Question

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 \(11.2 \mathrm{~cm}\). How many photons are required to heat \(200 \mathrm{~mL}\) of coffee from 23 to \(60^{\circ} \mathrm{C}\) ? (b) Suppose the microwave's power is \(900 \mathrm{~W}\) ( 1 watt \(=1\) joule-second). How long would you have to heat the coffee in part (a)?

Short answer

Approximately \(1.558 \times 10^{27}\) photons are required to heat 200 mL of coffee from 23°C to 60°C using microwave radiation with a wavelength of 11.2 cm. It would take approximately 34.37 seconds to heat the coffee using a microwave with a power of 900 W.

Step-by-step solution

Calculate the energy needed to heat the coffee

First, we need to find the energy needed to heat the coffee. The formula for calculating the energy required to change the temperature of a substance is: \[Q = mcΔT\] where \(Q\) is the energy needed, \(m\) is the mass of the substance, \(c\) is the specific heat capacity, and \(ΔT\) is the change in temperature. We have the volume of coffee (200 mL), which we can convert to mass, as the density of water is approximately 1 g/mL. The specific heat capacity of water is 4.186 J/g°C, and the change in temperature is from 23°C to 60°C. Mass of coffee, \(m = 200 \: \text{mL} \times 1 \: \frac{\text{g}}{\text{mL}} = 200 \: \text{g}\) Change in temperature, \(ΔT = 60°C - 23°C = 37°C\) Now, we can find the energy needed: \[Q = (200 \: \text{g}) \times (4.186 \: \frac{\text{J}}{\text{g°C}}) \times (37°C) = 30933.2 \: \text{J}\]

Calculate the energy of a single photon

Next, we need to find the energy of a single photon. The energy of a photon is given by: \[E = \frac{hc}{λ}\] where \(E\) is the energy of the photon, \(h\) is Planck's constant (\(6.63 \times 10^{-34} \: \text{J} \cdot \text{s}\)), \(c\) is the speed of light in a vacuum (\(3 \times 10^8 \: \frac{\text{m}}{\text{s}} \)), and \(λ\) is the wavelength of the microwave radiation. We are given the wavelength as 11.2 cm, which we need to convert to meters: \[λ = 11.2 \: \text{cm} \times \frac{1 \: \text{m}}{100 \: \text{cm}} = 0.112 \: \text{m}\] Now we can find the energy of a single photon: \[E = \frac{6.63 \times 10^{-34} \: \text{J} \cdot \text{s}}{0.112 \: \text{m} \times 3 \times 10^8 \: \frac{\text{m}}{\text{s}}} = 1.985 \times 10^{-24} \: \text{J}\]

Calculate the number of photons required

Now that we have both the energy needed to heat the coffee and the energy of a single photon, we can find the number of photons required: \[N = \frac{Q}{E}\] \[N = \frac{30933.2 \: \text{J}}{1.985 \times 10^{-24} \: \text{J}} = 1.558 \times 10^{27}\] Approximately \(1.558 \times 10^{27}\) photons are required to heat the coffee from 23°C to 60°C.

Calculate the time required to heat the coffee

In this part, we will calculate the time required to heat the coffee using the microwave with a power of 900 W. The formula to find the time is: \[t = \frac{Q}{P}\] where \(t\) is the time, \(Q\) is the energy needed, and \(P\) is the power. We are given the power as 900 W, which is equivalent to 900 joule/second: \[t = \frac{30933.2 \: \text{J}}{900 \: \frac{\text{J}}{\text{s}}} = 34.37 \: \text{s}\] It would take approximately 34.37 seconds to heat the coffee from 23°C to 60°C at a power of 900 W.

Key concepts

Photon Energy

When we talk about the energy of a photon, it refers to the packet of electromagnetic energy. Microwaves, like all electromagnetic waves, are composed of these tiny packets. The energy of a single photon can be determined using the formula: \[E = \frac{hc}{\lambda}\]where:

• \(E\) is the energy of the photon.
• \(h\) is Planck’s constant, valued at \(6.63 \times 10^{-34} \text{ J} \cdot \text{s}\).
• \(c\) is the speed of light in a vacuum, approximately \(3 \times 10^8 \text{ m/s}\).
• \(\lambda\) is the wavelength of the microwave.

Knowing the wavelength helps calculate how much energy each photon carries. For instance, a wavelength of \(11.2\) cm will yield a tiny energy value, since longer wavelengths indicate lower energy per photon. This small energy is why many photons are needed to heat even a small amount of water in a microwave.

Specific Heat Capacity

Specific heat capacity is a property that tells us how much energy is needed to raise the temperature of a substance. It is defined as the amount of heat required to increase the temperature of one gram of a substance by one degree Celsius. The formula used to calculate the energy needed for a temperature change is:\[Q = mc\Delta T\]where:

• \(Q\) is the energy in joules.
• \(m\) is the mass of the substance in grams.
• \(c\) is the specific heat capacity (for water it is \(4.186 \; \text{J/g°C}\)).
• \(\Delta T\) is the change in temperature in degrees Celsius.

In the case of heating coffee, the specific heat capacity of water is used because coffee is mainly water. Thus, the specific heat capacity assists in quantifying the energy required, showcasing why so many photons need to be absorbed to cause significant temperature increases.

Temperature Change

Temperature change is an essential part of understanding how energy affects substances. When you add energy to a substance, its temperature will change according to its specific heat capacity. This relationship is expressed as:

• \(\Delta T = \frac{Q}{mc}\)

Here, the change in temperature, \(\Delta T\), depends on how much energy, \(Q\), is added or removed, and the material's mass and specific heat capacity. For our coffee example, increasing the temperature from 23°C to 60°C requires calculating the energy needed with the temperature change \((\Delta T = 37°C)\). With this, you can determine how effectively the microwave transfers energy and how long it will take to reach the desired temperature.