Clausius Inequality Check Calculator

Example Data Table

Formula Used

The Clausius inequality expresses a strict constraint on heat exchange and entropy. For any cyclic process, ∮ δQ/T ≤ 0. With discrete heat transfers against reservoirs at temperatures T_i, this becomes: Σ(Q_i/T_i) ≤ 0.

For a general process between two states, the entropy change satisfies: ΔS ≥ ∫ δQ/T. Using segment data, the calculator checks: ΔS − Σ(Q_i/T_i) ≥ 0. Near equality indicates a nearly reversible process under the given data.

How to Use This Calculator

1. Select a check mode: cycle inequality or entropy bound.
2. Choose the temperature unit for your entered values.
3. Add one row per heat-transfer segment in the table.
4. Enter heat Q and reservoir temperature T.
5. If using entropy mode, enter the overall ΔS.
6. Submit to view results above the form and header.
7. Export CSV or PDF for documentation and sharing.

Interpretation Tips

• Use Q > 0 for heat added to the system.
• Always ensure temperatures are positive in Kelvin.
• Small positive sums can arise from rounding or mismatched data.
• Negative cycle sums indicate net entropy production.

Clausius Inequality Compliance Guide

1) Why this check matters in practice

Clausius inequality is a fast reality check for thermodynamic data. In experiments and audits, segmenting heat exchanges by reservoir temperature helps detect inconsistent signs, unit mistakes, and impossible “free energy” claims. This calculator turns your segment list into a single diagnostic quantity, Σ(Q/T).

2) Understanding the sign convention for Q

The calculator uses Q > 0 for heat added to the system and Q < 0 for heat rejected. For cycles, an incorrect sign on one segment can flip the conclusion. When troubleshooting, compare your segment list against the device boundary and measurement direction.

3) Temperature handling and Kelvin conversion

Clausius checks require absolute temperature. If you enter Celsius, the tool converts each segment using T(K)=T(°C)+273.15. Any segment with T≤0 K is invalid, because it would make Q/T undefined for physical reservoirs.

4) Cycle inequality with discrete segment data

For a cyclic process, the discrete form is Σ(Q_i/T_i) ≤ 0. If your computed sum is strongly negative, the data are consistent with irreversibility and entropy production. If the sum is near zero, the cycle is close to reversible within your tolerance and rounding.

5) Entropy bound mode when ΔS is available

For a process between states, the bound is ΔS − Σ(Q_i/T_i) ≥ 0. The calculator reports the gap. A positive gap indicates allowable entropy production; a negative gap indicates that the stated ΔS is too small for the supplied heat transfers at the given temperatures.

6) Typical magnitudes and what “small” means

Since Q/T has units of J/K, values depend on both energy scale and reservoir temperature. For example, Q=500 J at T=500 K contributes 1.0 J/K. A tolerance like 1e−9 is strict; many lab datasets prefer 1e−6 to 1e−3 depending on sensors.

7) Segment breakdown improves debugging

The segment table shows each Q/T term so you can spot outliers. Large positive terms often come from high heat input at low temperature. If a single row dominates the sum, re-check that row’s temperature unit, sign, and whether the reservoir is steady.

8) Reporting results with exports

Use CSV to archive computations with your raw segment list, and PDF to share a clean summary. For compliance notes, include the mode, tolerance, computed sum, and final status. In entropy mode, include the reported gap because it quantifies the required entropy production margin.

FAQs

1) What does Σ(Q/T) represent?

It is the discrete approximation to the integral of heat divided by absolute temperature. For cycles it must be non‑positive; for state changes it cannot exceed the entropy change ΔS.

2) Why must temperature be in Kelvin?

Because the inequality uses absolute temperature. Celsius is shifted and can be negative, which would distort Q/T. The calculator converts Celsius to Kelvin before computing each segment.

3) What should I do if the cycle check shows a violation?

Verify sign convention, check that each segment uses the correct reservoir temperature, and confirm units. A single wrong row can make Σ(Q/T) positive and trigger a violation.

4) How do I choose a reasonable tolerance?

Pick a tolerance consistent with measurement uncertainty. Many practical datasets use 1e−6 to 1e−3 J/K. Very small tolerances can classify rounding noise as a meaningful deviation.

5) What does “near‑reversible” mean here?

It means the inequality is satisfied and the computed sum or gap is close to zero within the chosen tolerance. It does not prove perfect reversibility; it indicates consistency with low dissipation.

6) Can Σ(Q/T) be negative even for good devices?

Yes. Negative values are expected for irreversible cycles and real processes. They indicate entropy production. Only positive values for a cycle, or negative gaps in entropy mode, are problematic.

7) Do I need to include work or pressure‑volume data?

No. This check uses only heat transfers and reservoir temperatures (and optionally ΔS). Work affects energy balance, but Clausius inequality evaluates entropy consistency of heat exchange.