Question

1. click on the visible/full button below the graph to zoom to only the visible spectrum. if you hover your cursor over a peak, it will identify the wavelength and intensity. record the wavelengths of the four peaks in the visible hydrogen spectrum in the data table. (round to whole numbers.)

1. the rydberg equation has the form \\(\frac{1}{\lambda} = r_h \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \

ight)\\) where \\(\lambda\\) is the wavelength in meters, \\(r_h\\) is the rydberg constant, \\(n_f\\) is the final principal quantum (for the balmer series, which is in the visible spectrum, \\(n_f = 2\\)), and \\(n_i\\) is the initial principal quantum number (\\(n = 3, 4, 5, 6, ...\\)). calculate from your experimental data the wavelength in meters and \\(1/\lambda\\) in \\(m^{-1}\\). record your answers in the data table.

note: (for the \\(\lambda\\)m and \\(1/\lambda\\) columns, enter your answers in scientific notation. formatted like: 3e5 or 3*10^5.

data table

	\\(\lambda\\) (nm)	\\(\lambda\\) (m)	\\(1/\lambda\\) (1/m)
line #2	486
line #3	434
line #4 (right)	410

Explanation:

To solve for \(\lambda\) (in meters) and \(\frac{1}{\lambda}\) (in \(m^{-1}\)) for each line, we use the conversion factor \(1\space nm = 10^{-9}\space m\).

Line #1 (left): \(\lambda = 656\space nm\)

Step 1: Convert \(\lambda\) from nm to m

To convert nanometers to meters, we use the conversion \(1\space nm = 10^{-9}\space m\). So, for \(\lambda = 656\space nm\):
\[
\lambda (m)=656\times 10^{-9}\space m = 6.56\times 10^{-7}\space m
\]

Step 2: Calculate \(\frac{1}{\lambda}\)

Now, we find the reciprocal of \(\lambda\) (in meters) to get \(\frac{1}{\lambda}\) (in \(m^{-1}\)):
\[
\frac{1}{\lambda}=\frac{1}{6.56\times 10^{-7}\space m}\approx 1.524\times 10^{6}\space m^{-1}
\]

Line #2: \(\lambda = 486\space nm\)

Step 1: Convert \(\lambda\) from nm to m

Using the conversion \(1\space nm = 10^{-9}\space m\) for \(\lambda = 486\space nm\):
\[
\lambda (m)=486\times 10^{-9}\space m = 4.86\times 10^{-7}\space m
\]

Step 2: Calculate \(\frac{1}{\lambda}\)

Find the reciprocal of \(\lambda\) (in meters):
\[
\frac{1}{\lambda}=\frac{1}{4.86\times 10^{-7}\space m}\approx 2.058\times 10^{6}\space m^{-1}
\]

Line #3: \(\lambda = 434\space nm\)

Step 1: Convert \(\lambda\) from nm to m

Using the conversion \(1\space nm = 10^{-9}\space m\) for \(\lambda = 434\space nm\):
\[
\lambda (m)=434\times 10^{-9}\space m = 4.34\times 10^{-7}\space m
\]

Step 2: Calculate \(\frac{1}{\lambda}\)

Find the reciprocal of \(\lambda\) (in meters):
\[
\frac{1}{\lambda}=\frac{1}{4.34\times 10^{-7}\space m}\approx 2.304\times 10^{6}\space m^{-1}
\]

Line #4 (right): \(\lambda = 410\space nm\)

Step 1: Convert \(\lambda\) from nm to m

Using the conversion \(1\space nm = 10^{-9}\space m\) for \(\lambda = 410\space nm\):
\[
\lambda (m)=410\times 10^{-9}\space m = 4.10\times 10^{-7}\space m
\]

Step 2: Calculate \(\frac{1}{\lambda}\)

Find the reciprocal of \(\lambda\) (in meters):
\[
\frac{1}{\lambda}=\frac{1}{4.10\times 10^{-7}\space m}\approx 2.439\times 10^{6}\space m^{-1}
\]

Filling the Data Table:

	\(\lambda\) (nm)	\(\lambda\) (m)	\(\frac{1}{\lambda}\) (\(m^{-1}\))
Line #2	486	\(4.86\times 10^{-7}\)	\(2.06\times 10^{6}\) (approx)
Line #3	434	\(4.34\times 10^{-7}\)	\(2.30\times 10^{6}\) (approx)
Line #4	410	\(4.10\times 10^{-7}\)	\(2.44\times 10^{6}\) (approx)

(Note: The values for \(\frac{1}{\lambda}\) are rounded to a reasonable number of significant figures.)

