# Regarding the infinitesimal form of the First Law of Thermodynamics

First Law of Thermodynamics is expressed as[1]

The internal energy of an isolated system is constant.

If $$w$$ denotes the work done on a system, $$q$$ for the energy transferred as heat to the system, and $$\Delta U$$ for the resulting change in internal energy, then it follows that $$\Delta U = q + w \label{1}\tag{1}$$

$$\mathrm{d}U = \mathrm{d}q + \mathrm{d}w \label{2}\tag{2}$$

What I don't understand is how does $$\eqref{2}$$ follow mathematically from $$\eqref{1}$$. Taking the limit $$\Delta U \rightarrow 0$$ gives the left-hand side but how does $$q$$ change into $$\mathrm{d}q$$ and $$w$$ into $$\mathrm{d}w$$? I understand the interpretation of both the equations physically. Is this related to the fact that $$U$$ is a state function whereas $$q$$ and $$w$$ are not?

Reference:

[1] Atkins, P.; de Paula, J. Physical Chemistry, 10th ed; Oxford UP: Oxford, U.K., 2014.

• This is the reason why some authors do not write dq and dw, but use a sort of crossed symbol of d, that I am unable to represent here. It looks like a d with a superimposed / bar. – Maurice Feb 9 at 10:47
• It's just the authors playing fast and loose (inventing their own version) with the mathematics. – Chet Miller Feb 9 at 12:53
• Mathematically, neither dq neither dw are total differentials, but their sum is. – Poutnik Feb 9 at 12:56
• Some texts use $\delta$ to indicate inexact differentials. Not to be confused with $\partial$. – Andrew Feb 28 at 19:35

The first law of thermodynamics is a statement of energy conservation thus and defines the internal energy E as an extensive state function. In an infinitesimal transformation, the first then law reduces to $$\mathrm{d}E = \mathrm{d}Q + \mathrm{d}W$$
where $$\mathrm{d}E$$ is a total (exact) differential for infinitesimal transformation.
However, $$\mathrm{d}Q$$ and $$\mathrm{d}W$$ are not exact ($$Q$$ and $$W$$ are not state functions); $$Q$$ and $$W$$ in a thermodynamics transformation are process-dependent. All of these are properties of functions of more than one variables