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A $20 \mathrm{~cm}$ length of a certain solution causes right handed rotation of $38^{\circ}$. A $30 \mathrm{~cm}$ length of another solution causes left handed rotation of $24^{\circ}$. The optical rotation caused by $30 \mathrm{~cm}$ length of a mixture of the above solutions in the volume ratio $1: 2$ is
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The correct answer is:
right handed rotation of $3^{\circ}$
For liquid $\mathrm{A}$
$L_{1}=20 \mathrm{~cm}, \theta_{1}=38^{\circ}$; concentration $=C_{1}$
$\begin{aligned} \text { Specific rotation } \alpha_{1} &=\frac{\theta_{1}}{L_{1} C_{1}} \\ &=\frac{38^{\circ}}{20 \times C_{1}} \end{aligned}$
Similarly, for liquid $B$
$L_{2}=30 \mathrm{~cm} \theta_{2}=-24^{\circ}$, concentration $=C_{2}$
Specific rotation $\alpha_{2}=\frac{\theta_{2}}{L_{2} C_{2}}$
$$
=\frac{\left(-24^{\circ}\right)}{30 \times C_{2}}
$$
The mixture has 1 part of liquid $A$ and 2 parts of liquid $B$.
$$
\begin{aligned}
\therefore \quad C_{1}^{\prime} &: C_{2}^{\prime}=1: 2 \\
\theta &=\left[\alpha_{1} C_{1}^{\prime}+\alpha_{2} C_{2}^{\prime}\right] l \\
&=\left\{\frac{38^{\circ}}{20 \times C_{1}} \times \frac{C_{1}}{3}+\frac{\left(-24^{\circ}\right)}{30 \times C_{2}} \times \frac{2 C_{2}}{3}\right\} \times 30 \\
&=19^{\circ}-16^{\circ}=3^{\circ}
\end{aligned}
$$
Thus, the optical rotation of mixture is $+3^{\circ}$ in right hand direction
$L_{1}=20 \mathrm{~cm}, \theta_{1}=38^{\circ}$; concentration $=C_{1}$
$\begin{aligned} \text { Specific rotation } \alpha_{1} &=\frac{\theta_{1}}{L_{1} C_{1}} \\ &=\frac{38^{\circ}}{20 \times C_{1}} \end{aligned}$
Similarly, for liquid $B$
$L_{2}=30 \mathrm{~cm} \theta_{2}=-24^{\circ}$, concentration $=C_{2}$
Specific rotation $\alpha_{2}=\frac{\theta_{2}}{L_{2} C_{2}}$
$$
=\frac{\left(-24^{\circ}\right)}{30 \times C_{2}}
$$
The mixture has 1 part of liquid $A$ and 2 parts of liquid $B$.
$$
\begin{aligned}
\therefore \quad C_{1}^{\prime} &: C_{2}^{\prime}=1: 2 \\
\theta &=\left[\alpha_{1} C_{1}^{\prime}+\alpha_{2} C_{2}^{\prime}\right] l \\
&=\left\{\frac{38^{\circ}}{20 \times C_{1}} \times \frac{C_{1}}{3}+\frac{\left(-24^{\circ}\right)}{30 \times C_{2}} \times \frac{2 C_{2}}{3}\right\} \times 30 \\
&=19^{\circ}-16^{\circ}=3^{\circ}
\end{aligned}
$$
Thus, the optical rotation of mixture is $+3^{\circ}$ in right hand direction
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