Measure internal energy from heat and work. Switch models for gases, mixtures, or experiments fast. Download results instantly and keep calculations consistent every time.
| Scenario | Inputs | Expected output |
|---|---|---|
| First law | Q = 1200 J, W = 300 J (work by system) | ΔU = 900 J |
| Ideal gas (dof) | n = 1 mol, T = 300 K, f = 5 | U ≈ 6236 J |
| Mass heating | m = 2 kg, cv = 718 J/(kg·K), ΔT = 20 K | ΔU = 28,720 J |
Internal energy, U, summarizes microscopic energy stored in a system: translational, rotational, vibrational, electronic, and interaction energies. This calculator supports four common workflows used in thermodynamics labs and engineering analysis: the first law balance, two ideal-gas models, and a mass-based constant-volume heating model.
Unlike kinetic or potential energy of the whole system, U is an extensive property tied to the system’s state. For many problems you need only a change, ΔU, because absolute reference values depend on a chosen zero. For ideal gases, relative changes are often captured accurately with temperature-based relations.
For a closed system, the first law can be written as ΔU = Q − W when W is work done by the system. In many experiments, Q is inferred from heaters or calorimeters, while W comes from boundary work or shaft work. The sign convention selector prevents common bookkeeping errors.
For ideal gases, internal energy depends mainly on temperature, so a practical estimate is U = n·Cv·T. The calculator offers standard approximations: monatomic Cv = 3/2 R, diatomic Cv ≈ 5/2 R, and polyatomic Cv ≈ 3R. The gas constant is R = 8.314462618 J/(mol·K).
When you know the active degrees of freedom f, equipartition gives U = (f/2)·n·R·T. Typical values near room temperature are f = 3 for monatomic gases, f = 5 for diatomic gases, and f ≥ 6 for many polyatomic gases. At higher temperatures, vibrational modes can increase f and raise U.
For solids, liquids, or gases treated with a specific heat at constant volume, a common estimate is ΔU = m·cv·ΔT. This is useful when you have measured a temperature rise and a material property in J/(kg·K). For air near room temperature, a typical reference value is about cv ≈ 718 J/(kg·K) (use your dataset if available).
Inputs can be provided in joules, kilojoules, calories, kilocalories, electron-volts, or BTU. Temperatures accept kelvin, Celsius, or Fahrenheit, and temperature differences correctly scale for Fahrenheit. After calculation, the result is converted to your selected output unit and can be exported as a CSV table or a compact PDF summary.
For best results, keep units consistent with your measurement method and document the sign convention. If Cv varies strongly with temperature, compute piecewise or use a custom value. For real gases, phase changes, or reacting mixtures, temperature-only ideal-gas formulas may underpredict energy changes.
U is a state property that depends on the chosen reference zero. ΔU is the change between two states and is independent of the reference. Most engineering balances and experiments rely on ΔU.
Use the first law if you measured heat and work. Use the ideal-gas Cv or degrees-of-freedom models when temperature and amount are known and the gas behavior is close to ideal.
If work is defined positive when done by the system, use ΔU = Q − W. If your data defines work done on the system as positive, select the alternative option so the calculator uses ΔU = Q + W.
Internal energy relates to microscopic energy and, for ideal gases, depends on temperature through Cv. Cp is used for enthalpy changes. For constant-pressure heating, enthalpy is often the more convenient property.
Yes. For temperature differences, kelvin and Celsius increments are identical. For Fahrenheit differences, the calculator applies the 5/9 scaling automatically so energy results remain consistent.
They are common approximations near moderate temperatures. Vibrational modes, high temperatures, or unusual gases can change Cv. If you have tabulated Cv data, select custom Cv and enter your value.
Exports include the computed result table shown on the page: the main internal energy value in your selected unit plus the base value in joules. This is designed for quick reporting and spreadsheet logging.
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.