The Form of the Fundamental Thermodynamic Relation per Unit Mass In the book Understanding Non-Equilibrium Thermodynamics written by Georgy Lebon and David Jou, the fundamental relation per unit mass is written in the form,
$$u=u(s,v,c_{1},...,c_{n})$$
where $c_{k}=m_{k}/m$ and $m$ is the total mass.
But after differentiation it reads as follows,
$$du=Tds-pdv+\sum_{k=1}^{n-1} (\bar \mu_{k}-\bar \mu_{n})dc_{k}$$
I have no idea why the last term is written like that instead of this
$$\sum_{k=1}^{n} \bar \mu_{k}dc_{k}$$
Thanks.
 A: The authors appear to be imposing the constraint that $\sum_k c_k = 1$ (which constraint follows from the conservation of mass.)  In particular, this means that
$$
\sum_{k=1}^n \mathrm{d}c_k = 0 \quad \Rightarrow \quad \mathrm{d}c_n = -\sum_{k=1}^{n-1} \mathrm{d}c_k.
$$
This relation allows us to show that the form you thought should be there is equivalent to the form used by the authors:
$$
\sum_{k=1}^{n} \bar \mu_{k}\mathrm{d}c_{k} = \left[\sum_{k=1}^{n-1} \bar \mu_{k}\mathrm{d}c_{k} \right] + \bar{\mu}_{n}\mathrm{d}c_{n} = \left[\sum_{k=1}^{n-1} \bar \mu_{k}\mathrm{d}c_{k} \right] - \bar{\mu}_{n}\sum_{k=1}^{n-1} \mathrm{d}c_k = \sum_{k=1}^{n-1} (\bar \mu_{k} - \bar{\mu}_n)\mathrm{d}c_{k} 
$$
as claimed.
(I should note that I do not have access to the full book, merely the previews available on Google Books.  In particular, I don't know why this latter form is used, and if the authors explain their motivation later in the book I may not be able to read it.  Thankfully, the relation itself was on p. 18, which is one of the pages I have access to.)
