In statistical mechanics the total chemical potential $\mu _{\mbox{Total}}=\mu _{\mbox{Internal}}+\mu_{\mbox{External}}$. In the examples I have done so far $\mu_{\mbox{External}}$ is the external (effective) potential energy acting on the system. My question is why do you sometimes add this quantity, and sometimes subtract it?

For example, on the centrifuge problem, the centrifugal potential energy is $-mr^2\omega ^2/2$, so we have the equilibrium chemical potentials: $$\tau \ln \left[\frac{n(0)}{n_Q}\right]=\tau \ln \left[\frac{n(r)}{n_Q}\right]-\frac{mr^2\omega ^2}{2}$$

On the other-hand on the tree-sap problem, the gravitational potential is $Mgh$, and we have the equilibrium chemical potentials: $$\tau \ln \left[\frac{n(0)}{n_Q}\right]=\tau \ln \left[\frac{n(r)}{n_Q}\right]+Mgh$$

Is it something as simple as I am getting confused on directions, or what am I missing here?

In statistical mechanics the total chemical potential $\mu _{\text{Total}}=\mu _{\text{Internal}}+\mu_{\text{External}}$. In the examples I have done so far $\mu_{\text{External}}$ is the external (effective) potential energy acting on the system. My question is why do you sometimes add this quantity, and sometimes subtract it?

For example, on the centrifuge problem, the centrifugal potential energy is $-mr^2\omega ^2/2$, so we have the equilibrium chemical potentials: $$\tau \ln \left[\frac{n(0)}{n_Q}\right]=\tau \ln \left[\frac{n(r)}{n_Q}\right]-\frac{mr^2\omega ^2}{2}$$

On the other-hand on the tree-sap problem, the gravitational potential is $Mgh$, and we have the equilibrium chemical potentials: $$\tau \ln \left[\frac{n(0)}{n_Q}\right]=\tau \ln \left[\frac{n(r)}{n_Q}\right]+Mgh$$

Is it something as simple as I am getting confused on directions, or what am I missing here?