# Why is the value of specific rotation different in different cases?

An unknown compound with a mass of $4.5~\mathrm{g}$ is dissolved in enough carbon tetrachloride to make a total volume of $250~\mathrm{cm^3}$. The observed rotation of this solution is $+357.75^\circ$ in a $25~\mathrm{cm}$ cell using the sodium D line. But if $4.5~\mathrm{g}$ is dissolved in $125~\mathrm{cm^3}$ we observed rotation is $+355.50^\circ$. Calculate specific rotation for this compound. (assuming length of polarimeter tube is $1~\mathrm{dm}$)

My approach is starting with the formula stated on Wikipedia.

$$\text{Specific rotation (in first case)} = \frac{+357.75^\circ}{\left(\frac{4.5}{250}\right)(1)}=19875^\circ=75^\circ$$

$$\text{Specific rotation (in second case)} = \frac{+355.50^\circ}{\left(\frac{4.5}{125}\right)(1)}=9875^\circ=155^\circ$$

Any idea why the two answers are different? Did I miss out something?

• Note that the concentration is doubled in the seconds case. Therefore, the expected rotation should be $2\times357.75^\circ=715.50^\circ$, which is more than a full circle. Substracting one circle $(360^\circ)$ gives the indicated $715.50^\circ-360.00^\circ=355.50^\circ$. – Faded Giant Sep 18 '16 at 16:21
• @Loong but the length changes in the second case too!You did'nt take that into account...Initially it was 2.5 dm but finally 1 dm. – user14857 Sep 18 '16 at 16:23
• No, an experimentally observed rotation of +181° may actually be +181, +1, +361, -179, etc. That is why the experiment is run at 2 concentrations, then you can solve the equation and determine the "true" rotation. – ron Sep 18 '16 at 16:40
• If indeed the path lengths are equal, then rot_1=357.75 + (n*180) and, due to concentration 2*rot_1 = rot_2. Solving we find that n=-2, therefore rot_1 = 357.75 +(-2*180) or -2.25. This "true" observed value may now be converted to the specific rotation using your formula. – ron Sep 18 '16 at 17:37
• @Loong I took the liberty to correct that O.o – Martin - マーチン Sep 27 '16 at 12:43

A polarimeter is typically a long cylindrical tube with flat ends. I think the 25 cm was referring to the diameter of the tube. The other wrinkle here is that the angle measurement may be + or -. So 357.75 may in fact be -2.25.

Specific Rotation (in first case)=$$\frac{-2.25^{0}}{(\frac{4.5}{250})(1)}= - 125^{0}$$

Specific Rotation (in second case)=$$\frac{-4.50^{0}}{(\frac{4.5}{125})(1)}=-125^{0}$$

• Could you check my calculation in the original question please?Why is the answer not matching if I take the original values with +ve sign? – user14857 Sep 18 '16 at 17:03
• @ZOZ - How can I do the calculations wrong and get a good answer? – MaxW Sep 18 '16 at 17:10
• I never said your answer is wrong or your calculation is wrong.I'm asking why is my calculation wrong?Where did I go wrong?See the question... – user14857 Sep 18 '16 at 17:11
• Your problem was that you assumed that the reading for the solutions were positive when the are in fact negative. Think of a 360 degree compass. If you draw an angle of +357.75 degrees that is the same as drawing an angle of -2.25 degrees. – MaxW Sep 18 '16 at 17:17
• In other words think of a 360 degree compass from geometry class. They are graduated from 0 to 360. There are no negative markings. A polarimeter is setup the same way. There are no negative angles on the actual dial. – MaxW Sep 18 '16 at 17:24

If indeed the path lengths are equal, then $rot_1=357.75 + (n*180)$ and, due to concentration $2*rot_1 = rot_2$. Solving we find that $n=-2$, therefore $rot_1 = 357.75 +(-2*180)$ or $-2.25$. This "true" observed value may now be converted to the specific rotation using your formula.

-by @ron in the comments above.