Entropy form of dry potential temperature ($ds=c_{p}d\ln \theta$)

For a dry fixed-composition ideal gas with constant specific heats, $ds=c_{p}d\ln T-R_{d}d\ln p$. Defining $\theta =T(p_0/p{)}^{R_d/c_p}$ gives the displayed identity. Reversible adiabatic parcel motion conserves entropy and theta simultaneously. Irreversible entropy production or moist latent heating can change ordinary dry theta even without heat exchanged through the parcel boundary.