Internal Energy Calculator
Enter degrees of freedom, moles and temperature to compute the internal energy of an ideal gas U = (f/2)·n·R·T.
Input Data
Results
At a glance:For an ideal gas, internal energy U is the total microscopic kinetic energy of its molecules: U = (f/2)·n·R·T, where f is the number of quadratic degrees of freedom (3 for monatomic, 5 for linear diatomic at room temp, 6 or 7 for nonlinear/polyatomic with vibration), n moles, R the gas constant and T temperature. U depends only on temperature for an ideal gas (not on volume). The constant-volume molar heat capacity is Cv = (f/2)·R, constant-pressure Cp = Cv + R (Mayer's relation), and the specific heat ratio γ = Cp/Cv = (f+2)/f. These govern adiabatic relations (PV^γ = const).
Formula
U = (f/2)·n·R·T
Cv = (f/2)·R
Cp = Cv + R
γ = Cp/Cv = (f+2)/f
$$U = \frac{f}{2} n R T, \quad C_v = \frac{f}{2} R, \quad C_p = C_v + R, \quad \gamma = \frac{C_p}{C_v}$$How to Use
- Enter degrees of freedom f, moles n and temperature T (K).
- The calculator returns U, Cv, Cp and γ.
Case Studies
Monatomic gas
f = 3, n = 2 mol, T = 300 K.
U = 1.5×2×8.314×300 ≈ 7483 J.
Cv = 12.5 J/mol·K, γ = 5/3.
FAQ
Why is internal energy independent of volume for an ideal gas?
Ideal gas molecules have no intermolecular potential energy; U is purely kinetic and set by temperature alone. Real gases gain volume dependence from interactions.
What is γ used for?
γ = Cp/Cv appears in adiabatic processes (PV^γ = const, TV^(γ−1) = const) and the speed of sound in gases (c = √(γRT/M)).
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References
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.