2.5 Enthalpy
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 hμ
is the molar enthalpy, and
H = hμM. (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
dqp
= 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 cp, 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 cp 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
= cpdT. (2.52)
In thermodynamics, internal energy,
enthalpy and heat capacity are referred to as the calorific properties of
substance[3], and the specific volume,
pressure and temperature, are known as the thermal properties of
substance.
Previously, in Chapter 1, we introduced
the concept of the equation of state of substance, interrelating any three
properties of a pure substance, i.e. representing one of the properties as a
function of the other two.
The equation of state, relating two
thermal properties of substance, is called the thermal equation of state,
and the equation of state, which interrelates the three variables of which at
least one is a calorific property, is referred to as the calorific equation
of state. It is clear, that
functional dependences of the kind h = f (p, T), u = φ (v, T), etc. can be regarded as calorific
equations of state. It will be recalled that the equations of state are
strictly individual for each substance.
Since, as it was shown above, the internal
energy and the enthalpy of an ideal gas only depend on temperature, it is clear
that when applied to an ideal gas, the partial derivatives in the relationships
(2.31)
and
(2.46)
should be replaced by the total derivatives
(2.31a)
and
(2.46a)
From this it follows that for an ideal
gas, cv and cp, just as u and i, depend only on temperature.
Let us consider now relationship (2.41)
i
= u + pv
applying it to an ideal gas. Differentiating this
relationship with respect to temperature, we obtain:
(2.53)
From Clayperon’s
equation
pv = RT
it follows that
With account taken of this relationship
and also of Eqs. (2.31a) and (2.46a), we find from Eq. (2.53) that for an ideal
gas
cp
- cv = R (2.54)
which is referred to as Mayer's relation.
Making use of this relation, J. R. Mayer
as far back as 1842, before Joule's work appeared, attempted to calculate the
mechanical equivalent of heat. Having found the magnitude of cp —
cv expressed in units of kcal/(kg·K) by measuring the heat capacities of gases at low
pressures, and having calculated R in kgf·m/(kg·K) from Clapeyron's equation
Mayer equated the two quantities and, proceeding in
this way, calculated the mechanical equivalent of heat. The numerical value of J found by Mayer was not very accurate
due to the low accuracy of the experimental values of the heat capacities of
the gases, cp and cv.