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Chapter 7: Problem 138

A certain microwave oven delivers 750 . watts \((\mathrm{J} / \mathrm{s})\) of power to a coffee cup containing \(50.0 \mathrm{~g}\) water at \(25.0^{\circ} \mathrm{C}\). If the wavelength of microwaves in the oven is \(9.75 \mathrm{~cm}\), how long does it take, and how many photons must be absorbed, to make the water boil? The specific heat capacity of water is \(4.18 \mathrm{~J} /{ }^{\circ} \mathrm{C} \cdot \mathrm{g}\), and assume only the water absorbs the energy of the microwaves.

Short Answer

Expert verified
The water requires 15,675 Joules of energy to reach boiling point, and it takes approximately 20.9 seconds for the microwave to deliver that amount of energy. The energy of a single microwave photon with a wavelength of 9.75 cm is approximately \(2.042 \times 10^{-23}\) Joules. To make the water boil, approximately \(7.67 \times 10^{23}\) photons need to be absorbed by the water.

Step by step solution

01

Calculate energy needed to heat the water to boiling point

First, we'll find the energy needed to heat the water from 25°C to 100°C using the specific heat capacity formula: Energy = m × c × ΔT where: - Energy is the energy required to heat the water (J) - m is the mass of the water (g) - c is the specific heat capacity of water (J/(°C·g)) - ΔT is the change in temperature (°C) Energy = (50.0 g) × (4.18 J/(°C·g)) × (100°C - 25°C) Energy = 50.0 × 4.18 × 75 = 15675 J The energy required to heat the water is 15,675 Joules.
02

Determine the time required for the heating process

Now, we will divide the energy required to heat the water by the microwave's power to find the time it takes: Time = Energy / Power where: - Time is the time required to heat the water (s) - Energy is the energy required to heat the water (J) - Power is the power of the microwave (J/s) Time = 15675 J / 750 J/s = 20.9 s It takes approximately 20.9 seconds to heat the water to boiling point.
03

Calculate the energy of a single microwave photon

Using the wavelength of the microwave and the speed of light (c), we can calculate the energy of a single microwave photon using the following formula: E_photon = (h × c) / λ where: - E_photon is the energy of a single microwave photon (J) - h is the Planck's constant (6.63 × 10⁻³⁴ Js) - c is the speed of light (3 × 10⁸ m/s) - λ is the wavelength of microwaves (9.75 cm or 0.0975 m) E_photon = (6.63 × 10⁻³⁴ Js × 3 × 10⁸ m/s) / 0.0975 m = 2.042 × 10⁻²³ J The energy of a single microwave photon is approximately 2.042 × 10⁻²³ Joules.
04

Calculate the number of absorbed photons

Finally, we can calculate the number of photons required to heat the water by dividing the total energy required by the energy of a single photon: Number of photons = Energy / E_photon where: - Number of photons is the number of photons required to heat the water - Energy is the energy required to heat the water (J) - E_photon is the energy of a single microwave photon (J) Number of photons = 15675 J / 2.042 × 10⁻²³ J = 7.67 × 10²³ photons Approximately 7.67 × 10²³ photons need to be absorbed by the water to make it boil.

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Most popular questions from this chapter

Which of the following statements is(are) true? a. \(\mathrm{F}\) has a larger first ionization energy than does \(\mathrm{Li}\). b. Cations are larger than their parent atoms. c. The removal of the first electron from a lithium atom (electron configuration is \(1 s^{2} 2 s^{1}\) ) is exothermic - that is, removing this electron gives off energy. d. The He atom is larger than the \(\mathrm{H}^{+}\) ion. e. The \(\mathrm{Al}\) atom is smaller than the \(\mathrm{Li}\) atom.

The ionization energy for a \(1 s\) electron in a silver atom is \(2.462 \times 10^{6} \mathrm{~kJ} / \mathrm{mol}\) a. Determine an approximate value for \(Z_{\text {eff }}\) for the Ag \(1 s\) electron. Assume the Bohr model applies to the \(1 s\) electron. \(Z_{\text {eff }}\) is the apparent nuclear charge experienced by the electrons. b. How does \(Z_{\text {eff }}\) from part a compare to \(Z\) for Ag? Rationalize the relative numbers.

A certain oxygen atom has the electron configuration \(1 s^{2} 2 s^{2} 2 p_{x}^{2} 2 p_{y}^{2} .\) How many unpaired electrons are present? Is this an excited state of oxygen? In going from this state to the ground state, would energy be released or absorbed?

Draw atomic orbital diagrams representing the ground-state electron configuration for each of the following elements. a. Na b. Co c. \(\mathrm{Kr}\) How many unpaired electrons are present in each element?

Order the atoms in each of the following sets from the least exothermic electron affinity to the most. a. \(\mathrm{N}, \mathrm{O}, \mathrm{F}\) b. Al, Si, \(\mathrm{P}\)
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