Ch.7 - Quantum-Mechanical Model of the Atom

Tro - Chemistry: A Molecular Approach 4th Edition

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Tro 4th Edition

Ch.7 - Quantum-Mechanical Model of the Atom

Problem 41

Chapter 7, Problem 41

Calculate the energy of a photon of electromagnetic radiation at each of the wavelengths indicated in Problem 39. a. 632.8 nm (wavelength of red light from helium–neon laser) b. 503 nm (wavelength of maximum solar radiation) c. 0.052 nm (wavelength contained in medical X-rays)

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1. The energy of a photon can be calculated using the formula: E = h * c / λ, where E is the energy, h is Planck's constant (6.626 x 10^-34 J*s), c is the speed of light (3.00 x 10^8 m/s), and λ is the wavelength. Note that the wavelength must be in meters for the units to cancel out correctly.

2. Convert the given wavelengths from nanometers to meters. Remember that 1 nm = 1 x 10^-9 m. So, for example, 632.8 nm would be 632.8 x 10^-9 m.

3. Substitute the values of h, c, and λ into the formula for each of the given wavelengths. For example, for a wavelength of 632.8 nm, the calculation would be E = (6.626 x 10^-34 J*s) * (3.00 x 10^8 m/s) / (632.8 x 10^-9 m).

4. Repeat the calculation for the other given wavelengths (503 nm and 0.052 nm), remembering to convert the wavelengths to meters before substituting into the formula.

5. The results of these calculations will give the energy of a photon at each of the given wavelengths. Remember that the energy will be in joules (J) because of the units of Planck's constant (J*s) and the speed of light (m/s).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Photon Energy

The energy of a photon is directly proportional to its frequency and inversely proportional to its wavelength. This relationship is described by the equation E = hν, where E is energy, h is Planck's constant (6.626 x 10^-34 J·s), and ν (nu) is the frequency of the electromagnetic radiation. Since frequency and wavelength are related by the speed of light (c = λν), this can also be expressed as E = hc/λ, allowing for energy calculations based on wavelength.

Electromagnetic Spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation, ranging from radio waves to gamma rays. Each type of radiation is characterized by its wavelength and frequency. In the context of the question, visible light (like red and green light) and X-rays are part of this spectrum, with X-rays having much shorter wavelengths and higher energy compared to visible light.

Units of Measurement

In calculations involving photon energy, it is crucial to use consistent units. Wavelength is often given in nanometers (nm), where 1 nm = 10^-9 meters. Energy is typically expressed in joules (J). When calculating energy from wavelength, ensure to convert units appropriately, such as converting nm to meters, to maintain accuracy in the results.

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Textbook Question

List these types of electromagnetic radiation in order of (i) increasing frequency and (ii) decreasing energy per photon. a. gamma rays b. radio waves c. microwaves d. visible light

List these types of electromagnetic radiation in order of (ii) increasing energy per photon. a. radio waves b. microwaves c. infrared radiation d. ultraviolet radiation

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Textbook Question

A laser pulse with wavelength 532 nm contains 3.85 mJ of energy. How many photons are in the laser pulse?

A heat lamp produces 32.8 watts of power at a wavelength of 6.5 µm. How many photons are emitted per second? (1 watt = 1 J/s)

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Textbook Question

Determine the energy of 1 mol of photons for each kind of light. (Assume three significant figures.) a. infrared radiation (1500 nm) b. visible light (500 nm) c. ultraviolet radiation (150 nm)

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Textbook Question

The nearest star to our sun is Proxima Centauri, at a distance of 4.3 light-years from the sun. A light-year is the distance that light travels in one year (365 days). How far away, in km, is Proxima Centauri from the sun?

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