Answer:

To solve for \(\lambda\) (in meters) and \(\frac{1}{\lambda}\) (in \(m^{-1}\)) for each line, we use the conversion factor \(1\space nm = 10^{-9}\space m\).

Line #1 (left): \(\lambda = 656\space nm\)

Step 1: Convert \(\lambda\) from nm to m

To convert nanometers to meters, we use the conversion \(1\space nm = 10^{-9}\space m\). So, for \(\lambda = 656\space nm\):
\[
\lambda (m)=656\times 10^{-9}\space m = 6.56\times 10^{-7}\space m
\]

Step 2: Calculate \(\frac{1}{\lambda}\)

Now, we find the reciprocal of \(\lambda\) (in meters) to get \(\frac{1}{\lambda}\) (in \(m^{-1}\)):
\[
\frac{1}{\lambda}=\frac{1}{6.56\times 10^{-7}\space m}\approx 1.524\times 10^{6}\space m^{-1}
\]

Line #2: \(\lambda = 486\space nm\)

Step 1: Convert \(\lambda\) from nm to m

Using the conversion \(1\space nm = 10^{-9}\space m\) for \(\lambda = 486\space nm\):
\[
\lambda (m)=486\times 10^{-9}\space m = 4.86\times 10^{-7}\space m
\]

Step 2: Calculate \(\frac{1}{\lambda}\)

Find the reciprocal of \(\lambda\) (in meters):
\[
\frac{1}{\lambda}=\frac{1}{4.86\times 10^{-7}\space m}\approx 2.058\times 10^{6}\space m^{-1}
\]

Line #3: \(\lambda = 434\space nm\)

Step 1: Convert \(\lambda\) from nm to m

Using the conversion \(1\space nm = 10^{-9}\space m\) for \(\lambda = 434\space nm\):
\[
\lambda (m)=434\times 10^{-9}\space m = 4.34\times 10^{-7}\space m
\]

Step 2: Calculate \(\frac{1}{\lambda}\)

Find the reciprocal of \(\lambda\) (in meters):
\[
\frac{1}{\lambda}=\frac{1}{4.34\times 10^{-7}\space m}\approx 2.304\times 10^{6}\space m^{-1}
\]

Line #4 (right): \(\lambda = 410\space nm\)

Step 1: Convert \(\lambda\) from nm to m

Using the conversion \(1\space nm = 10^{-9}\space m\) for \(\lambda = 410\space nm\):
\[
\lambda (m)=410\times 10^{-9}\space m = 4.10\times 10^{-7}\space m
\]

Step 2: Calculate \(\frac{1}{\lambda}\)

Find the reciprocal of \(\lambda\) (in meters):
\[
\frac{1}{\lambda}=\frac{1}{4.10\times 10^{-7}\space m}\approx 2.439\times 10^{6}\space m^{-1}
\]

Filling the Data Table:

	\(\lambda\) (nm)	\(\lambda\) (m)	\(\frac{1}{\lambda}\) (\(m^{-1}\))
Line #2	486	\(4.86\times 10^{-7}\)	\(2.06\times 10^{6}\) (approx)
Line #3	434	\(4.34\times 10^{-7}\)	\(2.30\times 10^{6}\) (approx)
Line #4	410	\(4.10\times 10^{-7}\)	\(2.44\times 10^{6}\) (approx)

(Note: The values for \(\frac{1}{\lambda}\) are rounded to a reasonable number of significant figures.)