# Why was a Plimsoll symbol chosen to indicate standard state?

Historically, the Plimsoll symbol (aka Plimsoll line) was created as hull mark that would serve as a ready indicator of whether a ship was overloaded and thus running too low in the water. It was later adopted by chemists to indicate standard state (currently defined by IUPAC as 1 bar).

Why was the Plimsoll symbol chose for this purpose? According to Wikipedia,

The plimsoll symbol (⦵ or o) ... is used as a superscript in the notation of thermodynamics to indicate an arbitrarily chosen non-zero reference point ("standard state").

The language used by Wikipedia suggests it was chosen because it looks like a zero with a line through it, thus communicating "not zero". Applying this to enthalpy or free energy seems redundant and unnecessary, though, since by definition there is no true zero reference point for either (among state functions, this exists only for entropy). It's like having a special symbol attached to $$ΔU$$ to indicate that $$U$$ is conserved, when, of course, $$U$$ is always conserved.

Indeed, the problem with attaching a symbol to $$U$$ saying it's conserved is that it suggests that sometimes $$U$$ is not conserved (otherwise why would you need the symbol?). Likewise, attaching a symbol to $$U$$, $$H$$, or $$G$$ saying that the reference state is non-zero suggests sometimes it could be zero. I.e., I understand they needed a symbol for reference state. But I'm puzzled why they chose one that communicated "non-zero" instead of just choosing something that meant "reference", period.

Nevertheless, I'm sure these people knew what they were doing, and had good reasons for what they did. I just don't understand what that those reasons would be. I'd thus be interested to hear from anyone out there with knowledge of the history of this.

Perhaps the answer (here I speculate) is that they wanted to communicate the obvious (that thermodynamic reference states can never be true zero reference states) — and that the reason they wanted to do this was because they anticipated tables of thermodynamic values would be used by a broad audience, some of whom might not understand the underlying thermodynamic theory.

• @TryHard Note that in the citation, there was missing strike (horizontal bar) in the second symbol. (Now fixed.) – mykhal Jan 27 at 11:20
• Plimsoll symbol was commonly used in the literature in XIX century to show the non-zero nature of standard values. It's ironic that due to typographical reasons that plimsoll symbol is often replaced by a zero sign in the superscript (ligatures for upper 0s were everywhere, but upper ⦵s were not so ubiquitous). – andselisk Jan 27 at 11:20
• The reference in Wikipedia for the history of this notation is "Prigogine, I. & Defay, R. (1954) Chemical thermodynamics, p. xxiv". Does anyone have a copy of this text? Does it cite a primary source? – Andrew Jan 27 at 12:28
• Also, is there a geographical preference for the Plimsoll? I've never seen it in my US texts, which use the open circle "$^\circ$". – Andrew Jan 27 at 12:29
• Atkins (Oxford U.) seems to use the plimsoll exclusively in his textbook. – Night Writer Jan 27 at 12:56

Prigogine's Chemical Thermodynamics gives a bit more context as to why a plimsoll symbol is somewhat more preferable and is indeed used to denote non-zero nature of standard values [1, pp. xxiv, 86]:

The properties of a pure substance are denoted by a superscript $$^0$$. The use of this symbol to denote "standard" thermodynamic quantities is then permissible only when these refer to the pure substance as standard state. Some other convention is needed to denote the more general "standard quantities" defined in equation (7.51), where the standard state may for example refer to an infinitely dilute solution. After much consideration the symbol $$^⦵$$ has been introduced. This symbol is based on the circle, and is thus closely related to the more common (but occasionally confusing) notation for standard quantities. It seems useful, however, to associate it with the plimsoll mark which, appropriately enough, refers to a reference state of loading of a ship; this use of an ideogram has, we believe, some value in keeping the essential nature of "standard states" in prominence.

Quantities denoted by the plimsoll symbol are still, in general, functions of temperature and pressure; the idea of "standardization" of temperature and pressure is not stressed. When necessary we denote quantities at the standard pressure (which for convenience is nearly always chosen as the unit of pressure) by the superscript $$^†$$, quantities at a standard volume (which for convenience is always chosen as the unit of volume) are denoted by $$^‡$$. Quantities referring to a (usually hypothetical) ideal system are denoted by $$^\mathrm{id}$$.

[...]

$$P = P^⦵(T,p) + P^M(T,p,x_1\ldots x_c) \tag{7.51}$$

It's also interesting that Lewis and Randall, known for defining the standard state in the beginning of XX century as we know it, used an upper circle [2, p.623]:

$$G^\circ$$ A property of a substance in its standard state.

However, they referred to it as to a standard reference state (distinguished from a standard state) throughout the textbook [2, p. 88, 99]:

While we cannot give the absolute magnitude of $$\bar{H}$$ for a substance in solution, we may ascertain how much greater or less this is than the heat content of the same substance in some chosen state. Thus, at any temperature, if we are dealing with the partial molal heat content of water in some solution in which water is the solvent, we may choose pure liquid water as the reference state, and denote its molal heat content by $$H_1^\circ$$.

[...]

It is highly convenient, especially for purposes of concise tabulation, to know the heats of reaction when various substances are formed from their elements. It is therefore desirable to choose some one standard reference state for each element. For this purpose, we shall, at all temperatures, take the element at a pressure of one atmosphere, and in that form which is most stable or most common at room temperature. Thus liquid mercury, gaseous oxygen, solid iodine, rhombic sulfur and graphitic carbon, all under a pressure of one atmos­phere, will be considered to be in their standard reference state.

### References

1. Prigogine, I.; Defay, R. Chemical Thermodynamics; Treatise On Thermodynamics; Longmans, Green & Co Ltd: Glasgow, 1954; Vol. 1.
2. Lewis, G. N.; Randall, M. Thermodynamics and the Free Energy of Chemical Substances, First Edition later Printing edition.; McGraw-Hill Book Company, 1923.
• What's the difference between a "standard state" and a "standard reference state"? Isn't use of the word "reference" redundant in that context? To quote eg Atkins: "the standard state of a substance at a specified temperature is its pure form at 1 bar pressure." – Night Writer Jan 27 at 12:58
• Does that mean that "reference" implies a different stated pressure? – Night Writer Jan 27 at 12:58
• @TryHard The term "standard reference state" arose from the need to use some starting point to quantitatively describe thermodynamic functions of state such as heat of formation. Regarding the pressure, well, Lewis used 1 atm instead of these days' 1 bar ( I added additional quote on that). – andselisk Jan 27 at 13:06
• Summarizing what Prigogine wrote, it seems the open circle meant "pure substance", and the horizontal bar was then added to indicate a "standard quantity" of that pure substance, rather than standard pressure (the latter instead being designated with a single dagger). Hence there was no notion of "not-zero" when the bar was originally introduced, and the Plimsoll was chosen simply because they needed to add something to the zero and, hey, there's already this good-looking compact symbol that does that and, in another well-known application, already has the connotation of "reference value". – theorist Jan 27 at 16:26
• And if my summary is correct, it must then be the case that the horizontal symbol's meaning was subsequently (between when Prigogine wrote this in 1954 and IUPAC's formal action in 1970) migrated from "standard quantity" to "standard state". And perhaps at that time the "not-zero" association was introduced? – theorist Jan 27 at 16:40