Compute unknown thermodynamic quantity (Change in Internal Energy, Heat Transfer, or Work Done) using the energy conservation principle.
The First Law of Thermodynamics is a formulation of the principle of energy conservation for thermodynamic systems. It states that the increase in internal energy of a closed system equals the heat added to the system minus the work done by the system: ΔU = Q – W. This law bridges heat transfer and mechanical work, establishing that energy cannot be created or destroyed — only converted from one form to another.
ΔU = Q − W
Where: ΔU = Internal energy change [J], Q = Net heat transfer [J], W = Work done by system [J]
Pioneering work by James Prescott Joule, Julius Robert von Mayer, and Hermann von Helmholtz in the 1840s established the equivalence between heat and work. Joule's paddle-wheel experiment demonstrated that mechanical work can be converted into heat, proving the mechanical equivalent of heat. The First Law formally rejects the possibility of a perpetual motion machine of the first kind (a device producing work without energy input). Today, it is fundamental to engine design, refrigeration cycles, chemical reactions, and atmospheric science.
| Process | Constraint | First Law Simplification | Engineering Example |
|---|---|---|---|
| Isochoric | Constant volume → W = 0 | ΔU = Q | Heating gas in a rigid container (engine cylinder at top dead center). |
| Isobaric | Constant pressure | Q = ΔU + PΔV | Boiling water, piston-cylinder with constant load. e.g., ΔU = 200 J, W = 80 J → Q = 280 J. |
| Isothermal (ideal gas) | ΔU = 0 (ideal gas only) | Q = W | Slow expansion/compression in contact with heat reservoir. |
| Adiabatic | Q = 0 | ΔU = –W | Rapid expansion in a gas turbine, compression stroke in diesel engines. |
During the power stroke, hot gases expand nearly adiabatically (Q ≈ 0). If the gas does 750 J of work on the piston, the internal energy decreases by 750 J (ΔU = –W). This converted energy becomes mechanical work propelling the vehicle. Using our calculator: set unknown ΔU, Q = 0, W = 750 J → ΔU = –750 J, confirming energy conservation. Design of thermal efficiency directly depends on this principle.
From ΔU = Q – W, solving for any unknown is straightforward algebra:
⚠️ Important: Many textbooks use the sign convention “W = work done BY system”. Some engineering contexts may define W as work done ON system. Our calculator follows the IUPAC/IUPAP convention: ΔU = Q – W (W > 0 means system does work on surroundings). If your problem uses opposite sign, simply invert the sign of W accordingly.
Ideal Gas Internal Energy: For an ideal monatomic gas, ΔU = (3/2)nR ΔT. For diatomic, ΔU = (5/2)nR ΔT. Combined with first law, you can derive temperature changes or heat capacity.
For open systems (mass flow across boundaries), the First Law extends to include flow work and enthalpy. The steady-flow energy equation (SFEE) is: Q – W = ṁ(h₂ – h₁ + ½Δv² + gΔz). Our calculator focuses on closed systems, but the fundamental conservation principle remains identical. For steady‑flow devices (turbines, compressors, nozzles), the same calculator can be used if you substitute ΔU with ΔH (enthalpy change) and include kinetic/potential energy corrections where needed.