Question

what is the wavelength of light with 2.89 × 10⁻¹⁹ j of energy? (the speed of light in a vacuum is 3.00 × 10⁸ m/s, and planck’s constant is 6.626 × 10⁻³⁴ j•s.)

a. 1450 nm

b. 688 nm

c. 1310 nm

d. 764 nm

Explanation:

Step1: Recall the energy - wavelength formula

The formula that relates the energy \(E\) of a photon to its wavelength \(\lambda\) is \(E = h
u\), and the relationship between the speed of light \(c\), frequency \(
u\) and wavelength \(\lambda\) is \(c=\lambda
u\), so \(
u=\frac{c}{\lambda}\). Substituting \(
u\) into the energy formula, we get \(E=\frac{hc}{\lambda}\), and then we can solve for \(\lambda\) as \(\lambda=\frac{hc}{E}\).

Step2: Identify the given values

We are given that \(E = 2.89\times10^{- 19}\space J\), \(h = 6.626\times10^{-34}\space J\cdot s\), and \(c=3.00\times10^{8}\space m/s\).

Step3: Substitute the values into the formula

Substitute the values of \(h\), \(c\) and \(E\) into the formula \(\lambda=\frac{hc}{E}\):
\[

$$\begin{align*} \lambda&=\frac{(6.626\times 10^{-34}\space J\cdot s)\times(3.00\times 10^{8}\space m/s)}{2.89\times 10^{-19}\space J}\\ &=\frac{6.626\times3.00\times10^{-34 + 8}}{2.89}\space m\\ &=\frac{19.878\times10^{-26}}{2.89}\space m\\ &\approx6.878\times 10^{-26}\space m\\ &=6.878\times10^{-26}\times10^{9}\space nm\quad(\text{since }1\space m = 10^{9}\space nm)\\ &\approx688\space nm \end{align*}$$

\]

Answer:

B. 688 nm