Page 165 - Advanced Thermodynamics for Engineers, Second Edition

152 CHAPTER 7 GENERAL THERMODYNAMIC RELATIONSHIPS

Now, if it is required to calculate the total change in internal energy, u, for a change in volume and
temperature, it is necessary to use Eqn (7.46)

vu vu
du ¼ dT þ dv: (7.46)
vT vv
v T
For an ideal gas

vu
du ¼ dT ¼ c v dT: (7.55)
vT
v
i.e. for an ideal gas
u ¼ fðTÞ:
It is also possible to use the following approach to show whether u ¼ f (p). This may be done in the
following way.
vu vu vp

¼ ¼ 0:
vv vp vv
T T T
Now

vp p
¼ ;
vv T v
and since

vu
ps0; ¼ 0:
vp
T
Hence, for an ideal gas
usfðpÞ:
For a van der Waals gas
a
du ¼ c v dT þ 2 dv: (7.56)
v
Hence, for a van der Waals gas u ¼ f (T,v).

7.4 RELATIONSHIPS BETWEEN SPECIFIC HEAT CAPACITIES
These relationships enable the value of one specific heat capacity to be calculated if the other is known.
This is useful because it is much easier to measure the specific heat capacity at constant pressure, c p ,
than that at constant volume, c v .
Using the two-property rule it is possible to write

s ¼ sðT; vÞ (7.57)