Question #654cf

May 11, 2017

Question 1: $E = 5.0 \cdot {10}^{-} 18$ $\text{J}$
Question 2: $E = 3.0 \cdot {10}^{6}$ $\text{J/mol}$

Explanation:

Use the formulas:
$E = h \nu$, where $E$ is energy (J), $h$ is Planck's constant (Js), and $\nu$ is the frequency (Hz).
and
$c = \lambda \nu$, where $c$ is the speed of light in a vacuum (m/s), $\lambda$ is the length of a wavelength (m), and $\nu$ is the frequency (Hz).
Solving the second formula for $\nu$, we get $\nu = \frac{c}{\lambda}$

Combining these formulas, we get:
$E = \frac{h c}{\lambda}$

For the first question, we are given that $\lambda = 40$ nm or $40 \cdot {10}^{-} 9$ m. By substituting into the above formula, we get:

$E = \frac{\left(6.626 \cdot {10}^{-} 34\right) \left(3 \cdot {10}^{8}\right)}{40 \cdot {10}^{-} 9} = 4.9695 \cdot {10}^{-} 18$
$E = 5.0 \cdot {10}^{-} 18$ $\text{J}$ with proper significant figures

For the second question, we can use our answer for the first question (the amount of energy for a single photon) and Avogadro's number to find the the amount of energy in a mole of photons with wavelength of $40 \cdot {10}^{-} 9$ m:

$E = \left(4.9695 \cdot {10}^{-} 18\right) \cdot \left(6.02 \cdot {10}^{23}\right) = 2.99164 \cdot {10}^{6}$
$E = 3.0 \cdot {10}^{6}$ $\text{J/mol}$ with proper significant figures