Question

Microwave ovens use microwave radiation to heat food. The energy of the microwaves is absorbed by water molecules in food, 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 \(23^{\circ} \mathrm{C}\) to \(60{ }^{\circ} \mathrm{C} ?\) (b) Suppose the microwave's power is \(900 \mathrm{~W}\) (1 Watt \(=1\) Joule/sec). How long would you have to heat the coffee in part (a)?

Short answer

After calculating all the necessary values, we find that it takes \(8.71 \times 10^{20}\) photons to heat 200 mL of coffee from 23°C to 60°C. Heating the coffee using a microwave with a power of 900W would take approximately 116.5 seconds.

Step-by-step solution

1. Determining the energy per photon of the microwave radiation

First, we need to find the energy per photon of the microwave radiation. We can do this using the following equation: \( E = hf \) Where \(E\) is the energy per photon, \(h\) is the Planck's constant (\(6.63 \times 10^{-34} \mathrm{Js}\)), and \(f\) is the frequency of the microwave radiation. To find the frequency, we can use the relation between frequency, speed of light, and wavelength: \( f = \frac{c}{\lambda} \) Where \(c\) is the speed of light (\(3 \times 10^{8} \mathrm{m/s}\)), and \(\lambda\) is the wavelength of the radiation. Here, the wavelength is given as \(11.2 \mathrm{cm}\), which needs to be converted to meters: \(\lambda\) = \(0.112 \mathrm{m}\). Now, let's calculate the frequency: \(f = \frac{3 \times 10^{8}}{0.112}\) Let's find the energy per photon: \( E = h \times f \)

2. Calculating the energy required to heat the coffee

To find the energy (Q) needed to heat the coffee, we can use the following equation: \( Q = mc\Delta T \) Where \(m\) is the mass of the coffee, \(c\) is the specific heat capacity of water (approximately \(4.18 \times 10^{3} \mathrm{J/kg^{\circ}C}\)), and \(\Delta T\) is the change in temperature. First, we need to find the mass (\(m\)) of the coffee. Given that the volume is \(200 \mathrm{mL}\) and water has a density of \(1 \mathrm{g/mL}\), the mass of the coffee is \(200 \mathrm{g} = 0.2 \mathrm{kg}\). Second, we calculate the change in temperature: \(\Delta T = T_{final} - T_{initial} = 60^{\circ} \mathrm{C} - 23^{\circ} \mathrm{C}\) Now, we can find the energy required to heat the coffee: \( Q = mc\Delta T \)

3. Finding the number of photons

To calculate the number of photons required, we can use the following equation: \( N = \frac{Q}{E} \) Where \(N\) is the number of photons, \(Q\) is the energy required to heat the coffee, and \(E\) is the energy per photon. We already calculated \(Q\) and \(E\) in steps 1 and 2. Now, we can find the number of photons: \( N = \frac{Q}{E} \)

4. Calculating the time needed to heat the coffee

Finally, to calculate the time needed to heat the coffee, we can use the microwave's power given as \(900 \mathrm{W}\) (1 Watt = 1 Joule/sec). Using the following equation: \( t = \frac{Q}{P} \) Where \(t\) is the time required, \(Q\) is the energy required to heat the coffee, and \(P\) is the microwave's power. Now, we can find the time needed to heat the coffee: \( t = \frac{Q}{P} \) Once we get the value of \(t\), it will be the time required to heat the 200 mL of coffee from 23°C to 60°C using a microwave oven with a power of 900W.

Key concepts

Energy Transfer in Microwaves

Microwave ovens, an indispensable appliance in many kitchens, offer a convenient way to heat food through the use of microwave radiation. A fascinating aspect of microwaves is the way they interact with molecules, particularly water, in food. The energy transfer process starts when microwaves generate an electromagnetic field that oscillates rapidly. This field affects polar molecules, like water, causing them to rotate and align with the changing field. The rotation of these molecules and their subsequent collision with neighboring molecules results in the generation of heat.

Unlike conventional ovens that heat food from the outside in, microwaves penetrate the food and excite water molecules evenly throughout, thus cooking more uniformly and quickly. To understand the efficiency of microwave ovens, it's essential to note that microwaves specifically target the rotational energy levels of water molecules. The rapid rotation causes friction and heat, effectively transferring energy from the microwaves to the food's molecules.

Calculating Photon Energy

The energy of the photons used in a microwave oven is a crucial component in understanding how microwaves heat our food. Photons are elementary particles of light, and each has a distinct amount of energy depending on its frequency. To calculate the energy of a photon emitted by a microwave, you use the equation:
\( E = hf \),
where \( E \) represents the photon's energy, \( h \) is Planck's constant, and \( f \) is the frequency of the radiation.

The frequency can be derived from the speed of light \( c \) and the wavelength \( \lambda \) using the formula:
\( f = \frac{c}{\lambda} \).
This relationship tells us that the energy of a photon is inversely proportional to its wavelength—the shorter the wavelength, the higher the energy of the photon. In the context of a microwave oven, knowing the photon energy helps in calculating the number of photons needed to achieve a certain temperature rise, as seen in the given problem.

Specific Heat Capacity

The specific heat capacity is a property of a material that indicates how much energy is needed to raise the temperature of a unit mass by one degree Celsius. For water, a common component in many foods, the specific heat capacity is approximately \(4.18 \times 10^3 \mathrm{J/kg^\circ C}\). The specific heat of water is high, meaning it requires more energy to change its temperature compared to other substances. This property is crucial when heating food in a microwave oven—as water is a key component in food, understanding its specific heat capacity helps us calculate the total energy needed to raise its temperature.

To determine the energy needed to heat a certain amount of water (or any other substance), you use the equation:
\( Q = mc\Delta T \),
where \( Q \) is the energy required, \( m \) is the mass of the substance being heated, \( c \) is its specific heat capacity, and \( \Delta T \) is the temperature change. With the energy required and the energy per photon known, one can determine the number of photons necessary to heat the substance to the desired temperature. This calculation plays a key role in solving problems related to microwave ovens and the heating of food.