At least one common quantity I can think of has dimension with a non-integer exponent. The specific detectivity, $\text{D}^*$ is a common descriptor of photodiodes, and I'm sure one could make an analogous figure of merit for other types of sensor.

The unit of $\text{D}^*$ is the "Jones," which is equal to

$$\frac{cm \cdot \sqrt{Hz}}{W}$$

Watts in SI decompose into

$$Kg \cdot m^2 \cdot s^{-3}$$

which makes one Jones equal to:

$$\frac{s^{2.5}}{Kg \cdot m} \times 10^{-2}$$

with dimension:

$$\text{time}^{2.5} \text{mass}^{-1} \text{length}^{-1}$$

There are other quantities related to $\text{D}^*$, such as Noise Equivalent Power (NEP), which come up a lot in radiometry. Basically, any measurement which is normalized by frequency bandwidth will end up with that $\sqrt{Hz}$ in the units.

I don't think that you can assume that $I_c=\ln (p_0)$

A lot of equations have logs of dimensioned quantities in them. It's usually not hard to get rid of the log:

$$
\mathrm{ln}\left(\frac{p_1}{p_2}\right) = I_{c,1}-I_{c,2} - \left(\frac{\Delta H^{\circ}_{v,1}}{RT_1}-\frac{\Delta H^{\circ}_{v,2}}{RT_2}\right) + \mathrm{ln}\left(\frac{T_1^{\frac{\left(C^{g}_{p,1} - C_{p,1}^{l}\right)}{R}}}{T_2^{\frac{\left(C^{g}_{p,2} - C_{p,2}^{l}\right)}{R}}}\right)
$$

It may also be that $I_c$ is just a gas-specific constant, of the form $$I_c=\ln p_0-\frac{\Delta H^{\circ}_{v,0}}{R} -\frac{\left(C^{g}_{p,0} - C_{p}^{l,0}\right)}{RT_0}\mathrm{ln}\left(T_0\right) $$

for some $p_0,T_0$.

You can do the same for the other equations as well. When there's a log of a dimensioned quantity, one of two things is possible:

- The equation has the units implicitly assumed (this is generally not done in physics, but you see it a lot in chemistry)
- The equation is a change equation, and you need to subtract one copy of the equation at a different point from it.

The latter form arises when one takes an indefinite integral in the last step while solving a differential equation (Which is why subtracting is the right way to fix this, since that corresponds to tacking limits onto the integral)

## Best Answer

If you have n moles of a substance and it receives Q kJ of heat, how much heat does each mole of the substance receive? What are the units of this?