Van der Waals molar volume calculator

When the iteration fails to converge—singular derivative, escape from the physical domain Vm greater than b, or residual still large after the iteration budget—the API returns a clear invalid_input error instead of a fabricated volume.

What the van der Waals equation solves for molar volume

Convergence yields a physical root with Vm greater than b; failure to converge is reported as invalid_input so you never treat a stalled solver as a measured volume.

How to call the API and what comes back

Pressure, temperature, b, and R must be strictly positive; a must be non-negative. Send a JSON body with four required finite numbers: p (aliases pressure and P), t (aliases temperature, T, temp, and kelvin), a (aliases vdw_a, constant_a, and A), and b (aliases vdw_b, constant_b, and B). Optional r (aliases R, gas_constant, and gasConstant) defaults to 0.0831446261815324 for litre–bar chemistry problems; pass 8.314462618 when your a, b, and P are in SI. Absolute values must stay within the published limit of one trillion so intermediate products remain finite in IEEE double precision on the Worker and in the browser. On success the response echoes rounded p, t, a, b, and r, sets molar_volume and vm to the same converged Vm, ideal_molar_volume to R T over P, compressibility_factor and z to P times Vm over R T, iterations to the Newton count, formula and equation to the van der Waals string, message to a short English summary, and steps to an ordered array of substitution lines. Failed validation—missing fields, non-finite numbers, values outside absolute limits, non-positive P or T or b or R, negative a, or iteration that fails to converge—returns invalid_input and is never charged. Pricing is a flat $0.002 per successful request with unit item and zero marginal unit price.

Practical tips, limits, and related searches

Prefer sending numbers rather than pre-formatted strings when you control the client; string numbers are accepted only when they match a strict finite numeric pattern. Near critical points or at very high pressure the physical root can sit close to b, so the solver damps Newton steps that would jump below the excluded volume and still reports non-convergence if no stable root appears within one hundred iterations. Prefer this capability when you already have P, T, a, and b and need one deterministic molar volume; if your source only has critical temperature and pressure, convert to a and b with the usual van der Waals critical relations first. Keep P, T, a, b, and R in one consistent unit system before you call the API; mixing bar pressures with SI a and b is the most common reason residual stays large and the iteration fails to converge. When teaching, pair examples: carbon dioxide near room temperature with tabulated a and b against the ideal Vm equals R T over P to show Z not equal to one; a deliberate missing pressure or zero temperature to demonstrate invalid_input; and a second gas such as nitrogen or methane from the same table so students see that a and b, not the algorithm, carry substance identity. This landing targets people searching for a van der Waals molar volume calculator, real gas Vm from a and b, or van der Waals equation of state solver online; it is not a multi-root cubic root finder that lists all three mathematical roots, not a full equation-of-state library (Peng–Robinson, Redlich–Kwong), and not an ideal-gas-only rearrange. Failed tasks that return invalid_input do not consume prepaid balance, which keeps exploratory scripts cheap when learners deliberately break convergence. Round-trip checks are easy: recompute (P plus a over Vm squared) times (Vm minus b) and confirm it equals R T within the same twelve-decimal rounding, and confirm that Z equals P times Vm over R T matches the published compressibility_factor field.