1) Fundamental units of
are ![[\frac{mol}{m\cdot s^2}]](https://tex.z-dn.net/?f=%5B%5Cfrac%7Bmol%7D%7Bm%5Ccdot%20s%5E2%7D%5D)
2) Fundamental units of
are ![[\frac{mol}{m^3}]](https://tex.z-dn.net/?f=%5B%5Cfrac%7Bmol%7D%7Bm%5E3%7D%5D)
Explanation:
The equation for the variable
is

where we have:
measured in ![[\frac{mol}{ft^3}]](https://tex.z-dn.net/?f=%5B%5Cfrac%7Bmol%7D%7Bft%5E3%7D%5D)
measured in ![[\frac{J}{kg}]](https://tex.z-dn.net/?f=%5B%5Cfrac%7BJ%7D%7Bkg%7D%5D)
measured in ![[in]](https://tex.z-dn.net/?f=%5Bin%5D)
measured in ![[\frac{m}{s^2}]](https://tex.z-dn.net/?f=%5B%5Cfrac%7Bm%7D%7Bs%5E2%7D%5D)
We can re-write the equation as

And we notice that the units of the term on the left must be equal to the units of the term on the right.
This means that:
1) First of all,
must have the same units of
. So,
![[\rho r g]=[\frac{mol}{ft^3}][in][\frac{m}{s^2}]](https://tex.z-dn.net/?f=%5B%5Crho%20r%20g%5D%3D%5B%5Cfrac%7Bmol%7D%7Bft%5E3%7D%5D%5Bin%5D%5B%5Cfrac%7Bm%7D%7Bs%5E2%7D%5D)
However, both ft (feet) and in (inches) are not fundamental dimensions: this means that they can be expressed as meters. Therefore, the fundamental units of
are
![[\Psi]=[\frac{mol}{m^3}][m][\frac{m}{s^2}]=[\frac{mol}{m\cdot s^2}]](https://tex.z-dn.net/?f=%5B%5CPsi%5D%3D%5B%5Cfrac%7Bmol%7D%7Bm%5E3%7D%5D%5Bm%5D%5B%5Cfrac%7Bm%7D%7Bs%5E2%7D%5D%3D%5B%5Cfrac%7Bmol%7D%7Bm%5Ccdot%20s%5E2%7D%5D)
2)
The term
must have the same units of
in order to be added to it. Therefore,
![[\gamma \Phi] = [\frac{mol}{m\cdot s^2}]](https://tex.z-dn.net/?f=%5B%5Cgamma%20%5CPhi%5D%20%3D%20%5B%5Cfrac%7Bmol%7D%7Bm%5Ccdot%20s%5E2%7D%5D)
We also know that the units of
are
, therefore
![[\frac{J}{kg}][\Phi]= [\frac{mol}{m\cdot s^2}]](https://tex.z-dn.net/?f=%5B%5Cfrac%7BJ%7D%7Bkg%7D%5D%5B%5CPhi%5D%3D%20%5B%5Cfrac%7Bmol%7D%7Bm%5Ccdot%20s%5E2%7D%5D)
And so, the fundamental units of
are
![[\Phi]= [\frac{mol\cdot kg}{J\cdot m\cdot s^2}]](https://tex.z-dn.net/?f=%5B%5CPhi%5D%3D%20%5B%5Cfrac%7Bmol%5Ccdot%20kg%7D%7BJ%5Ccdot%20m%5Ccdot%20s%5E2%7D%5D)
However, the Joules can be written as
![[J]=[kg][\frac{m^2}{s^2}]](https://tex.z-dn.net/?f=%5BJ%5D%3D%5Bkg%5D%5B%5Cfrac%7Bm%5E2%7D%7Bs%5E2%7D%5D)
Therefore
![[\Phi]= [\frac{mol\cdot kg}{(kg \frac{m^2}{s^2})\cdot m\cdot s^2}]=[\Phi]= [\frac{mol}{m^3}]](https://tex.z-dn.net/?f=%5B%5CPhi%5D%3D%20%5B%5Cfrac%7Bmol%5Ccdot%20kg%7D%7B%28kg%20%5Cfrac%7Bm%5E2%7D%7Bs%5E2%7D%29%5Ccdot%20m%5Ccdot%20s%5E2%7D%5D%3D%5B%5CPhi%5D%3D%20%5B%5Cfrac%7Bmol%7D%7Bm%5E3%7D%5D)
#LearnwithBrainly