# Potential Temperature

Derived directly from integration of the 1st law
of thermodynamics. It is the temperature
a parcel of air at *P* and *T* would have if it were at *P _{s}*. It is
conserved for adiabatic motions, (

*i.e.*,

*d*/

*dt*= 0).

*dQ*/(

*n*· <mw> ·

*T*) = (

*C*/

_{p}dT*T*) - (

*R*/

_{g}dP*P*) = 0

*T*· (

*P*/

_{s}*P*)

^{}=

*P*/(

*R*) [(

_{g}*P*/

_{s}*P*)

^{}] =

*R*/

_{g}*C*

_{p}
For earth = 0.286 (<mw> = 28.96, *C _{p}* =
1.004 Joules/gram/K). Some authors write this equation with
=

*C*/

_{p}*C*= 1/(1-)

_{v}*P*/

*z*= - ·

*g*= (

*P*·

*g*)/(

*R*·

_{g}*T*) = -

*P*/

*H*(

*z*)

*P*) = -

*H*(

*z*) ·

*z*

therefore,

*P*/

*P*= e

_{s}^{- dz'/H(z')}

*T*· e

^{- dz'/H(z')}

if

*H*(

*z*) =

*H*

_{0}*T*· e

^{- · z/H0}