Page 178 - Advanced thermodynamics for engineers

164 CHAPTER 8 EQUATIONS OF STATE

constant composition mixture can be defined as a function of two independent properties. The actual
mathematical relationship has to be found from experiment (or a simulation of the molecular prop-
erties of the gas molecules), and this can be derived by knowing that, if the property, v, is a continuous
function of the other two properties, p and T, as discussed in Chapter 7, then

vv vv
dv ¼ dp þ dT: (8.4)
vp vT
T p

vv vv
Hence, if the partial derivatives ; and can be evaluated then the gas law will be
vp vT
T p
defined. It is possible to evaluate the first derivative by a Boyle’s law experiment, and the second one
by a Charles’ law experiment. It is found from Boyle’s law that
pv ¼ constant; (8.5)
giving
vp p

¼ : (8.6)
vv v
T
Similarly, it is found from Charles’ law that
v
¼ constant; (8.7)
T
giving
vv v

¼ : (8.8)
vT T
p
Substituting Eqns (8.6) and (8.8) into Eqn (8.4) gives
v v
dv ¼ dp þ dT; (8.9)
p T
which may be integrated to give
pv
¼ constant: (8.10)
T
Equation (8.10) is known as the ideal gas law. This equation contains no information about the
internal energy of the fluid, and does not define the specific heat capacities. If the specific heat
capacities are not functions of temperature then the gas is said to obey the perfect gas law: if the
specific heat capacities are functions of temperature (i.e. the internal energy and enthalpy do not vary
linearly with temperature) then the gas is called an ideal gas.
It is possible to define two coefficients from Eqn (8.4), which are analogous to concepts used for
describing the properties of materials. The first is called the isothermal compressibility,or isothermal
bulk modulus, k. This is defined as the ‘volumetric strain’ produced by a change in pressure, giving

1 vv 1 vV
k ¼ ¼ : (8.11)
v vp V vp
T T