Question

a photon has a frequency of 3.30 × 10⁷ hz. calculate the energy (in joules) of 1 mole of photons with this frequency. be sure your answer has the correct number of significant digits.

Explanation:

Step1: Recall the formula for photon energy

The energy of a single photon is given by \( E = h
u \), where \( h \) is Planck's constant (\( h = 6.626\times 10^{-34}\ \text{J·s} \)) and \(
u \) is the frequency.

Step2: Calculate energy of one photon

Substitute \( h = 6.626\times 10^{-34}\ \text{J·s} \) and \(
u = 3.30\times 10^{7}\ \text{Hz} \) into the formula:
\( E_{\text{photon}} = 6.626\times 10^{-34}\ \text{J·s} \times 3.30\times 10^{7}\ \text{Hz} \)
\( E_{\text{photon}} = 6.626\times 3.30\times 10^{-34 + 7}\ \text{J} \)
\( E_{\text{photon}} = 21.8658\times 10^{-27}\ \text{J} = 2.18658\times 10^{-26}\ \text{J} \)

Step3: Calculate energy of 1 mole of photons

One mole of photons contains Avogadro's number (\( N_A = 6.022\times 10^{23}\ \text{mol}^{-1} \)) of photons. So the energy per mole (\( E_{\text{mole}} \)) is:
\( E_{\text{mole}} = E_{\text{photon}} \times N_A \)
\( E_{\text{mole}} = 2.18658\times 10^{-26}\ \text{J/photon} \times 6.022\times 10^{23}\ \text{photons/mol} \)
\( E_{\text{mole}} = 2.18658\times 6.022\times 10^{-26 + 23}\ \text{J/mol} \)
\( E_{\text{mole}} = 13.1676\times 10^{-3}\ \text{J/mol} = 0.0131676\ \text{J/mol} \approx 0.0132\ \text{J/mol} \) (or in scientific notation \( 1.32\times 10^{-2}\ \text{J/mol} \))

Answer:

\( 1.32\times 10^{-2}\) (or \( 0.0132\))