Calculator
Formula used
For an ideal gas with constant degrees of freedom, internal energy depends on temperature: U = (f/2) · n · R · T. Here n is moles, R is the gas constant, and T is absolute temperature.
Using the ideal-gas relation nRT = PV, you may also compute: U = (f/2) · P · V. This is useful when pressure and volume are known directly.
How to use this calculator
- Select an input method: n,T, m,M,T, or P,V.
- Choose degrees of freedom using gas type, or enter a custom value.
- Enter your values with units, then press Calculate.
- Review the result shown above the form after submission.
- Use Download CSV or Download PDF to export.
Example data table
| Case | Gas type (f) | n (mol) | T (K) | U (J) |
|---|---|---|---|---|
| 1 | Monoatomic (3) | 1.0 | 300 | 3741.51 |
| 2 | Diatomic (5) | 2.0 | 350 | 14550.31 |
| 3 | Polyatomic (6) | 0.5 | 500 | 6235.85 |
Values use R = 8.314462618 J/(mol·K) and U = (f/2) nRT.
1) Ideal-gas internal energy in one equation
This calculator estimates internal energy for an ideal gas using U = (f/2) nRT, with R = 8.314462618 J/(mol·K). It assumes a constant f over the temperature range and reports energy in joules. Use the rounding control to match lab precision and the method selector to compute from either n,T or P,V.
2) Degrees of freedom and gas selection
The gas-type choice sets the degrees of freedom f: monatomic (~3), diatomic (~5), and polyatomic (~6) are common teaching defaults. At higher temperatures, vibrational modes may increase the effective f, raising U beyond the constant‑f estimate.
3) Temperature units and why kelvin matters
Internal energy depends on absolute temperature, so °C and °F inputs are converted to K. A 10 K rise changes energy by (f/2) nR·10. For 2.0 mol diatomic gas, that is about 416 J per 10 K, useful for quick sanity checks.
4) Example values you can compare
︎For n = 1.00 mol at 300 K, a monatomic estimate (f=3) gives U ≈ 3741.5 J, while a diatomic estimate (f=5) gives U ≈ 6235.9 J. Doubling moles or temperature doubles internal energy.
5) Using pressure–volume data directly
If pressure and volume are known, the ideal-gas identity nRT = PV allows U = (f/2)PV. With P = 200 kPa and V = 0.010 m³, PV = 2000 J, so a diatomic estimate is U ≈ 5000 J.
6) Sensitivity, rounding, and uncertainty
Because U is proportional to n and T, relative errors add in a simple way: 2% uncertainty in moles produces 2% uncertainty in energy. If T is uncertain by ±2 K at 300 K, that is about ±0.67% in U. Use the precision control to avoid reporting digits your inputs cannot justify.
7) Thermodynamics connections
Internal energy supports first‑law calculations: ΔU = Q − W. You can compute U at two temperatures and take the difference to estimate heating or cooling effects. Combine U with PV trends when discussing enthalpy in ideal processes. This is helpful for constant‑volume and constant‑pressure comparisons.
8) Limits of the ideal-gas model
The approach works best at moderate pressures and temperatures well above liquefaction. Near phase change or at high pressure, real-gas effects and temperature‑dependent heat capacities matter. For higher accuracy, use tabulated Cv(T) and integrate U(T).
Frequently asked questions
1) Why does internal energy depend only on temperature here?
For an ideal gas, intermolecular potential energy is neglected, so energy comes from molecular motion. With constant degrees of freedom, average kinetic energy scales with absolute temperature, making U a function of T only.
2) Which gas type should I choose?
Use monatomic for noble gases, diatomic for many common gases near room temperature, and polyatomic as a simple approximation for complex molecules. At high temperatures, vibrational modes can require a larger effective f.
3) Can I use °C or °F directly?
Yes. The calculator converts °C and °F to kelvin before computing. Internal energy must use an absolute temperature scale; using °C or °F without conversion would produce incorrect results and even negative values.
4) What is the PV method used for?
If pressure and volume are known, the identity nRT = PV lets you compute U without separately finding n or T. This is useful when you have measured P and V and want a fast energy estimate.
5) What units does the calculator output?
Internal energy is reported in joules. If you use kPa for pressure and m³ for volume, PV naturally equals joules. The tool also shows intermediate values, helping you validate unit conversions.
6) Is internal energy the same as heat added?
Not always. Heat Q and internal energy change ΔU are related by ΔU = Q − W. If the gas expands and does work, some heat becomes work and ΔU is smaller than Q.
7) When should I avoid the ideal-gas assumption?
Avoid it near condensation, at very high pressures, or when high accuracy is required. In those cases, real‑gas behavior and temperature‑dependent heat capacities can shift U from the constant‑f estimate.