First Law Solver

• •Use the First Law Solver to check your hand calculations on homework or exam problems — comparing its result with your own working helps identify arithmetic errors before they propagate.
• •Use it during the early design phase to rapidly iterate on parameters and narrow down feasible configurations before committing time to detailed finite element simulations or full design packages.
• •Use it when reviewing a colleague's calculation or checking a vendor's data sheet for plausibility — a quick sanity check can prevent costly downstream errors.
• •Use it to generate reference data for a technical report or presentation without manual computation, ensuring consistent, reproducible numbers throughout the document.
• •Use it in the field when a quick estimate is needed and a full engineering software package is not available.

The First Law Solver is a precision engineering calculation tool designed for students, engineers, and technical professionals. Solve Q - W = ΔU (closed) or Q - W = ΔH (open) with IAPWS-IF97 water and steam states and process type selection (adiabatic, isentropic, isothermal, isobaric, isochoric) All calculations are performed using established engineering formulas from the relevant scientific literature and standards. Follow the units printed beside each field. Where a unit selector is provided, the selected unit is converted internally. Results are computed in your browser without sending data to a server, which keeps your inputs private. This calculator is intended as a supplementary tool for learning and design exploration; always verify results against authoritative references for safety-critical applications.

The first law of thermodynamics for a closed system is Q − W = ΔU, where Q is heat added (positive in, negative out), W is work done BY the system (positive out, negative in), and ΔU is the change in internal energy of the contents. For an open system (control volume) with flow in and out, the steady-state form is Q − W = Δh + ΔKE + ΔPE (per unit mass), where Δh is the enthalpy change across the control volume, ΔKE is the kinetic energy change, and ΔPE is the potential energy change. Kinetic and potential energy terms are often negligible in power and refrigeration analysis and are commonly dropped: Q − W = Δh. The first law is conservation of energy applied to thermodynamic systems — it is always true and is one of the two fundamental laws of thermodynamics (along with the second law). Applying it to a problem requires: (1) identifying the system (closed or open, control mass or control volume), (2) identifying the heat and work interactions at the boundaries, (3) computing the changes in state properties between initial and final states (or inlet and outlet for steady-flow), and (4) solving for the unknown. Common problems include: heating water in a closed tank (ΔU = Q); work done by expanding gas (W = ∫pdv for a piston-cylinder); turbine and compressor work analysis (W_s = −Δh for adiabatic steady flow); heat exchanger energy balance (Q = ṁ·Δh). The calculator handles closed-system and open-system first-law problems for water and steam, evaluating each state with IAPWS-IF97 from any two of T, P and quality x. Isentropic processes fix s₂ = s₁, isochoric processes fix v₂ = v₁, and isothermal and isobaric processes fix T or P; given the states and one of Q or W (when the process does not fix it), it solves for the other.

• •Closed-system heat addition: for water heated in a rigid sealed tank (choose Isochoric), enter the initial state and the final temperature; the calculator holds v₂ = v₁, finds the final pressure and returns Q = m·(u₂ − u₁).
• •Boundary work in piston-cylinder processes: at constant pressure the boundary work is W = m·P·(v₂ − v₁), using the v₁ and v₂ the calculator reports; enter that W to get the heat transfer from Q = ΔU + W.
• •Turbine steady-flow analysis: compute turbine work output W_s = h_inlet − h_outlet per unit mass flow, assuming adiabatic operation and negligible kinetic/potential energy changes.
• •Heat exchanger energy balance: on each side of a heat exchanger, Q = ṁ·Δh. For a counter-flow heat exchanger transferring heat from hot to cold stream, the two energy balances are equal in magnitude and opposite in sign.
• •Thermodynamic cycle analysis: apply the first law to each process in a cycle (compression, heat addition, expansion, heat rejection) to track the energy flow and compute cycle efficiency.