
First Law—The E = Mc
2
of Thermodynamics 43
2.4.5 Isothermal changes
We now turn to a discussion of the implications of first law for changes under con-
stant temperature. Most generally
¯
dQ = C
V
(V,T )dT +
∂
U
∂
V
T
+ P
dV →
∂
U
∂
V
T
+ P
dV (2.21)
For an ideal gas many simple consequences follow. Firstly, pressure is inversely pro-
portional to volume, i.e PV = const.; this is nothing but Boyle’s law. As we saw
earlier, U for ideal gases is independent of volume. Even for the general case for
which C
V
depends on temperature, for isothermal changes there is no change in in-
ternal energy and the first law gives dQ = PdV. In this case too, the heat Q is a state ...