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Speculation on Gibbs Paradox for an Ideal Gas and the Ideal Gas Law Francesco R. Ruggeri Hanwell, N.B. Nov. 4, 2025 Following (1), we consider the Gibb’s Paradox for a system with two adjacent volumes V each with N particles and temperature T. We assume an ideal gas. If the two gases consist of different molecules, then using the Sackur-Tetrode equation, and lifting the wall/partition separating the two V’s leads to no work being done and no change the overall (total) volume, but changes the total entropy by 2N ln(2). If the gases are the same, however, there should be no change in entropy according to (1). We suggest that the Sackur-Tetrode equation is based on the first law of thermodynamics: dE= TdS - PdV as well as the ideal gas equations E=3/2 N RT and PV = NkT. We argue that the Sackur-Tetrode equation contains the variable V (volume), but this volume is only applicable if it is linked to a PV=nRT action. The reason for this is that one may derive the Sackur-Tetrode equation directly from the first law and the two idea gas equations and so an V in entropy must be consistent with a V which appears in a PV=nRT process. If the volume is changed in a manner inconsistent with a PV=nRT process, one should not use the Sackur-Tetrode equation, we argue as this is inconsistent with its derivation. In other words, one may have a gas in one of the V boxes and no gas in the other. Lifting the wall leads to an irreversible process, but it can be mapped to an infinitesimal PV=nRT one which is often done in textbooks. One starts with one system and ends with one system. In the case of gases in one box or the other with the same T and N, we argue that lifting the wall is not linked with any kind of PV=nRT process. In such a case, one starts with two systems and ends up with one system, but this is not linked to the first law of thermodynamics because one has a change in volume which is not linked in any to a PV=nRT process. One cannot move the same wall between the two volumes to the right and left simultaneously. Thus, we argue that there is no reason to compare S1+S2 using the Sackur-Tetrode equation for the two system situation to the S for the single 2V system. The two scenarios are not linked by a PV=nRT process. Sackur-Tetrode Equation The Sackur-Tetrode equation is given in (2) as: S/(kN) = { V/N (b U/N) power 3/2 } + 5/2 ((1)) This holds for an ideal gas for which: E = 3/NRT ((2)) and PV = NkT ((3)) In addition, the first law of thermodynamics must also hold: dE = T dS - P dV ((4))
By using ((2)) and ((3)) together with ((4)), one may obtain the essential structure of ((1)). If dV=0, and dN=0 then: dE = 3/2 N dT = T dS → dS goes as 3/2 dT/T or S goes as ln(T) power 3 /2 ((5)) For dE=0, dN=0 —> TdS = PdV = NkT dV/V so S goes as ln(V) ((6)) The point we make for ((6)) is that the volume V which appears in the Sackur-Tetrode equation must be directly linked to a PV = nRT process for an ideal gas. One cannot simply use any change in volume, we argue, and this we suggest is the reason for Gibb’s paradox. In particular, if one has a volume 2V separated in the middle by a partition and N molecules of one gas with T sit in the right hand V and a different gas with N molecules with T, in the left, then using the Sackur-Tetrode equation, one might write: Change in S if the partition is removed = (S1+S2) final - (S1+S2) initial = 2N ln(2V) - 2Nln(V) = 2N ln(2) ((6)) This result, however, holds for both the case of identical molecules and different ones in each box, leading to a paradox as noted in (1). As a result, some suggest considering the notion of identical particles when defining entropy. We suggest here that given that the Sackur-Tetrode equation follows form the first law of thermodynamics and the equations of an ideal case ((2)), ((3)), one may only consider V changes in S (using the Sackur-Tetrode equation) which are linked to PV=nRT scenarios. Otherwise one contradicts the derivation of the Sackur-Tetrode equation. As an example, one may consider a gas in one V box and none in the second. Removing the partition is an irreversible process, but one may arrive at the same end state by considering small PV=nRT changes with T kept constant (i.e. heat exchange). In such a case, one has a single state at the beginning and a single one at the end and there is a way to link the two using PV=nRT for infinitesimal dV changes. Such is not the case for the removal of a partition separating the two V boxes. There is no PV=nRT link and so one should not use the Sackur-Tetrode equation with respect to V because such a change in V is not one compatible with PV=nRT. One cannot move the partition both to the left and right to accommodate both gases. As a result, we suggest that in the ideal gas case and given the derivation of the Sackur-Tetrode equation, it is incompatible to consider S1+S2, each with V, and then S with 2V. The volume change is not one associated with a PV=nRT set of changes in volume in any way, we argue. Conclusion In conclusion, we speculate that the Gibb’s paradox arises because one considers volume changes which are not compatible with the derivation of the Sackur-Tetrode equation. We suggest that the Sackur-Tetrode equation for an ideal gas must follow from E=3/2NRT, PV=NkT
and dE= TdS - PdV. This leads directly to the S being proportional to ln(V) and ln(T power 3/2) as we show above. The point we make is that ln(V) in S only applies if one is able to consider PV=nRT consistent changes in V. This may be done for a gas in a V with no gas in the adjacent V. Removing the partition leads to an irreversible process, but one may consider the end state in terms of infinitesimal dV changes compatible with PV=nRT. Such cannot be done with the removal of a partition between two gases with the same N and T whether they are identical or not. We suggest that one cannot simply use different volume values in the Sackur-Tetrode equation unless there is a PV=nRT compatible V changing process which may be used to obtain the end state. In the case of lifting the partition it does not seem that there is any such PV=nRT process and so we question why different V values are used in the Sackur-Tetrode equation when they are linked to PV=nRT which is required to derive the V dependence of S in the first place. References 1.McGlynn, E. Reminder of Jaynes’ Views on two key issues of Gibbsian Statistical Mechanics (2025) https://iopscience.iop.org/article/10.1088/1361-6404/ae0dca/pdf 2 .https://en.wikipedia.org/wiki/Sackur%E2%80%93Tetrode_equation
