Question

what is the wavelength (in nm) of a photon if the energy is 6.85e - 19 j? (h = 6.626×10^(-34) j·s)

Explanation:

Step1: Recall energy - wavelength formula

The energy of a photon is given by $E = h\frac{c}{\lambda}$, where $E$ is energy, $h = 6.626\times10^{-34}\ J\cdot s$ is Planck's constant, $c= 3\times 10^{8}\ m/s$ is the speed of light, and $\lambda$ is the wavelength. We need to solve for $\lambda$, so $\lambda=\frac{hc}{E}$.

Step2: Substitute the given values

$h = 6.626\times 10^{-34}\ J\cdot s$, $c = 3\times 10^{8}\ m/s$, and $E=6.85\times 10^{-19}\ J$.
$\lambda=\frac{6.626\times 10^{-34}\ J\cdot s\times3\times 10^{8}\ m/s}{6.85\times 10^{-19}\ J}$

Step3: Calculate the wavelength in meters

$\lambda=\frac{6.626\times3\times 10^{-34 + 8}}{6.85\times 10^{-19}}\ m=\frac{19.878\times 10^{-26}}{6.85\times 10^{-19}}\ m$.
Using the rule of exponents $\frac{a\times10^{m}}{b\times10^{n}}=\frac{a}{b}\times10^{m - n}$, we get $\lambda=\frac{19.878}{6.85}\times10^{-26+19}\ m\approx2.902\times 10^{-7}\ m$.

Step4: Convert wavelength to nanometers

Since $1\ m = 10^{9}\ nm$, then $\lambda=2.902\times 10^{-7}\ m\times10^{9}\ nm/m = 290.2\ nm$.

Answer:

$290.2$