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.