Fundamental thermodynamic relation

If we now express the differential of the entropy w.r.t. the variables $E,V,N$

$\displaystyle dS = \left ( \frac{\partial S}{\partial E}\right )_{V,N} dE + \le... ...l V}\right )_{E,N} dV + \left ( \frac{\partial S}{\partial N}\right )_{E,V} dN,$

(3.21)

using the definitions

for the partial derivatives, we obtain:

$\displaystyle TdS = dE + P dV -\mu dN.$

(3.22)

which is known as the fundamental thermodynamic relation. In its restricted form for systems with constant number of particles Eq.

takes the form:

$\displaystyle TdS = dE + P dV.$

(3.23)

Combining

with the first law

and with

for reversible changes we also obtain:

$\displaystyle TdS = {\mathchar'26\mkern-12mu d}Q,$

(3.24)

which shows that in a adiabatic transformation the entropy remains constant. If we refer back to Eq.

we therefore see that:

$\displaystyle P = -\left( \frac{\partial E}{\partial V}\right )_{adiabatic} = -\left( \frac{\partial E}{\partial V}\right )_{S}.$

(3.25)

The definition of the pressure in

is then simply a consequence of

and of that of temperature in

, $\frac{1}{T}=\left( \frac{\partial S}{\partial E}\right )_V$ , as we can easily verify by writing:

$\displaystyle dS = \left( \frac{\partial S}{\partial V}\right )_E dV + \left( \frac{\partial S}{\partial E}\right )_V dE.$

(3.26)

Dividing everything by $dV$

, and imposing the adiabatic condition of constant entropy, $dS=0$

, we obtain:

$\displaystyle 0 = \left( \frac{\partial S}{\partial V}\right )_E + \left( \frac{\partial S}{\partial E}\right )_V \left( \frac{\partial E}{\partial V}\right )_S,$

(3.27)

which gives:

$\displaystyle \left( \frac{\partial S}{\partial V}\right )_E = - \left( \frac{\... ...tial E}\right )_V \left( \frac{\partial E}{\partial V}\right )_S = \frac{P}{T}.$

(3.28)