온위 방정식에 자연로그를 취한다.
$\theta = T(\frac{1000hPa}{p})^\kappa$
$\ln \theta = \ln T + \kappa \ln (1000hPa) - \kappa \ln p$
$z$에 대해 미분한다.
$\frac{\partial}{\partial z}(\ln \theta) = \frac{\partial}{\partial z}(\ln T) - \kappa\frac{\partial}{\partial z} \ln p$
$\frac{1}{\theta}\frac{\partial \theta}{\partial z} = \frac{1}{T}\frac{\partial T}{\partial z} - \kappa \frac{1}{p} \frac{\partial p}{\partial z}$
이상기체방정식과 정역학 방정식을 대입한다.
$p = \rho RT,\; \frac{\partial p}{\partial z} = -\rho g,\; \kappa = \frac{R}{c_p}$
$\frac{1}{\theta}\frac{\partial \theta}{\partial z} = \frac{1}{T}\frac{\partial T}{\partial z}+\frac{g}{c_p}\frac{1}{T}$
양변에 $T$를 곱하면
$\frac{T}{\theta}\frac{\partial \theta}{\partial z} = \frac{\partial T}{\partial z} + \frac{g}{c_p}$
건조단열과정에 의해 $\frac{\partial \theta}{\partial z}=0$
$-\frac{\partial T}{\partial z} = \frac{g}{c_p} = \Gamma _d$
$\Gamma _d = \frac{g}{c_p} = \frac{9.8\; m\; s^{-2}}{1004\;J\;kg^{-1}\;K^{-1}}=9.76 \; [\;K\;km^{-1}\;]$
