Similarly, taking the logarithm of eq.(56) and differentiating following the motion gives:
using the fact that is sufficiently small for the expansion
to be accurate and where it has been assumed that the air is saturated. Now using the simplified thermodynamic equation (51) to eliminate from the above gives:
if the process is moist adiabatic. Substituting this into eq.(56) to eliminate the term then leads to:
A further simplifying assumption is to ignore the temperature dependence of the term by setting it equal to a constant value (correct to order ) and then writing the above equation as:
For moist air, the kinetic energy equation (43) becomes:
but
and therefore: