ABSTRACT
Joule's law states that internal energy is a function of temperature only. It can be derived from the experimental free expansion of a gas, illustrated in Figure 9.1, in which no work is done and no heat is transferred, so that
0=+= ðWðQdU The result of the experiment is that the temperature change is too small to be detected, to a first approximation, and can be expressed as
0 V VU T
T T U U
∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − =⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠ where the reciprocity theorem (8.37) has been used. Since ( ) VV/ CTU =∂∂ is the heat capacity at constant volume and is finite, it follows that
0 V
=⎟⎠ ⎞⎜⎝
⎛ ∂ ∂
U (9.2)
Figure 9.1. Free expansion of a gas in a thermally isolated vessel with rigid walls We may then write
p U
⎟⎟⎠ ⎞
⎜⎜⎝ ⎛ ∂ ∂⎟⎠
⎞⎜⎝ ⎛ ∂ ∂=⎟⎟⎠
⎞ ⎜⎜⎝ ⎛ ∂ ∂ V
V
where is a finite quantity, defined in Eq.(8.42), so that Eq.(9.2) becomes
( ) TT kp V/V −=∂∂
0=⎟⎟⎠ ⎞
⎜⎜⎝ ⎛ ∂ ∂
Tp U (9.3)
Therefore the internal energy U of a perfect gas is independent of p and V, and thus it is a function of T only. Joule's law, as expressed by both Eqs.(9.2) and (9.3), describes the limiting behaviour of any real gas at low pressures.
