Answer to Problem 48
If we rewrite equation (5.3) as

Equation 56
the quadratic formula yields

Equation 57
Since the temperature is a single-valued function of
,
only one of the square
roots should give a physically valid temperature
function.
For fixed
,
,
and
(given by Equation (5.4)) we take
positive and small enough that

Equation 58
Then using the Taylor approximation

Equation 59
we write the above expression for
as

Equation 60
If the plus sign is used, we have

Equation 61
which implies that as
,
rather than
, which is required by the physics.
Therefore the plus sign must be rejected. Retaining
the minus sign in (5.5) yields the solution to this
problem as

Equation 62
In this case, we have, for small
,

Equation 63
which is physically sensible.