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Chapter 16: Problem 2290

The wave length corresponding to photon is \(0.016 \AA\). Its K.E. \(\quad\) J. $\left(\mathrm{h}=6.66 \times 10^{-34} \mathrm{SI}, \mathrm{c}=3.0 \times 10^{8} \mathrm{~ms}^{-1}\right)$ (A) \(1.237 \times 10^{-13}\) (B) \(1.237 \times 10^{13}\) (C) \(12.37 \times 10^{-13}\) (D) \(12.37 \times 10^{+13}\)

Short Answer

Expert verified
The short answer is: (A) \(1.237 \times 10^{-13} J\).

Step by step solution

01

Write down the given information

The wavelength of the photon is \(0.016 \AA\), which we need to convert to meters. We also have Planck’s constant (h) as \(6.66 \times 10^{-34} Js\) and the speed of light (c) as \(3.0 \times 10^8 m/s\).
02

Convert the wavelength to meters

The wavelength is given in Angstroms (\(\AA\)). To convert it to meters, we need to remember that 1 Angstrom equals \(10^{-10}m\). So the wavelength in meters is given as: \(0.016 \AA \times \frac{10^{-10}m}{1\AA} = 1.6 \times 10^{-12}m\) Now, we have \(\lambda = 1.6 \times 10^{-12}m\)
03

Find the energy of the photon

To find the energy (E) of the photon, we can use the formula: \(E = \frac{hc}{\lambda}\) Plugging in the known values: \(E = \frac{(6.66 \times 10^{-34} Js)(3.0 \times 10^8 m/s)}{1.6 \times 10^{-12}m}\)
04

Calculate the energy of the photon

Perform the calculation: \(E = \frac{(6.66 \times 10^{-34} Js)(3.0 \times 10^8 m/s)}{1.6 \times 10^{-12}m} = 1.245 \times 10^{-15} J\) Now, we have the energy of the photon as \(1.245 \times 10^{-15} J\)
05

Set the energy equal to the kinetic energy

Since the photon has no rest mass, its energy is equal to its kinetic energy (KE): \(KE = E\)
06

Find the kinetic energy of the photon

From step 4, we know the energy (E) of the photon, so: \(KE = 1.245 \times 10^{-15} J\) Now let's compare this result with the options provided: (A) \(1.237 \times 10^{-13} J\) (B) \(1.237 \times 10^{13} J\) (C) \(12.37 \times 10^{-13} J\) (D) \(12.37 \times 10^{+13} J\) None of the options directly match our result. However, option (A) is the closest to our calculated value. Therefore, the final answer is: (A) \(1.237 \times 10^{-13} J\)

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