Question

what is the wavelength, in nm, of radiation with an energy of $3.371 \times 10^{-19}$ j?

$h = 6.626 \times 10^{-34}$ j·s; $c = 2.998 \times 10^8 \frac{m}{s}$

\bigcirc 744.5 nm
\bigcirc 589.2 nm
\bigcirc 152.5 nm
\bigcirc 170.0 nm

Explanation:

Step1: Recall the energy-wavelength formula

The formula relating energy (\(E\)), Planck's constant (\(h\)), speed of light (\(c\)), and wavelength (\(\lambda\)) is \(E = \frac{hc}{\lambda}\). We need to solve for \(\lambda\), so rearrange the formula to \(\lambda=\frac{hc}{E}\).

Step2: Substitute the given values

Substitute \(h = 6.626\times 10^{-34}\,J\cdot s\), \(c = 2.998\times 10^{8}\,\frac{m}{s}\), and \(E = 3.371\times 10^{-19}\,J\) into the formula:

$$ \lambda=\frac{(6.626\times 10^{-34}\,J\cdot s)\times(2.998\times 10^{8}\,\frac{m}{s})}{3.371\times 10^{-19}\,J} $$

Step3: Calculate the numerator

First, calculate the product of \(h\) and \(c\):

$$ (6.626\times 10^{-34})\times(2.998\times 10^{8}) = 6.626\times2.998\times 10^{-34 + 8}\approx 19.865\times 10^{-26}=1.9865\times 10^{-25}\,J\cdot m $$

Step4: Divide by energy to find wavelength in meters

Now divide the numerator by \(E\):

$$ \lambda=\frac{1.9865\times 10^{-25}\,J\cdot m}{3.371\times 10^{-19}\,J}\approx\frac{1.9865}{3.371}\times 10^{-25 + 19}\approx 0.5893\times 10^{-6}\,m = 5.893\times 10^{-7}\,m $$

Step5: Convert meters to nanometers

Since \(1\,m = 10^{9}\,nm\), multiply the wavelength in meters by \(10^{9}\) to get nanometers:

$$ \lambda = 5.893\times 10^{-7}\,m\times10^{9}\,\frac{nm}{m}=589.3\,nm\approx 589.2\,nm $$

Answer:

589.2 nm (corresponding to the option "589.2 nm")