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Chapter 7: Problem 97

A particular microwave oven delivers 750 watts. (A watt is a unit of power, which is the joules of energy delivered, or used, per second.) If the oven uses microwave radiation of wavelength \(12.6 \mathrm{~cm}\), how many photons of this radiation are required to heat \(1.00 \mathrm{~g}\) of water \(1.00^{\circ} \mathrm{C}\) assuming that all of the photons are absorbed?

Short Answer

Expert verified
Approximately \(2.65 \times 10^{24}\) photons are required.

Step by step solution

01

Determine the Energy Required to Heat Water

To heat 1.00 g of water by 1.00掳C, we use the specific heat capacity of water, which is 4.18 J/g掳C. The energy required is given by the product of the mass, specific heat capacity, and temperature change: \[ \text{Energy} = m \times c \times \Delta T = 1.00 \, \text{g} \times 4.18 \, \text{J/g掳C} \times 1.00 \, 掳\text{C} = 4.18 \, \text{J} \]
02

Calculate the Energy of a Single Photon

We can find the energy of a single photon using the equation \( E = \frac{hc}{\lambda} \), where \( h = 6.626 \times 10^{-34} \, \text{J鈰卻} \) is Planck's constant and \( c = 3.00 \times 10^8 \, \text{m/s} \) is the speed of light. The wavelength \( \lambda \) is given as 12.6 cm or 0.126 m \[ E = \frac{6.626 \times 10^{-34} \, \text{J鈰卻} \times 3.00 \times 10^8 \, \text{m/s}}{0.126 \, \text{m}} = 1.578 \times 10^{-24} \, \text{J} \]
03

Find the Number of Photons Required

To find out how many photons are needed to provide the required energy (4.18 J), we divide the total energy by the energy of one photon:\[ \text{Number of photons} = \frac{4.18 \, \text{J}}{1.578 \times 10^{-24} \, \text{J/photon}} \approx 2.65 \times 10^{24} \text{ photons} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Microwave Radiation
Microwave radiation is a form of electromagnetic energy. This type of radiation falls on the electromagnetic spectrum between radio waves and infrared light. Microwaves are widely used in various technologies, including microwave ovens. They are particularly useful for heating food and liquids, as they interact strongly with molecules like water to generate heat.

In a microwave oven, microwaves penetrate food and excite water molecules, causing them to vibrate and generate heat. This is why food heats efficiently in a microwave. In the context of our exercise, the microwave radiation has a wavelength of 12.6 cm. The energy of these waves is used to increase the temperature of water.
  • Microwaves cause water molecules to vibrate.
  • This vibration results in heat, which warms food or liquid.
  • The energy from microwaves is absorbed by the water.
Specific Heat Capacity of Water
The specific heat capacity of water is a fundamental property that measures the amount of heat required to raise the temperature of a given mass of water by one degree Celsius. For water, this value is 4.18 J/g掳C.

This means that to increase the temperature of 1 gram of water by 1掳C, 4.18 Joules of energy are needed. In our exercise, the microwave radiation provides the energy necessary to change the water's temperature.
  • Water has a high specific heat capacity.
  • It requires 4.18 Joules per gram per Celsius degree increase.
  • This property makes water effective at storing heat.
Planck's Constant
Planck's constant is crucial in physics for quantifying the energy of photons based on their frequency. The constant is denoted by \( h \) and has a value of \( 6.626 \times 10^{-34} \) J路s.

In the exercise, Planck's constant is used to find the energy of a single microwave photon using the formula \( E = \frac{hc}{\lambda} \), where \( c \) is the speed of light and \( \lambda \) is the wavelength. This formula shows how energy is proportional to the frequency of the radiation.
  • Planck's constant relates energy and frequency.
  • It is fundamental in quantum mechanics.
  • Helps calculate energy for photons in electromagnetic radiation.
Energy Conversion
Energy conversion involves the transformation of energy from one form to another. In this scenario, the energy provided by the microwave radiation is converted into thermal energy that heats the water.

The conversion begins when the microwaves excite water molecules, leading to increased molecular motion and, thus, an increase in temperature. To understand the total energy required, we use this principle to convert microwave radiation energy into heat to achieve the desired temperature increase in the water.
  • The microwaves convert energy into heat.
  • This conversion heats the material.
  • Understanding energy conversion is key in many technological processes.

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Most popular questions from this chapter

The surface of a metal was illuminated with light whose wavelength is \(327 \mathrm{nm}\). Upon illumination, the metal ejected an electron with a speed of \(3.46 \times 10^{5} \mathrm{~m} / \mathrm{s}\). What is the photoelectric work function of the metal (in \(\mathrm{eV}\) )? Assuming that the metal is an element, use values given in the CRC Handbook of Chemistry and Physics to discover the element.

List the possible subshells for the \(n=6\) shell.

An atom emits yellow light when an electron makes the transition from the \(n=5\) to the \(n=1\) level. In separate experiments, suppose you bombarded the \(n=1\) level of this atom with red light, yellow light (obtained from the previous emission), and blue light. In which experiment or experiments would the electron be promoted to the \(n=5\) level?

In X-ray fluorescence spectroscopy, a material can be analyzed for its constituent elements by radiating the material with short-wavelength \(\mathrm{X}\) rays, which induce the atoms to emit longer-wavelength \(\mathrm{X}\) rays characteristic of those atoms. Tungsten, for example, emits characteristic \(\mathrm{X}\) rays of wavelength \(0.1476 \mathrm{nm}\). If an electron has an equivalent wavelength, what is its kinetic energy?

The term degeneracy means the number of different quantum states of an atom or molecule having the same energy. For example, the degeneracy of the \(n=2\) level of the hydrogen atom is 4 (a \(2 s\) quantum state, and three different \(2 p\) states). What is the degeneracy of the \(n=5\) level?
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