If molar extinction coefficient is unknown, can concentration still be found in grams rather than moles?

Question

Only the absorbance is known (as wel as l)

Answer 1

A=E*d*C

(d=length of the path of the light through the solution)

If you have only one value of absorbance then obviously there is no way of determining the concentration without the molar extiction coefficient (E).

In practice even if you know E, you don't use it to find the unknown concentration. Instead you measure the A of standard solutions and do a standard curve. From the equation of that curve (which you find using linear regression) and the value of A of your sample you calculate the unknown amount.
The standard curve doesn't need to be A vs mole/lt. It can be A vs anything e.g. g/ml, mg/ml, ppm or even just g, mole, etc. the only thing that changes is what the slope represents.
For example for A vs mole/lt the slope corresponds to E*d, for A vs g/lt the slope corresponds to E*d/MW, etc.

If you always have the same volume of sample you can do the curve A vs grams directly. Then if V is the volume of the sample, the slope corresponds to E*d*V/MW

Even if your standard curve is expressed in moles/lt you can convert it to grams. You just need the MW of the compound and the volume, not E...