechanics |
| Stefan-Boltzmann law | σ = 2π⁵k_B⁴ / (15h³c²) | Radiation from a black body | Thermal radiation |
Interactive Tools
NIST Physical Constants Reference
Official NIST value and uncertainty for the Boltzmann constant
Open ToolWolfram Alpha
Evaluate expressions involving the Boltzmann constant and thermodynamic quantities
Open ToolKhan Academy – Kinetic Molecular Theory
Learn how the Boltzmann constant connects temperature to molecular kinetic energy
Open Tool
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Related Terms
Entropy
Entropy is a thermodynamic state function that quantifies the degree of disorder, randomness, or the number of microstates available to a system at a given macrostate. Macroscopically, it is defined via the Clausius inequality as the ratio of reversible heat exchange to absolute temperature; microscopically, Boltzmann's formula connects it to the number of microscopic configurations. Entropy always increases in irreversible processes in isolated systems, driving systems toward equilibrium and explaining the thermodynamic arrow of time.
Absolute Zero
Absolute zero is the lowest theoretically possible temperature, defined as 0 K (−273.15°C or −459.67°F), at which a system would have minimum possible internal energy and all classical thermal motion ceases. At absolute zero, quantum mechanical effects dominate: particles occupy their lowest quantum energy states (zero-point energy), meaning even at 0 K some residual energy remains due to Heisenberg's uncertainty principle. The Third Law of Thermodynamics establishes that absolute zero can be approached asymptotically but never actually reached in a finite number of cooling steps.
Third Law of Thermodynamics
The Third Law of Thermodynamics, formulated by Walther Nernst, states that the entropy of a perfect crystalline substance approaches zero as the absolute temperature approaches zero kelvin. This means it is impossible to reach absolute zero in a finite number of steps, establishing a natural reference point for the entropy scale. The law has profound implications for low-temperature physics, quantum behavior of matter, and the calculation of absolute entropies used in chemical thermodynamics.
Named after Ludwig Eduard Boltzmann (1844–1906), the Austrian physicist who founded statistical mechanics. The constant was first calculated by Max Planck in 1900 from blackbody radiation data. The symbol k_B uses "B" for Boltzmann; some texts use simply "k".