# Notation for specific/volumetric entropy

I'm working on a problem related to entropy, and for this particular use case it's advantageous to write down properties in 'specific' units. In the case of entropy (with units Joules over Kelvin, I think is commonly referred to as: $[S] = \pu{J K-1}$), I want to know how to communicate volumetric entropy, i.e. Joules per cubic meter Kelvin: $[\color{red}{?}] = \pu{J K-1 m-3 }$.

Is it something like $\bar{S}$?

• Are you looking for an analogue to specific volume or the specific gas constant? Nov 25, 2017 at 23:58
• No I'm wondering if the notation is different for the intrinsic property of entropy. Specific entropy is useful in fields such as chemistry and thermodynamics to compare substances. Here's a link to a thermodynamic table using specific entropy (in that case, kg in the denominator) Nov 26, 2017 at 0:57
• I'd definitely go with the IUPAC answer as a default, but just so you know, a lot of times people will use a \hat character over the (capital) symbol to denote "specific". That's pretty common in introductory thermodynamics textbooks. Koretsky's text comes to mind. Nov 26, 2017 at 6:02

### Specific entropy

According to IUPAC "Green Book" Quantities, units, and symbols in physical chemistry, specific entropy is denoted as lowercase latin "s": $s$ [1, p. 56], whereas $\bar{S}$ would refer to molar entropy:

\begin{array}{lll} \text{Name} & \text{Symbol} & \text{Definition} & \text{SI unit} & \text{Notes} \\ \hline [...]\\ \text{molar quantity}~X & X_\mathrm{m}, (\bar{X}) & X_\mathrm{m} = X/n & [X]/\pu{mol} & 5,6 \\ \text{specific quantity}~X & x & x = X/m & [X]/\pu{kg} & 5,6 \\ [...]\\ \end{array} [...]

$(5)$ The definition applies to pure substance. However, the concept of molar and specific quantities (see Section 1.4. p. 6) may also be applied to mixtures, n is the amount of substance (see Section 2.10, notes 1 and 2, p. 47).

$(6)$ $X$ is an extensive quantity, whose SI unit is $[X]$. In the case of molar quantities the entities should be specified.

Example $V_\mathrm{m,\ce{B}} = V_\mathrm{m}(\ce{B}) = V/n$ denotes the molar volume of $\ce{В}$.

Just as specific heat capacity $c$, specific entropy $s$ is measured in $\pu{J K-1 kg-1}$ [1, p. 90].

General note [1, p. 6]:

The adjective specific before the name of an extensive quantity is used to mean divided by mass. When the symbol for the extensive quantity is a capital letter, the symbol used for the specific quantity is often the corresponding lower case letter.

### Volumetric entropy

Quite often there is no special notation used, and volumetric entropy is denoted with $S$. Specifically, volumetric entropy generation rate for the convective heat transfer in flowing viscous fluid is often denoted with S-prime notation, e.g. from classical paper "A Study of Entropy Generation in Fundamental Convective Heat Transfer" :

\begin{align} & S' & [\pu{W K-1 m-1}] \\ & S'' & [\pu{W K-1 m-2}] \\ & S''' & [\pu{W K-1 m-3}] \end{align}

### References

1. IUPAC “Green Book”. Quantities, units, and symbols in physical chemistry, 3rd ed.; Cohen, R. E., Mills, I., Eds.; IUPAC Recommendations; RSC Pub: Cambridge, UK, 2007. ISBN 978-0-85404-433-7.
2. Bejan, A. J. Heat Transfer 1979, 101 (4), 718–725. DOI: 10.1115/1.3451063.