[[Thermodynamics MOC]]
# First Law of Thermodynamics
The first law is a statement of [[Conservation of energy]] in a thermodynamic system.
> The change of internal energy of a system 
> is the sum of the [[Heat]] added to and [[Work]] done on the system, 
> and depends only on the initial and final states of the system 
> and not on the particular process involved.

For any infinitesimal process on a [[Thermodynamic system]] the change in internal energy is given by[^IUPAC]
$$
\begin{align*}
dE = \dj Q + \dj W + \sum_{i}\mu_{i} \,dN_{i}
\end{align*}
$$
where

- $\dj Q$ is the infinitesimal heat added to the system;
- $\dj W$ is the infinitesimal work done on the system;
- $\mu_{i}$ is the chemical potential of species $i$; and
- $dN_{i}$ is the infinitesimal change in the number of particles of species $i$.

In the case of a [[Quasistatic process]], the [[Quasistatic expression for work]] and definition of [[Entropy]] give
$$
\begin{align*}
dE = T\,dS - p\,dV + \sum_{i}\mu_{i} \,dN_{i}
\end{align*}
$$
^Quasistatic

[^IUPAC]: Here we use the IUPAC signing convention.

## Special cases

- [[First law for isovolumetric processes]]
-  [[First law for isobaric processes]]
-  [[First law for isothermal processes]] 

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