Answer:
[tex]\boxed{\text{A. } m = 2, n = 3; \text{B. } m = 2, n = 4; \text{C. } m = 2, n = 5; \text{D. } m = 2, n = 6}[/tex]
Explanation:
The Balmer equation is
[tex]\lambda = B\left(\dfrac{n^{2}}{n^{2} -m^{2}}\right)[/tex]
where B = 364.5 nm and m = 2
Thus, the Balmer equation reduces to
[tex]\lambda = 364.5\left(\dfrac{n^{2}}{n^{2} - 4}\right)[/tex]
We will be doing four separate calculations for n, so it will be convenient to solve the equation for n.
[tex]\lambda (n^{2} -4) = 364.5n^{2}\\\\\lambda n^{2} -4\lambda = 364.5n^{2}\\\\\lambda n^{2}- 364.5n^{2} = 4\lambda \\\\ n^{2}(\lambda - 364.5) = 4\lambda \\\\ n^{2}= \dfrac{4\lambda}{\lambda - 364.5}\\\\ n= \sqrt{\dfrac{4\lambda}{\lambda - 364.5}}[/tex]
A. λ = 656.5 nm
[tex]n= \sqrt{\dfrac{4 \times 656.5}{656.5 - 364.5}} = \sqrt{\dfrac{2626}{292}} =\sqrt{8.993} = 2.999 \approx \boxed{\mathbf{3}}[/tex]
B. λ = 486.3 nm
[tex]n= \sqrt{\dfrac{4 \times 486.3}{486.3 - 364.5}} = \sqrt{\dfrac{1945}{121.8}} =\sqrt{15.97} = 3.996 \approx \boxed{\mathbf{4}}[/tex]
C. λ = 434.2 nm
[tex]n= \sqrt{\dfrac{4 \times 434.2}{434.2 - 364.5}} = \sqrt{\dfrac{1737}{69.7}} =\sqrt{24.9} = 4.99 \approx \boxed{\mathbf{5}}[/tex]
D. λ = 410.3 nm
[tex]n= \sqrt{\dfrac{4 \times 410.3}{410.3 - 364.5}} = \sqrt{\dfrac{1641}{45.8}} =\sqrt{35.8} = 5.99 \approx \boxed{\mathbf{6}}[/tex]
