Question

Write the total differential for H as a function of T & P and derive the appropriate working equation.

Answer 1

As it will be seen below, of paramount importance to various thermodynamic calculations is the sum of the internal energy of a system U and the product of the systems pressure p by the volume of the system V; this quantity is called enthalpy[1] and is denoted by H:

                                                                     H = U + pV                                                           (2.39)

(previously, this thermodynamic quantity was referred to as heat content). It is clear that similar to internal energy enthalpy is an extensive pro­perty of substance:

                                                                        H = hG,                                                              (2.40)

where h is the specific enthalpy (per a unit mass of substance).

For the specific enthalpy we can write

                                                                       h = u+pv                                                             (2.41)

Denoting the number of moles of a substance in a system by M,

                                                                  

where h_μ is the molar enthalpy, and

                                                                       H = h_μM.                                                             (2.42)

Enthalpy is measured in the units used to measure heat, work and internal energy (see Table 2.1).

Since h and u are related unambiguously, the reference point for enthalpy is linked with the reference point for internal energy: at the point, assumed as a reference for internal energy (u = 0), enthalpy will be equal to h = pv. So, at the above-mentioned zero reference point for the internal energy of water (t = 0.01 °C, p = 610.8 Pa, v = 0.0010002 m³/kg) the enthalpy is equal to h = pv = 610.8 × 0.0010002 = 0.611 J/kg (0.000146 kcal/kg).

Inasmuch as the new function, enthalpy, is made up of quantities which are functions of state (u, p, v), enthalpy is also a function of state. Just as internal energy, the enthalpy of a pure substance can be represented as a function of any two properties, or parameters, of state, for instance, of pressure p and temperature T:

                                                                     h = f (p, T).

Further, as enthalpy is a function of state, its differential is a total diffe­rential:

                                                                                                 (2.43)

The mathematical formulation of the first law of thermodynamics for the case where the only kind of work is that of expansion,

                                                                   dq = du + pdv

with account taken of the obvious relationship[2]

                                                               pdv = d(pv) — vdp

takes the following form:

                                                           dq = du + d(pv) — vdp,

or, which is the same,

                                                            dq = d(u + pv) — vdp,

i.e.

                                                                  dq = dh — vdp.                                                        (2.44)

It follows from Eq. (2.41) that if the pressure of a system is maintained constant, i.e. an isobaric process is being realized (dp = 0), then

                                                                       dq_p= dh,                                                             (2.45)

i.e. the heat added to a system undergoing an isobaric process is only expended to change the enthalpy of the system. From this it follows that the expression for the isobaric heat capacity c_p, which in accordance with Eq. (1.69) is equal to

                                                                  

can be presented in the following form:

                                                                                                                         (2.46)

It is clear from Eq. (2.46) that the constant-pressure heat capacity c_p characterises the rate of increase in enthalpy with rising temperature.

With account taken of (2.46), Eq. (2.43), representing the total differential, of enthalpy, takes the following form:

                                                                                                         (2.47)

The partial derivative characterizes the dependence of enthalpy on pressure.

Using condition (2.34), we can show that for an ideal gas

                                                                                                                           (2.48)

i.e. the enthalpy of an ideal gas does not depend on pressure.

     By analogy, it can then be easily shown that

                                                                                                                           (2.49)

Relationships similar to (2.48) and (2.49) can be naturally written for a system as a whole:

                                                                                                                           (2.50)

and

                                                                                                                          (2.51)

It follows from relationships (2.50) and (2.51) that the enthalpy of an ideal gas, just like its internal energy, only depends on temperature: with account taken of Eq. (2.48), it follows from Eq. (2.47) that

                                                                      dh = c_pdT.                                                            (2.52)