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Helmholtz free energy (often written as A or F) is defined as: A = U − T·S
Under constant temperature and volume, systems tend to move toward lower Helmholtz free energy.
| Case | Internal Energy U (J) | Temperature T (K) | Entropy S (J/K) | Helmholtz A = U − T·S (J) |
|---|---|---|---|---|
| 1 | 5000 | 300 | 10 | 2000 |
| 2 | 12000 | 350 | 18 | 5700 |
| 3 | 8000 | 298 | 12.5 | 4275 |
These values are illustrative and depend on your chosen reference state.
Helmholtz free energy, A, measures the “useful” energy available to do work when a system is held at constant temperature and constant volume. It is central in thermal physics, statistical mechanics, and materials work. Lower values indicate a more favorable equilibrium state for the same T and V.
The calculator uses A = U − T·S. Internal energy U is commonly reported in joules, kilojoules, or electronvolts. Temperature T must be absolute; room conditions are about 298 K. Entropy S is often in J/K; for many macroscopic samples, values from 1 to 10,000 J/K are possible depending on size and units basis.
At fixed T and V, spontaneous change tends to reduce A. This makes Helmholtz free energy the natural potential for closed, rigid containers in contact with a heat reservoir, such as calorimetry cells and many solid‑state experiments.
The absolute value of A depends on the chosen reference (zero) for energy and entropy. Negative values are common and not “wrong.” The physically meaningful quantity is often a difference, ΔA, between two states at the same temperature.
Suppose a sample has U = 5000 J, T = 300 K, and S = 10 J/K. Then T·S = 3000 J, so the calculator returns A = 2000 J. If the temperature rises to 350 K while S stays 10 J/K, then A = 1500 J, showing how higher temperature can lower available free energy for the same entropy.
A common mistake is mixing “per mole” entropy with total internal energy. If you use molar entropy (J/mol·K), make sure U and A are also molar (J/mol). For quick sanity checks, note that 1 kJ = 1000 J, and 1 kcal ≈ 4184 J.
Helmholtz free energy is widely used for phase stability, lattice vibrations, and polymer elasticity. In statistical mechanics, an advanced link is A = −k·T·ln Z, where Z is the partition function and k is Boltzmann’s constant. This connects measured thermodynamics to microscopic models.
Ensure T is above 0 K, and avoid solving for T or S with a zero denominator. If your computed temperature becomes non‑physical, revisit signs, references, and whether your entropy basis matches your energy basis.
They are different potentials. Helmholtz applies naturally at constant temperature and volume. Gibbs applies at constant temperature and pressure, using G = H − T·S.
Yes. The zero reference for energy and entropy is arbitrary, so absolute A can be negative. Differences, such as ΔA between states at the same temperature, carry the physical meaning.
Use J/K (or compatible units). If you use molar entropy like J/mol·K, then internal energy and free energy should also be molar, such as J/mol, to keep the equation consistent.
The formula uses absolute temperature. Celsius or Fahrenheit must be converted to Kelvin before multiplying by entropy. The calculator does this conversion automatically when you select the input unit.
That result is non‑physical for ordinary equilibrium thermodynamics. It usually indicates inconsistent signs, mismatched unit basis (total vs molar), or an entropy value that should not be used for that state.
The exports capture the exact inputs, selected units, and computed outputs shown on the page. They are ideal for lab notes, reports, and quick comparisons between scenarios.
Solve for entropy when you have measured internal energy and free energy, and you know the temperature. This is helpful when comparing experimental data to models that predict free energy directly.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.