Question

a photon has a frequency of 6.50 × 10⁴ 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 \( h = 6.626\times 10^{-34}\ \text{J·s} \) (Planck's constant) and \(
u \) is the frequency.

Step2: Calculate energy of one photon

Given \(
u = 6.50\times 10^{4}\ \text{Hz} \), so energy of one photon \( E = 6.626\times 10^{-34}\ \text{J·s} \times 6.50\times 10^{4}\ \text{Hz} \)
\( E = 6.626\times 6.50\times 10^{-34 + 4}\ \text{J} \)
\( E = 43.069\times 10^{-30}\ \text{J} = 4.3069\times 10^{-29}\ \text{J} \)

Step3: Calculate energy of 1 mole of photons

1 mole of photons contains \( N_A = 6.022\times 10^{23}\) photons (Avogadro's number). So energy per mole \( E_{\text{mole}} = E\times N_A \)
\( E_{\text{mole}} = 4.3069\times 10^{-29}\ \text{J/photon} \times 6.022\times 10^{23}\ \text{photons/mol} \)
\( E_{\text{mole}} = 4.3069\times 6.022\times 10^{-29 + 23}\ \text{J/mol} \)
\( E_{\text{mole}} = 25.936\times 10^{-6}\ \text{J/mol} = 2.59\times 10^{-5}\ \text{J/mol} \) (rounded to three significant digits as the given frequency has three significant digits)

Answer:

\( 2.59\times 10^{-5} \)