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Chapter 8: Problem 73

Microwave ovens convert radiation to energy. A microwave oven uses radiation with a wavelength of \(12.5 \mathrm{~cm}\). Assuming that all the energy from the radiation is converted to heat without loss, how many moles of photons are required to raise the temperature of a cup of water \((350.0 \mathrm{~g}\), specific heat \(=4.18 \mathrm{~J} / \mathrm{g} \cdot{ }^{\circ} \mathrm{C}\) ) from \(23.0^{\circ} \mathrm{C}\) to \(99.0^{\circ} \mathrm{C} ?\)

Short Answer

Expert verified
Answer: 1. Calculate the energy required to heat the water: Q = (350 g)(4.18 J/g°C)(76°C) = 111124 J 2. Calculate the energy of one photon: E = (6.626 × 10^-34 J·s)(3.0 × 10^8 m/s) / 0.125 m = 1.59 × 10^-24 J 3. Calculate the number of photons required: n = 111124 J / 1.59 × 10^-24 J = 6.99 × 10^26 photons 4. Convert the number of photons to moles of photons: Moles of photons = 6.99 × 10^26 photons / 6.022 × 10^23 photons/mole = 1161.16 moles Therefore, 1161.16 moles of photons are required to heat the water from 23.0°C to 99.0°C.

Step by step solution

01

Calculate Energy Required to Heat the Water

First, calculate the energy (Q) required to heat the water from its initial temperature (23.0°C) to its final temperature (99.0°C) using the formula: \(Q = mcΔT\) where m is the mass of water (350 g), c is the specific heat of water (4.18 J/g°C), and ΔT is the change in temperature (99.0°C - 23.0°C). \(Q = (350 \mathrm{ ~g}) (4.18 \mathrm{ ~J}/\mathrm{g }^{\circ} \mathrm{C})(99.0^{\circ} \mathrm{C} - 23.0^{\circ} \mathrm{C})\) Calculate the energy required.
02

Calculate Energy of One Photon

Next, calculate the energy (E) of one photon using Planck's equation: \(E = \dfrac{hc}{λ}\) where h is Planck's constant (\(6.626 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}\)), c is the speed of light (\(3.0 \times 10^8 \mathrm{ ~m/s}\)), and λ is the wavelength of radiation given in meters (12.5 cm = 0.125 m). \(E = \dfrac{(6.626 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}) (3.0 \times 10^8 \mathrm{ ~m/s})}{0.125 \mathrm{ ~m}}\) Calculate the energy of one photon.
03

Calculate Number of Photons Required

Now find the number of photons (n) required to provide the energy found in step 1, using the energy of one photon calculated in step 2: \(n = \dfrac{Q}{E}\) Calculate the number of photons required.
04

Convert Number of Photons to Moles

Finally, convert the number of photons to moles of photons using Avogadro's number (\(6.022 \times 10^{23} \hspace{1mm} \mathrm{photons/mole}\)): Moles of photons = \(\dfrac{\mathrm{Number \hspace{1mm} of \hspace{1mm} Photons}}{6.022 \times 10^{23} \hspace{1mm} \mathrm{photons/mole}}\) Calculate the number of moles of photons required to heat the water.

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Most popular questions from this chapter

On a hot day, you take a six-pack of soda on a picnic, cooling it with ice. Each empty (aluminum) can weighs \(12.5 \mathrm{~g}\) and contains \(12.0\) oz of soda. The specific heat of aluminum is \(0.902 \mathrm{~J} / \mathrm{g} \cdot{ }^{\circ} \mathrm{C}\); take that of soda to be \(4.10 \mathrm{~J} / \mathrm{g} \cdot{ }^{\circ} \mathrm{C}\) (a) How much heat must be absorbed from the six-pack to lower the temperature from \(25.0^{\circ}\) to \(5.0^{\circ} \mathrm{C}\) ? (b) How much ice must be melted to absorb this amount of heat? \(\left(\Delta H_{\mathrm{fus}}\right.\) of ise is given in Table 8.2.)

Find (a) \(\Delta E\) when a gas absorbs \(18 \mathrm{~J}\) of heat and has \(13 \mathrm{~J}\) of work done on it. (b) \(q\) when 72 J of work is done on a system and its energy is increased by \(61 \mathrm{~J}\).

Determine whether the statements given below are true or false. Consider specific heat. (a) Specific heat represents the amount of heat required to raise the temperature of one gram of a substance by \(1^{\circ} \mathrm{C}\). (b) Specific heat is the amount of heat flowing into the system. (c) When 20 J of heat is added to equal masses of different materials at \(25^{\circ} \mathrm{C}\), the final temperature for all these materials will be the same. (d) Heat is measured in \({ }^{\circ} \mathrm{C}\).

Calcium carbide, \(\mathrm{CaC}_{2}\), is the raw material for the production of acetylene (used in welding torches). Calcium carbide is produced by reacting calcium oxide with carbon, producing carbon monoxide as a byproduct. When one mole of calcium carbide is formed, \(464.8 \mathrm{~kJ}\) is absorbed. (a) Write a thermochemical equation for this reaction. (b) Is the reaction exothermic or endothermic? (c) Draw an energy diagram showing the path of this reaction. (Figure \(8.4\) is an example of such an energy diagram.) (d) What is \(\Delta H\) when \(1.00 \mathrm{~g}\) of \(\mathrm{CaC}_{2}(\mathrm{~g})\) is formed? (e) How many grams of carbon are used up when \(20.00 \mathrm{~kJ}\) of heat is absorbed?

Strontium metal is responsible for the red color in fireworks. Fireworks manufacturers use strontium carbonate, which can be produced by combining strontium metal, graphite (C), and oxygen gas. The formation of one mole of \(\mathrm{SrCO}_{3}\) releases \(1.220 \times 10^{3} \mathrm{~kJ}\) of energy. (a) Write a balanced thermochemical equation for the reaction. (b) What is \(\Delta H\) when \(10.00 \mathrm{~L}\) of oxygen at \(25^{\circ} \mathrm{C}\) and \(1.00 \mathrm{~atm}\) is used by the reaction?
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