Physics and mechanics
Ideal Gas Energy Calculator
Calculate ideal-gas internal-energy and enthalpy change, then distinguish rigid-volume heat from constant-pressure heat and boundary work with one signed first-law ledger.
CURRENT MODEL
Enter the declared physical case
Thermodynamics students, laboratory teams, and preliminary process analysts comparing a rigid vessel with a quasi-static constant-pressure piston process.
Decision supportedDetermine how a specified temperature change divides heat input between stored internal energy and ideal-gas boundary work for one declared process constraint.
LIVE PHYSICAL ANALYSIS
Where the current heat transfer goes
The signed live energy ledger separates internal-energy change and boundary work, then reconciles their sum with heat into the gas.
| Quantity | Expression | Current value | Unit |
|---|
How to use
Choose the process before interpreting heat and work
- Enter the fixed amount of gas in mol or kmol.
- Enter initial and final equilibrium temperatures on one declared scale.
- Provide a molar Cv appropriate to the gas composition and temperature interval.
- Select rigid constant volume or quasi-static constant pressure.
- Use the signed cards to distinguish deltaU, work by the gas, heat into the gas, and deltaH.
- Verify Q - W - deltaU closes before exporting or comparing process duties.
Ideal-gas energy fundamentals
Five distinctions in the first-law ledger
- Internal energy U
- A state property whose ideal-gas change depends on temperature through Cv.
- Enthalpy H
- The state property U + PV, with ideal-gas change nCp deltaT.
- Heat Q
- Energy crossing the boundary because of a temperature difference; positive into the gas here.
- Boundary work W
- Energy transferred as a moving boundary; positive when done by the gas here.
- Rigid volume
- A fixed boundary with dV = 0 and therefore zero pressure-volume work.
- Constant pressure
- A quasi-static ideal-gas path where W = nR deltaT and Q = deltaH.
Calculation method
Hold endpoint state functions apart from path transfers
The model first converts both endpoint temperatures to Kelvin and forms deltaT. It calculates Cp from Cv + R, then obtains deltaU and deltaH regardless of process because they depend only on endpoint states under the constant-heat-capacity assumption.
The process choice affects boundary work and therefore heat. A rigid vessel has W = 0 and Q = deltaU; a quasi-static constant-pressure path has W = nR deltaT and Q = deltaH. The first-law residual independently reconciles the signed terms.
Heat capacity is a property model
The default Cv is near a room-temperature diatomic value, not a universal constant for all gases. Wide temperature intervals require composition-specific Cv(T) integration.
Negative energy terms
Cooling gives negative deltaT, so deltaU, deltaH, and heat are negative. At constant pressure, W is also negative because the surroundings do work on the contracting gas.
Closed system boundary
The page follows a fixed amount of gas. Open heaters, compressors, and turbines need mass-flow enthalpy, shaft work, and kinetic and potential energy terms.
Detailed calculation process
Symbols, current substitution, intermediate quantities, and reconciliation
deltaU = n Cv deltaT; Cp = Cv + R; Q = deltaU + W; W = n R deltaT at constant pressureThe model converts temperatures to Kelvin, retains unrounded joules, and applies the sign convention Q into the gas positive and W by the gas positive. Cards display kilojoules to four decimals.
| Symbol | Meaning | Default | Unit |
|---|---|---|---|
| n | Fixed amount of gas | 2 | mol |
| T1 / T2 | Endpoint temperatures | 25 / 125 | deg C |
| Cv | Molar heat capacity at constant volume | 20.786 | J/(mol K) |
| Cp | Cv + R | calculated | J/(mol K) |
| deltaU | Internal-energy change | calculated | J |
| W | Boundary work done by gas | calculated | J |
| Q | Heat transferred into gas | calculated | J |
| deltaH | Enthalpy change | calculated | J |
Waiting for valid inputs.
