Question

what is the momentum of a photon with a de broglie wavelength of 5.4 m? estimate h as 6.63 × 10⁻³⁴ j s and keep 3 significant figures.

a 1.01 × 10⁻³⁴ kg·m/s

b 1.23 × 10⁻³⁴ kg·m/s

c 1.23 × 10⁻³⁵ kg·m/s

d 2.15 × 10⁻³⁴ kg·m/s

e 1.45 × 10⁻³⁴ kg·m/s

Explanation:

Step1: Recall de Broglie relation

The de Broglie wavelength formula is $\lambda = \frac{h}{p}$, so momentum $p = \frac{h}{\lambda}$.

Step2: Substitute values

Given $h = 6.63 \times 10^{-34}\, \text{J s}$ and $\lambda = 5.4\, \text{m}$.
Calculate $p = \frac{6.63 \times 10^{-34}}{5.4}$.
First, $6.63\div5.4 \approx 1.2278$.
So $p \approx 1.23 \times 10^{-34}\, \text{kg·m/s}$? Wait, no: $6.63\times10^{-34} \div 5.4 = (6.63\div5.4)\times10^{-34} \approx 1.2278\times10^{-34} \approx 1.23\times10^{-34}$? Wait, no, wait: 6.63 divided by 5.4: 5.41.2 = 6.48, 6.63-6.48=0.15, 0.15/5.4≈0.0278, so total≈1.2278. So $1.2278\times10^{-34} \approx 1.23\times10^{-34}$? But wait, the options have c as $1.23\times10^{-35}$. Wait, did I miscalculate? Wait, 6.63e-34 divided by 5.4: 6.63/5.4 = (6.63/5.4) = let's do 663/540 = 221/180 ≈1.2278. So 1.2278e-34? But option b is 1.23e-34, option c is 1.23e-35. Wait, maybe I made a mistake. Wait, 6.63e-34 divided by 5.4: 5.4 is 5.4, so 6.63e-34 / 5.4 = (6.63 / 5.4) 10^{-34} = 1.2278 * 10^{-34} ≈1.23e-34. But option b is 1.23e-34, option c is 1.23e-35. Wait, maybe the question is about a photon, but de Broglie wavelength for photon: but photon's momentum is also $p = E/c = h
u/c = h/\lambda$, same formula. Wait, maybe I miscalculated the exponent. Wait, 6.63e-34 divided by 5.4: 5.4 is 5.4, so 6.63e-34 / 5.4 = (6.63 / 5.4) * 10^{-34} ≈1.2278e-34 ≈1.23e-34. So option b is 1.23e-34, which matches. Wait, but the original selected option was c. Wait, maybe I messed up. Wait, 6.63e-34 divided by 5.4: 5.4 * 1.23e-35 = 5.4*1.23e-35 = 6.642e-35, which is not 6.63e-34. Wait, no: 1.23e-34 *5.4 = 6.642e-34, which is close to 6.63e-34. So the correct calculation is $p = h/\lambda = 6.63e-34 /5.4 ≈1.23e-34$ kg·m/s, which is option b. Wait, but the user's selected option was c. Maybe a typo? Wait, let's recalculate: 6.63e-34 divided by 5.4: 6.63 ÷ 5.4 = 1.227777... So 1.227777e-34 ≈1.23e-34, so option b is correct.

Answer:

b. $1.23 \times 10^{-34}\, \text{kg·m/s}$