Answer
556 nm.
Work Step by Step
Solve equation 24–4 for the slit separation for each wavelength.
$$d=\frac{m_1\lambda_1}{sin\theta_1}=\frac{2(632.8nm)}{sin53.2^{\circ}}$$
$$d=\frac{m_2\lambda_2}{sin\theta_2}=\frac{1(\lambda)}{sin20.6^{\circ}}$$
The same diffraction grating, with the same slit spacing d, is used for both experiments. We equate the two expressions.
$$ \frac{2(632.8nm)}{sin53.2^{\circ}}=\frac{1(\lambda)}{sin20.6^{\circ}}$$
Solve for the unknown wavelength.
$$\lambda= \frac{2(632.8nm)}{sin53.2^{\circ}} sin20.6^{\circ}=556nm$$
