Question

what is the energy of light with a wavelength of 652 nm? (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. 3.28 × 10¹⁸ j
b. 3.28 × 10²⁷ j
c. 3.05 × 10⁻²⁸ j
d. 3.05 × 10⁻¹⁹ j

Explanation:

Step1: Recall the formula for energy of a photon

The energy \( E \) of a photon is given by \( E = h
u \), where \( h \) is Planck's constant and \(
u \) is the frequency. Also, the relationship between speed of light \( c \), wavelength \( \lambda \), and frequency \(
u \) is \( c=\lambda
u \), so \(
u=\frac{c}{\lambda} \). Substituting \(
u \) into the energy formula, we get \( E = \frac{hc}{\lambda} \).

Step2: Convert wavelength to meters

The wavelength \( \lambda = 652\space nm \). Since \( 1\space nm = 10^{-9}\space m \), we have \( \lambda = 652\times 10^{-9}\space m=6.52\times 10^{-7}\space m \).

Step3: Substitute values into the formula

We know \( h = 6.626\times 10^{-34}\space J\cdot s \), \( c = 3.00\times 10^{8}\space m/s \), and \( \lambda = 6.52\times 10^{-7}\space m \). Plugging these into \( E=\frac{hc}{\lambda} \):

\[

$$\begin{align*} E&=\frac{(6.626\times 10^{-34}\space J\cdot s)(3.00\times 10^{8}\space m/s)}{6.52\times 10^{-7}\space m}\\ &=\frac{6.626\times 3.00\times 10^{-34 + 8}}{6.52\times 10^{-7}}\space J\\ &=\frac{19.878\times 10^{-26}}{6.52\times 10^{-7}}\space J\\ &=\frac{19.878}{6.52}\times 10^{-26+7}\space J\\ &\approx 3.05\times 10^{-19}\space J \end{align*}$$

\]

Answer:

D. \( 3.05 \times 10^{-19}\space J \)