Question

The polarimeter readings in an experiment to measure the rate of inversion of cane sugar (1st order reaction) were as follows.
\[
  \begin{array}{*{20}{c}}
  {Time\left( {\min } \right):}&0&{30}&\infty
\end{array} \\
  \begin{array}{*{20}{c}}
  {Angle\left( {\deg } \right):}&{30}&{20}&{ - 15}
\end{array} \\
 \]
Identify the true statement(s)?
$\log 2 = 0.3$ , $\log 3 = 0.48$, $\log 7 = 0.84$, $\log {e^{10}} = 2.3$
A) The half life of the reaction is \[75\] min
B) The solution is optically inactive at \[120\] min
C) The equimolar mixture of the product is dextrorotatory
D) The angle would be \[7.5^\circ \] at half time

Answer 1

Hint: We know that a polarimeter is a logical instrument used to gauge the point of turn brought about by going energized light through an optically dynamic substance. Some synthetic substances are optically dynamic, and enraptured (uni-directional) light will pivot either to one side (counter-clockwise) or right (clockwise) when gone through these substances. The sum by which the light is pivoted is known as the point of revolution. The point of turn is fundamentally known as the noticed point.

Complete answer:
Given:
\[
  \begin{array}{*{20}{c}}
  {Time\left( {\min } \right):}&0&{30}&\infty
\end{array} \\
  \begin{array}{*{20}{c}}
  {Angle\left( {\deg } \right):}&{30}&{20}&{ - 15}
\end{array} \\
 \]
$K = \dfrac{1}{{30}}2.3\log \dfrac{{30 - \left( { - 15} \right)}}{{20 - \left( { - 15} \right)}} = \dfrac{1}{{30}}\log \dfrac{{45}}{{35}}$
${T_{1/2}} = 75\operatorname{mins} $
$K = \dfrac{1}{t}2.3\log \dfrac{{45}}{{0 - \left( { - 15} \right)}}$
$t = 120\min $
$2athalftime = \dfrac{{30 - \left( { - 15} \right)}}{{x - \left( {x - 15} \right)}}$
$45\left( {x + 15} \right) = 2$
$x = 22.5 - 15$
On simplification we get,
$ \Rightarrow x = 7.5^\circ $

Hence options A, B and D are correct.

Note:
Now we can discuss about the construction of polarimeter:
The polarimeter is composed of two Nicol crystals (the polarizer and analyzer). The polarizer is fixed and the analyzer can be pivoted. The crystals might be considered as cuts S1 and S2. The light waves might be considered to relate to waves in the string. The polarizer S1 permits just those light waves which move in a solitary plane. This makes the light get enraptured. At the point when the analyzer is likewise positioned in a comparable position it permits the light waves coming from the polarizer to go through it. At the point when it is turned through the correct point no waves can go through the correct point and the field has all the earmarks of being dim. In the event that now a glass tube containing an optically dynamic arrangement is set between the polarizer and analyzer the light presently pivots through the plane of polarization through a specific point, the analyzer should be turned in the same point.