Consider a cylinder closed by two sliding pistons separated by a permeable fixed wall. The cylinder contains N moles of an ideal gas passing through the wall under the effect of pistons 1 and 2 that keep the pressures p_1 and p_2 constant in the subsystems 1 and 2 on both sides of the wall (Fig. 4.9). The device is an adiabatic closed system.
1. Show that the enthalpy H is conserved if the external pressures exerted by the pistons are equal to the pressures in the corresponding subsystems at all times, i.e. p^{ext}_ 1 = p_1 and p^{ext}_ 2 = p_2. This is called the Joule–Thomson expansion.
2. For an arbitrary gas and an infinitesimal pressure difference d_p, show that the Joule– Thomson coefficient, defined as the partial derivative of temperature T with respect to the pressure p, is given by,
\frac{\partial T}{\partial p} =\frac{(T_\alpha-1)V }{C_p}.
where α is the thermal expansion coefficient, defined by,
\alpha=\frac{1}{V}\frac{\partial V}{\partial T}.
and C_p, the specific heat at constant pressure (see Chapter 5), defined as,
C_p=\frac{\partial H}{\partial T} \mid _p=T\frac{\partial S}{\partial T}.
The last equality is a consequence of the definitions (4.33) and (4.77). In these equations, the volume V and the entropy S are functions of the state variables temperature T and pressure p.
T(S,p,\left\{N_A\right\} )=\frac{\partial H(S,p,\left\{N_A\right\} ) }{\partial S }.
\frac{df}{dy}\mid _z ≡\frac{\partial f\Bigl(x(y,z),y\Bigr) }{\partial x (y,z)} \frac{\partial x(y,z) }{\partial y }+ \frac{\partial f\Bigl(x(y,z),y\Bigr) }{\partial y}.
\frac{df}{dz}\mid _y ≡\frac{\partial f\Bigl(x(y,z),y\Bigr) }{\partial x (y,z)} \frac{\partial x(y,z) }{\partial z }.