Evidence to retain
Keep process constraint and property basis
Record gas composition, amount basis, endpoint temperatures and uncertainty, Cv source and valid range, whether the boundary was truly rigid or maintained constant pressure, evidence of equilibrium, heat losses, pressure history, and any work modes excluded from the ledger.
Scope and limitations
What this energy ledger excludes
- Temperature-dependent heat capacities unless represented by a justified average Cv
- Real-gas residual internal energy and enthalpy
- Mass flow, shaft work, stirring, electrical work, and reactions
- Kinetic, potential, surface, and phase-change energy
- Irreversible non-quasi-static constant-pressure work
- Equipment sizing, relief, code, or safety certification
A closed fixed amount of calorically perfect ideal gas, constant entered molar Cv, Cp = Cv + R, uniform equilibrium endpoint temperatures, and either rigid constant volume or quasi-static constant pressure. No mass flow, reactions, phase change, kinetic or potential energy, or temperature-dependent heat capacity.
Key terminology
Ideal-gas energy glossary
- Calorically perfect gas
- An ideal-gas model with heat capacities treated as constant over the interval.
- State function
- A property change determined by endpoints rather than the detailed path.
- Isochoric process
- A constant-volume path with zero pressure-volume boundary work.
- Isobaric process
- A constant-pressure path; quasi-static boundary work equals nR deltaT for an ideal gas.
- Heat-capacity ratio gamma
- Cp/Cv, reported from the entered Cv and ideal relation Cp = Cv + R.
- First-law residual
- The numerical closure Q - W - deltaU under the declared sign convention.
Practical cases
Two heating paths with different duties
Rigid calibration vessel
A sealed fixed-volume gas cell is heated between two equilibrium temperatures. The boundary does no displacement work, so the idealized heat input equals deltaU while pressure changes with temperature.
Weighted piston heating
A low-friction piston rises slowly against a constant load. For the same amount, Cv, and deltaT, deltaU matches the rigid case, but extra heat supplies positive boundary work and Q equals deltaH.
Important note
Do not infer equipment duty without loss and path data
The reported Q is an idealized gas-boundary transfer. Real heater duty can include vessel thermal mass, losses, nonuniform states, property variation, controls, and transient operation. Obtain qualified thermodynamic and pressure-system review for design decisions.
Frequently asked questions
Why is internal-energy change the same for both process choices?
For a calorically perfect ideal gas, deltaU = n Cv deltaT depends only on endpoint temperature and amount. The process changes heat and work, not the state-function difference.
Why does constant-pressure heat equal enthalpy change?
For a closed ideal gas doing only quasi-static pressure-volume work at constant pressure, W = nR deltaT. Adding that work to deltaU gives n(Cv + R)deltaT = nCp deltaT = deltaH.
What sign convention does the page use?
Heat entering the gas is positive and work done by the gas is positive. Therefore the first law is deltaU = Q - W, or Q = deltaU + W.
Can Cv stay constant over any temperature range?
No. Real-gas heat capacities vary with temperature and composition, especially when vibrational modes become active. Integrate Cv(T) for a wide or high-accuracy interval.
Does constant volume mean pressure stays constant?
No. For fixed n and V, pressure changes in proportion to absolute temperature under the ideal-gas law. The boundary simply does no displacement work.
Can this model a compressor, turbine, or open heater?
Not directly. Those are control-volume problems with mass flow and usually enthalpy transport, shaft work, kinetic energy, and efficiency terms.
Authority and follow-on work
Reliable sources and related calculators
- OpenStax University Physics: Heat Capacities of an Ideal GasDerives deltaU = nCv deltaT and Cp = Cv + R for ideal gases.
- OpenStax College Physics: First Law of ThermodynamicsDefines deltaU = Q - W with work done by the system positive.
- NASA Glenn: Ideal Gases under Process ConstraintsRelates constant-volume and constant-pressure ideal-gas heat, work, and heat capacities.
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