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Chapter 6: Problem 105

Microwave ovens use microwave radiation to heat food. The energy of the microwaves is absorbed by water molecules in food and then transferred to other components of the food. (a) Suppose that the microwave radiation has a wavelength of $10 \mathrm{~cm} .\( How many photons are required to heat \)200 \mathrm{~mL}$ of water from 25 to \(75^{\circ} \mathrm{C} ?\) (b) Suppose the microwave's power is $1000 \mathrm{~W}\( ( 1 watt \)=1\( joule-second \)) .$ How long would you have to heat the water in part (a)?

Short Answer

Expert verified
\(a)\) First, calculate the frequency of the radiation: \(\nu = \frac{c}{\lambda} = \frac{3 \times 10^8 m/s}{0.1 m} = 3 \times 10^9 Hz\). Then, find the energy of one photon: \(E = h\nu = 6.63 \times 10^{-34} Js \times 3 \times 10^9 Hz = 1.99 \times 10^{-24} J\). Next, find the energy needed to heat the water: \(Q = mc\Delta T = (0.2 kg)(4.18 \times 10^3 J/kg·°C)(50^{\circ}C) = 4180 J\). Now, calculate the number of photons required: \(\text{Number of photons} = \frac{4180 J}{1.99 \times 10^{-24} J/photon} = 2.1 \times 10^{27} \text{ photons}\). \(b)\) Lastly, find the time taken to heat the water: \(t = \frac{Q}{P} = \frac{4180 J}{1000 W} = 4.18 \ \text{seconds}\).

Step by step solution

01

Calculate the energy of one photon

From the given wavelength (10 cm), first, we need to find the frequency of the microwave radiation using the formula: \(c = \lambda \nu\) Where: c = speed of light \(=3 \times 10^8 m/s\) \(\lambda\) = wavelength \(= 10 cm = 0.1 m\) \(\nu\) = frequency Divide both sides by \(\lambda\): \(\nu = \frac{c}{\lambda}\) Now, we can use Planck's formula to find the energy of one photon: \(E = h\nu\) Where: E = energy of one photon h = Planck's constant \(=6.63 \times 10^{-34} Js\)
02

Calculate the energy needed to heat the water

To find the energy required to heat the given amount of water, we can use the specific heat capacity formula: \(Q = mc\Delta T\) Where: Q = energy required to heat the water m = mass of water c = specific heat capacity of water \(=4.18 \times 10^3 J/kg·°C\) \(\Delta T = T_{final} - T_{initial}\) First, convert 200 mL of water to mass: 1 mL of water = 1 g 200 mL of water = 200 g = 0.2 kg Now, calculate the energy needed to heat the water from 25°C to 75°C: \(\Delta T = 75 - 25 = 50^{\circ}C\)
03

Calculate the number of photons required to heat the water

Now that we have the energy of one photon and the total energy needed to heat the water, we can calculate the number of photons required: \(\text{Number of photons} = \frac{\text{Total energy}}{\text{Energy of one photon}}\)
04

Calculate the time taken to heat the water using the microwave's power

Given the power of the microwave (1000 W = 1000 J/s), we can calculate the time taken to heat the water: \(P = \frac{Q}{t}\) Where: P = power Q = energy t = time Rearrange the formula to solve for time: \(t = \frac{Q}{P}\)

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Most popular questions from this chapter

Using Heisenberg's uncertainty principle, calculate the uncertainty in the position of (a) a 1.50 -mg mosquito moving at a speed of $1.40 \mathrm{~m} / \mathrm{s}\( if the speed is known to within \)\pm 0.01 \mathrm{~m} / \mathrm{s} ;\( (b) a proton moving at a speed of \)(5.00 \pm 0.01) \times 10^{4} \mathrm{~m} / \mathrm{s}$ (The mass of a proton is given in the table of fundamental constants in the inside cover of the text.)

Sketch the shape and orientation of the following types of orbitals: \((\mathbf{a}) s,(\mathbf{b}) p_{z},(\mathbf{c}) d_{x y}\)

Among the elementary subatomic particles of physics is the muon, which decays within a few microseconds after formation. The muon has a rest mass 206.8 times that of an electron. Calculate the de Broglie wavelength associated with a muon traveling at \(8.85 \times 10^{5} \mathrm{~cm} / \mathrm{s}\).

Identify the specific element that corresponds to each of the following electron configurations and indicate the number of unpaired electrons for each: (a) \(1 s^{2} 2 s^{2}\), (b) \(1 s^{2} 2 s^{2} 2 p^{4}\) (d) \([\mathrm{Kr}] 5 s^{2} 4 d^{10} 5 p^{4}\) (c) \([\mathrm{Ar}] 4 s^{1} 3 d^{5}\)

(a) What are the similarities of and differences between the \(1 s\) and \(2 s\) orbitals of the hydrogen atom? (b) In what sense does a \(2 p\) orbital have directional character? Compare the "directional" characteristics of the \(p_{x}\) and \(d_{x^{2}-y^{2}}\) orbitals. (That is, in what direction or region of space is the electron density concentrated?) (c) What can you say about the average distance from the nucleus of an electron in a \(2 s\) orbital as compared with a 3 s orbital? ( \(\mathbf{d}\) ) For the hydrogen atom, list the following orbitals in order of increasing energy (that is, most stable ones first): \(4 f, 6 s, 3 d, 1 s, 2 p\)
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