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Chapter 12: Problem 14

A photon of energy \(1.75 \mathrm{eV}\) has a wavelength of most nearly (A) \(710 \mathrm{~nm}\) (B) \(700 \mathrm{~nm}\) (C) \(650 \mathrm{~nm}\) (D) \(600 \mathrm{~nm}\) (E) \(550 \mathrm{~nm}\)

Short Answer

Expert verified
Answer: (A) 710 nm

Step by step solution

01

Use Planck's formula

Planck's formula relates the energy of a photon (E) to its wavelength (λ) and frequency (ν): \(E = hν\), where h is Planck's constant \((6.626 \times 10^{-34} \mathrm{J\cdot s})\). We also know the relationship between the wavelength and frequency of a photon is given by \(ν = \dfrac{c}{λ} \), where c is the speed of light \((3 \times 10^8 \mathrm{\,m/s})\). Combining these two equations, we can find the relationship between energy and wavelength: \(E = h\left(\dfrac{c}{λ}\right) \) Now solve for λ: \(λ = \dfrac{hc}{E}\)
02

Plug in the given values and convert units

Given energy E = 1.75 eV, we need to convert eV to Joules (J): \(E = 1.75 \mathrm{eV} \times \dfrac{1.6 \times 10^{-19} \mathrm{J}}{1 \mathrm{eV}} = 2.8 \times 10^{-19} \mathrm{J}\) Now plug in the values of h, c, E into the equation: \(λ = \dfrac{(6.626 \times 10^{-34}\mathrm{J\cdot s})(3 \times 10^8 \mathrm{m/s})}{2.8 \times 10^{-19}\mathrm{J}}\)
03

Calculate the wavelength and convert to nanometers

Calculate the wavelength in meters: \(λ = \dfrac{(6.626 \times 10^{-34}\mathrm{J\cdot s})(3 \times 10^8 \mathrm{m/s})}{2.8 \times 10^{-19}\mathrm{J}} = 7.115 \times 10^{-7}\mathrm{m}\) Convert this wavelength to nanometers: \(λ = 7.115 \times 10^{-7}\mathrm{m} \times \dfrac{10^9 \mathrm{nm}}{1 \mathrm{m}} = 711.5 \mathrm{nm}\)
04

Match the result to the given options

Compare the calculated wavelength (711.5 nm) to the options: (A) 710 nm (B) 700 nm (C) 650 nm (D) 600 nm (E) 550 nm The closest value to the calculated wavelength is \(710 \mathrm{nm}\). Therefore, the correct answer is (A).

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Most popular questions from this chapter

The half-life of the hydrogen isotope tritium is about 12 years. After a certain amount of time a fraction \(31 / 32\) of the atoms in the original sample has decayed. The time is most nearly equal to (A) 12 years (B) 24 years (C) 36 years (D) 48 years (E) 60 years

Meteorites created in the early solar system contaned aluminum-26, which is a radioactive isotope of aluminum with a half-life of \(7.2 \times 10^5 \mathrm{yrs}\). Aluminum-26 decays first into an excited state of magnesium-26 via the reaction \({ }_{13}^{26} \mathrm{Al} \rightarrow{ }_{12}^{26} \mathrm{Mg}^*+\mathrm{e}^{+}\), where the \(e^{+}\)has energy \(2.99 \mathrm{MeV}\). (The \(e^{+}\)is a positron; see previous problem. The asterisk (*) indicates "excited.") The \({ }_{12}^{26} \mathrm{Mg}^*\) then decays into the stable isotope magnesium- 26 via the reaction \({ }_{12}^{26} \mathrm{Mg}^* \rightarrow{ }_{12}^{26} \mathrm{Mg}+\gamma\). The \(\gamma\) has energy \(1.8 \mathrm{MeV}\). a) If you were asked to calculate the de Broglie wavelength of the positron, would it be permissible to use Newtonian physics? Justify your answer. b) What is the wavelength of the photon emitted when the excited magnesium-26 decays into its ground state? What is its momentum? c) \({ }_{12}^{26} \mathrm{Mg}\) has an atomic mass of \(25.9826 \mathrm{u}\). What is the speed of the recoiling nucleus when the photon is emitted? d) What is the nucleus' kinetic energy in electron volts? e) Precise measurements indicate that for a certain meteorite \(A\) the present ratio \({ }^{26} \mathrm{Mg} /{ }^{27} \mathrm{Al}=5 \times 10^{-5}\), where \({ }^{27} \mathrm{Al}\) is the common, stable isotope of aluminum. For a meteorite \(B\) the ratio is \({ }^{26} \mathrm{Mg} /{ }^{27} \mathrm{Al}=1.55 \times 10^{-7}\). Assuming that the different ratios are due to the difference in the meteorites' times of creation, how much older is meteorite \(B\) than \(A\) ?

Which of the following statements is true? The existence of the de Broglie wavelength \(\lambda_{d B}\) implies (A) that matter particles should undergo interference. (B) that matter waves travel at the speed of light. (C) that the frequency of matter waves is \(c / \lambda_{d B}\), where \(c\) is the speed of the particle. (D) that matter waves are given off by accelerating charges. (E) that matter waves are polarized.

Which of the following statements about the photon is true? i. The photon has a wavelength; ii. The photon has a mass; iii. The photon can undergo interference; iv. The photon is charged; \(\mathrm{v}\). The photon has a momentum. (A) i only (B) i, ii only (C) i, iii, v only (D) i, ii, iii, iv only (E) iii, v only

Two isotopes of an element ordinarily (A) have the same atomic number. (B) have the same atomic mass. (C) contain the same number of nucleons in the nucleus. (D) contain the same number of electrons. (E) (A) and (D)
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