REVIEW 4 major objections 5 minor 13 references
Revisiting the First, Second and Combined Laws of Thermodynamics
T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read Revising the first law for open systems puts a P V_i term into the chemical potential under hydrostatic work.
desk verdict A candid erratum that restores a standard textbook formula, but the argument oversells the conceptual change by attacking a strawman version of Hillert's first law. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the identity for hydrostatic work, $\mathrm{d}W = -P(\mathrm{d}V - \sum_i V_i\,\mathrm{d}N_i)$, derived by a two-step argument: first add matter to the system with no heat or work, so volume changes by $\sum_i V_i\,\mathrm{d}N_i$; then compress or expand the system to remove that volume change, doing work $P\sum_i V_i\,\mathrm{d}N_i$. Adding the two steps converts the mass-exchange term in the first law from $U_i\,\mathrm{d}N_i$ into $(U_i + P V_i)\,\mathrm{d}N_i$ and thereby places $P V_i$ inside the chemical potential.
What would settle it
Run a reversible open-system experiment at fixed temperature and pressure—for example, injecting a measured amount of component $i$ through a semipermeable boundary while measuring heat and work—and compare the inferred chemical potential with $U_i - T S_i + P V_i$ computed from separately measured partial molar energy, entropy, and volume; agreement with $U_i - T S_i$ instead would falsify Eq. 1 for that process.
Extended reading notes
Core claim
The central claim is that for a system under hydrostatic pressure the total work exchange is $\mathrm{d}W = -P(\mathrm{d}V - \sum_i V_i\,\mathrm{d}N_i)$, not $-P\,\mathrm{d}V$. Splitting mass addition and volume change into two steps—first adding matter with no heat or work, then compressing to restore the volume—shows that the first law for an open system is $\mathrm{d}U = \mathrm{d}Q - P\,\mathrm{d}V + \sum_i (U_i + P V_i)\,\mathrm{d}N_i$. Consequently the chemical potential appearing in the combined law $\mathrm{d}U = T\,\mathrm{d}S - P\,\mathrm{d}V + \sum_i \mu_i\,\mathrm{d}N_i - T\,\mathrm{d}_{\mathrm{ip}}S$ is $\mu_i = U_i - T S_i + P V_i$. The paper further claims that this chemical potential is not universal: when other kinds of work (mechanical, electric, magnetic) are present, the corresponding conjugate intensive variable times a partial extensive quantity joins the definition, with the most general form covering mechanical, electric, and magnetic work together.
Load-bearing premise
The derivation assumes that a system's volume is exactly the sum, over components, of each component's per-mole volume times its mole count, and that adding matter at pressure $P$ therefore does work $-P$ times that per-mole volume; if a real process changes volume differently, the $P V_i$ term in the chemical potential is not justified for that process.
Editorial extensions
If this is right
- For an open system under hydrostatic pressure, $\mu_i$ in the combined law must be read as $U_i - T S_i + P V_i$ when the work term is written $-P\,\mathrm{d}V$.
- The textbook form of the first law for open systems that treats all work as $-P\,\mathrm{d}V$ is inconsistent; the corrected form separates the $-P\,\mathrm{d}V$ term from the mass-addition work.
- The correction propagates into derived thermodynamic potentials: any quantity formed as $\mu_i - U_i + T S_i$ will differ from $P V_i$ for each component.
- Chemical potential becomes work-type-dependent: mechanical, electric, and magnetic work each add their intensive variable times a partial extensive property to the definition.
- The combined law still contains an entropy-production term $-T\,\mathrm{d}_{\mathrm{ip}}S$, so the revision does not change the second law's role in setting the direction of internal processes.
Reading between the lines
- One consequence the author leaves implicit is that chemical potentials tabulated from experiments under different work conditions (electrochemical cells versus high-pressure gas equilibria, for example) are not automatically comparable unless the $P V_i$ and analogous terms are removed.
- This suggests a direct high-pressure test: at pressures where $P V_i$ is a significant fraction of $\mu_i$, phase boundaries computed with and without the $P V_i$ term will diverge measurably, so existing high-pressure thermodynamic databases could be checked against this correction.
- The same two-step logic could be applied to systems where volume is not exactly additive under mixing (chemical contraction or expansion), revealing when the $P V_i$ correction needs to be replaced by an integral over the actual volume change.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This manuscript, authored by Z.-K. Liu, revisits the first, second, and combined laws of thermodynamics for open systems. It acknowledges an error in the author's earlier definition of chemical potential (Eq. 14 of ref. 6) and proposes to replace it by μ_i = U_i − T S_i + P V_i (Eq. 1). The paper defines the total work in hydrostatic processes as dW = −P(dV − Σ V_i dN_i) (Eq. 2) and writes the first law for open systems as dU = dQ − P dV + Σ (U_i + P V_i) dN_i (Eq. 4). It then presents a table of generalized chemical potentials for mechanical, electric, and magnetic work. The paper argues that because dW includes the flow-work term P V_i dN_i, the chemical potential gains the P V_i term, and it criticizes Hillert's formulation of the first law as incorrect.
Significance. The paper's central formula, Eq. 1, is the standard partial molar Gibbs energy for a homogeneous phase and is correct. The manuscript also honestly corrects the author's earlier publications, and the tabulations in Table 2 may be a useful pedagogical summary. However, the claimed novelty is largely a matter of bookkeeping: Eq. 4 is algebraically identical to the standard open-system first law dU = δQ − P dV + Σ H_i dN_i, with H_i = U_i + P V_i. The paper's derivation via the two-step thought experiment is not rigorous, and its assertion that Hillert's first law is incorrect is overstated. The contribution is therefore a clarification rather than a substantive revision of thermodynamics.
major comments (4)
- [Around Eq. 2 and Eq. 4] The redefinition of total work as dW = −P(dV − Σ V_i dN_i) is a bookkeeping convention, not a physical correction. The standard open-system first law dU = δQ − P dV + Σ H_i dN_i with H_i = U_i + P V_i is algebraically identical to Eq. 4. Therefore, the claim that Hillert's first law is 'incorrect' because it uses dW = −P dV for open systems is not supportable: Hillert's H_i dN_i term already contains the P V_i flow-work contribution. The manuscript should acknowledge that Eq. 2 is an alternative convention and that Eq. 4 changes no physical predictions relative to the standard form.
- [Around Eq. 3 and the two-step argument] The two-step Berry thought experiment, with the first step at dQ = 0 and dW = 0 and the second step a compression, is not a physically realizable sequence for a system at uniform T and P. Adding matter while keeping the system in equilibrium requires mass exchange with reservoirs and generally changes T and P; the steps are not independent equilibrium processes. The derivation of Eq. 2 from this thought experiment is therefore not a rigorous proof. The manuscript should either supply a more careful derivation from the extensive variables V(S, T, P, N_i) or state the assumed path and its limitations explicitly.
- [Table 2 and the universality of Eq. 1] The chemical potential μ_i = U_i − T S_i + P V_i is the standard partial molar Gibbs energy for a homogeneous phase, but its validity requires the additivity of partial molar volumes, V = Σ V_i N_i, and the existence of well-defined partial molar quantities. The manuscript presents Eq. 1 without this limitation and extends it to electric and magnetic work with new terms such as V E θ_i and V H B_i, which are not derived in the paper. These generalized expressions should be presented as conjectures or supported by explicit derivations, not asserted as established results.
- [Title, Abstract, and overall scope] The title and abstract claim a revision of the first, second, and combined laws of thermodynamics. The substantive content, however, is a correction of the author's own earlier Eq. 14 and a proposal to relabel the work term in the first law. The paper should state this scope at the outset, as the current framing overstates the novelty and may mislead readers into thinking that the fundamental equations themselves require revision.
minor comments (5)
- [Equations throughout] The equations are poorly typeset in the manuscript, with garbled symbols such as '−-𝑉!𝑑𝑁!' in Eq. 2 and unclear subscripts in several places; please ensure that all formulas are properly formatted and legible.
- [Table 1 footnotes] The definitions of U_i and S_i do not specify the independent variables held constant in the partial derivatives; please clarify, e.g., U_i = (∂U/∂N_i)_{T,P,N_j} or another appropriate set.
- [Reference 13] The reference to Berry et al. does not include a specific chapter, page, or equation number for the two-step argument; please provide a precise citation.
- [Notation for entropy production] The notation for entropy production, written as 'd_i^p S' or 'd_i^ip S' in different places, is inconsistent and should be unified and defined before first use.
- [Table 1 caption] The note about 'red and bold texts' is not meaningful in a black-and-white manuscript; please indicate the intended emphasis in the caption or replace the color reference with a clear explanation.
Circularity Check
No significant circularity: the derived chemical potential follows from standard thermodynamic identities and an explicitly stated work convention, not from fitting or from a load-bearing self-citation chain.
full rationale
The central relation, Eq. (1), is the standard Euler-relation identity μ_i = U_i − T S_i + P V_i for a homogeneous system with internal energy U(S,V,N). It is not an empirical prediction fitted to data, and it can be obtained directly from the combined law dU = T dS − P dV + Σ μ_i dN_i plus Euler's theorem, independently of the author's earlier Eqs. 13 and 14. Equation (2) is an explicitly stated convention for total hydrostatic work, dW = −P(dV − Σ V_i dN_i), justified by the partial-molar-volume change on mass exchange. Substituting that convention into the first and combined laws yields Eqs. (3), (4), and then Eq. (1) by algebra; the chain is input convention plus standard manipulation, not a definition of the output in terms of itself. The two-step argument from Berry et al. (ref. 13) is external textbook reasoning, not an author-uniqueness theorem, and the paper's abundant self-citations are historical/self-corrective: they identify the source of the earlier error rather than serve as authority for the corrected formula. The disagreement with Hillert is about whether PΣV_i dN_i is labeled as work or as part of the enthalpy flow; that is a convention dispute, and one may judge the labeling overstatement without finding circular reasoning. No equation reduces to its own input by construction, so the score is zero.
Assumptions & free parameters
assumptions (3)
- domain assumption Entropy S exists as a state function and the combined law dU = T dS - P dV + Σ μ_i dN_i - T d_i S holds for open systems.
- domain assumption The volume of a multicomponent system is additive over components with partial molar volumes V_i = (∂V/∂N_i)_{T,P,N_j}.
- domain assumption The work done by the surroundings when matter is added at constant pressure is -P V_i dN_i.
Cite this review
Pith. "Pith review of Revisiting the First, Second and Combined Laws of Thermodynamics." pith.science (2026). https://pith.science/paper/SQQMK2KB
@misc{pith2026250600055,
author = {Pith},
title = {Pith review of: Revisiting the First, Second and Combined Laws of Thermodynamics},
year = {2026},
howpublished = {\url{https://pith.science/paper/SQQMK2KB}},
note = {Machine review of arXiv:2506.00055}
}
read the original abstract
First, Second and Combined Laws of Thermodynamics are revised in terms of entropy change, partial entropy, partial volume, and chemical potential.
Reference graph
Works this paper leans on
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: Molar internal energy of the system • 𝐻
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Reviewed August 7, 2026 · model on record in the stance chip above.
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