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

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