Question

what is the wavelength (in nm) of a photon if the energy is 7.47×10^(-19) j? (h = 6.626×10^(-34) j·s and c = 3.00×10^8 m/s)

Explanation:

Step1: Recall energy - wavelength formula

The energy of a photon is given by $E = \frac{hc}{\lambda}$, where $E$ is energy, $h$ is Planck's constant, $c$ is the speed of light, and $\lambda$ is the wavelength. We can re - arrange this formula to solve for $\lambda$: $\lambda=\frac{hc}{E}$.

Step2: Substitute given values

We are given $h = 6.626\times10^{-34}\ J\cdot s$, $c = 3.00\times 10^{8}\ m/s$, and $E=7.47\times 10^{-19}\ J$.
$\lambda=\frac{(6.626\times 10^{-34}\ J\cdot s)\times(3.00\times 10^{8}\ m/s)}{7.47\times 10^{-19}\ J}$

Step3: Calculate the wavelength in meters

$\lambda=\frac{6.626\times3.00\times 10^{-34 + 8}}{7.47\times 10^{-19}}\ m=\frac{19.878\times 10^{-26}}{7.47\times 10^{-19}}\ m$
Using the rule of exponents $\frac{a\times10^{m}}{b\times10^{n}}=\frac{a}{b}\times10^{m - n}$, we have $\lambda=\frac{19.878}{7.47}\times10^{-26+19}\ m\approx2.66\times 10^{-7}\ m$

Step4: Convert wavelength to nanometers

Since $1\ m = 10^{9}\ nm$, then $\lambda=(2.66\times 10^{-7}\ m)\times(10^{9}\ nm/m)=266\ nm$

Answer:

266