Question

plant growth in various lights on the electromagnetic spectrum:

1. fill in the missing blanks in the chart. show your work!

use the equations: c = λf and e = hf
h = 6.626 x 10^{-34} js c = 3.0 x 10^{8} m/s

Explanation:

Step1: Calculate wavelength for Red

Using the formula \(c = \lambda f\), we can solve for \(\lambda\) (wavelength). Rearranging the formula gives \(\lambda=\frac{c}{f}\).
Given \(c = 3.0\times10^{8}\text{ m/s}\) and \(f = 4.38\times 10^{14}\text{ Hz}\)
\(\lambda=\frac{3.0\times 10^{8}}{4.38\times 10^{14}}=\frac{3.0}{4.38}\times10^{8 - 14}\approx0.685\times10^{- 6}=6.85\times10^{-7}\text{ m}\)

Step2: Calculate frequency for Orange

Using \(c=\lambda f\), we solve for \(f\). So \(f = \frac{c}{\lambda}\)
Given \(\lambda=6.05\times10^{-7}\text{ m}\) and \(c = 3.0\times10^{8}\text{ m/s}\)
\(f=\frac{3.0\times 10^{8}}{6.05\times 10^{-7}}=\frac{3.0}{6.05}\times10^{8+7}\approx0.496\times10^{15}=4.96\times10^{14}\text{ Hz}\)

Step3: Calculate frequency for Green

Using \(E = hf\), we first need to find \(f\). Rearranging gives \(f=\frac{E}{h}\)
Given \(E = 3.82\times10^{-19}\text{ J}\) and \(h = 6.626\times10^{-34}\text{ Js}\)
\(f=\frac{3.82\times 10^{-19}}{6.626\times 10^{-34}}=\frac{3.82}{6.626}\times10^{-19 + 34}\approx0.576\times10^{15}=5.76\times10^{14}\text{ Hz}\)
Then using \(c=\lambda f\) to find \(\lambda=\frac{c}{f}=\frac{3.0\times 10^{8}}{5.76\times 10^{14}}=\frac{3.0}{5.76}\times10^{8-14}\approx0.521\times10^{-6}=5.21\times10^{-7}\text{ m}\)

Step4: Calculate frequency for Blue

Using \(c=\lambda f\), with \(\lambda = 4.73\times10^{-7}\text{ m}\) and \(c = 3.0\times10^{8}\text{ m/s}\)
\(f=\frac{3.0\times 10^{8}}{4.73\times 10^{-7}}=\frac{3.0}{4.73}\times10^{8 + 7}\approx0.634\times10^{15}=6.34\times10^{14}\text{ Hz}\)
Then using \(E=hf\), with \(h = 6.626\times10^{-34}\text{ Js}\) and \(f = 6.34\times10^{14}\text{ Hz}\)
\(E=6.626\times10^{-34}\times6.34\times10^{14}=6.626\times6.34\times10^{-34 + 14}\approx4.20\times10^{-19}\text{ J}\)

Step5: Calculate frequency for Violet

Using \(c=\lambda f\), with \(\lambda=4.15\times10^{-7}\text{ m}\) and \(c = 3.0\times10^{8}\text{ m/s}\)
\(f=\frac{3.0\times 10^{8}}{4.15\times 10^{-7}}=\frac{3.0}{4.15}\times10^{8+7}\approx0.723\times10^{15}=7.23\times10^{14}\text{ Hz}\)
Using \(E = hf\), with \(h = 6.626\times10^{-34}\text{ Js}\) and \(f = 7.23\times10^{14}\text{ Hz}\)
\(E=6.626\times10^{-34}\times7.23\times10^{14}=6.626\times7.23\times10^{-34 + 14}\approx4.80\times10^{-19}\text{ J}\)

Answer:

	Wavelength (m)	Frequency (hertz)	Photon Energy (J)	Spinach Avg. Growth (cm)	Radish Avg. Growth (cm)	Lettuce Avg. Growth (cm)
Orange	\(6.05\times 10^{-7}\)	\(4.96\times 10^{14}\)	\(3.29\times 10^{-19}\)	15	9	7
Green	\(5.21\times 10^{-7}\)	\(5.76\times 10^{14}\)	\(3.82\times 10^{-19}\)	3	2	4
Blue	\(4.73\times 10^{-7}\)	\(6.34\times 10^{14}\)	\(4.20\times 10^{-19}\)	20	15	13
Violet	\(4.15\times 10^{-7}\)	\(7.23\times 10^{14}\)	\(4.80\times 10^{-19}\)	17	11	9