First Law of Thermodynamics

The first law of thermodynamics has been first proposed by MAYER^2.8 in 1841 [74]: ``Heat is a kind of energy and can therefore neither be created nor destroyed.'' Hence, heat has been defined as the transfered thermal energy between two systems if they are brought into thermal contact [75]. The infinitesimal change of the internal energy $U$ of a system $\mathcal{S}$ can be expressed as

$$\mathrm{d}U {=}\partial{Q} - \partial{W},$$ (2.28)

where $\partial{Q}$ is the infinitesimal heat added to the system and $\partial{W}$ represents the infinitesimal work performed by the system $\mathcal{S}$ . This equation follows directly from the total differential of the entropy $\mathrm{d}{S}$

$$\mathrm{d}S {=}\mathrm{d}S(U,V,N) {=}$$

$${=}{\frac{\partial{S}}{\partial{U}}} \mathrm{d}U + {\frac{\partial{S}}{\partial{V}}}\mathrm{d}V + {\frac{\partial{S}}{\partial{N}}}\mathrm{d}N {=}$$

$${=}{\frac{\partial{S}}{\partial{U}}} \mathrm{d}U + {\frac{\partia... ... {\frac{\partial{S}}{\partial{U}}}{\frac{\partial{U}}{\partial{N}}}\mathrm{d}N.$$ (2.29)

With the equations (2.22)-(2.24), the total differential of the entropy becomes

$$\mathrm{d}S {=}\displaystyle \frac{1}{T} \mathrm{d}U + \frac{1}{T}p\mathrm{d}V + \frac{1}{T}(-\mu)\mathrm{d}N,$$ (2.30)

where the pressure ${p^{\mathrm{mech}}}$ can be found as

$${p^{\mathrm{mech}}}{=}T {\frac{\partial{S}}{\partial{V}}} {=}{\fr... ...}{\partial{V}}}%\ist \frac{1}{T}p\mathrm{d}V % + \frac{1}{T}(-\mu)\mathrm{d}N$$ (2.31)

and the chemical potential ${\mu^{\mathrm{chem}}}$ as

$${\mu^{\mathrm{chem}}}{=}- T{\frac{\partial{S}}{\partial{N}}} {=}-... ...{\partial{N}}}.%\ist \frac{1}{T}p\mathrm{d}V % + \frac{1}{T}(-\mu)\mathrm{d}N$$ (2.32)

By comparison of the coefficients from (2.28) and (2.30)-(2.32), the net heat flow $Q$ and the work $W$ done by the system $\mathcal{S}$ can be found as

$$\partial{Q} {=}T \mathrm{d}S$$ (2.33)

$$\partial{W} {=}{p^{\mathrm{mech}}}\mathrm{d}V - {\mu^{\mathrm{chem}}}\mathrm{d}N.$$ (2.34)