Problem 97 - Chemistry The Central Science

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 \(23^{\circ} \mathrm{C}\) 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)?

Step-by-Step Solution

Answer

In conclusion, \(1.74 × 10^{27}\) photons of microwave radiation are required to heat 200 mL of coffee from 23°C to 60°C, and it would take about 34.36 seconds to do so with a microwave power of 900 W.

Step 1: Calculate the energy of a single photon.

First, we need to find out the energy of a single photon of microwave radiation. We are given the wavelength of the microwave radiation which is 11.2 cm or 0.112 m. We can use the formula \( E = h \times c / \lambda \), where E is the energy of the photon, h is the Planck's constant (6.63 × 10^{-34} Js), c is the speed of light (3 × 10^8 m/s), and λ is the wavelength. So, \( E = \frac{(6.63 × 10^{-34} Js) × (3 × 10^8 m/s)}{0.112 m}\) Calculating the value of E gives: \( E = 1.78 × 10^{-24} J \)

Step 2: Calculate the energy required to heat the coffee.

To find out how many photons are required to heat the coffee, we first need to find out how much energy is needed to heat the coffee. For this, we can use the specific heat formula: \( Q = mcΔT \), where Q is the energy, m is the mass of the coffee, c is the specific heat capacity of water (about 4.18 J/g°C), and ΔT is the change in temperature. We are given the volume of the coffee (200 mL), which is equivalent to 200 g since the density of water is approximately 1 g/mL. So, the mass of the coffee is 200 g, and the change in temperature is (60 - 23)°C = 37°C. Therefore, the energy required (Q) is: \( Q = (200 g) × (4.18 J/ g °C) × (37°C)\) Calculating the value of Q gives: \( Q = 30924 J \)

Step 3: Calculate the number of photons required.

Now that we have the energy required to heat the coffee and the energy of a single photon, we can find out how many photons are needed: Number of photons = Required energy / Energy of a single photon Number of photons = \( \frac{30924 J}{1.78 × 10^{-24} J}\) Calculating the number of photons gives: Number of photons = \(1.74 × 10^{27}\) photons.

Step 4: Calculate the time required to heat the coffee.

Finally, we need to find out how long it would take to heat the coffee with the microwave power of 900 W. We are given the power (P) in Watts, which is 900 J/s. We can use the formula: Time = Required energy / Power Time = \( \frac{30924 J}{900 J/s}\) Calculating the value of the time gives: Time = 34.36 s The time required to heat the coffee to the desired temperature is approximately 34.36 seconds. In conclusion, \(1.74 × 10^{27}\) photons of microwave radiation are required to heat 200 mL of coffee from 23°C to 60°C, and it would take about 34.36 seconds to do so with a microwave power of 900 W.

Key Concepts

Photon Energy Calculation

Understanding how photon energy is calculated can be quite interesting, especially when it comes to everyday applications like heating food in a microwave oven. The energy of a photon, which is a particle of light, can be determined by the equation \( E = h \times c / \lambda \), where \(h\) represents Planck's constant, \(c\) is the speed of light, and \(\lambda\) is the wavelength of the photon.

For microwave radiation with a wavelength of 11.2 cm, using Planck's constant \(6.63 \times 10^{-34} Js\) and the speed of light \(3 \times 10^8 m/s\), we can calculate the energy of a single microwave photon. This information not only helps in scientific studies but also gives us a glimpse into how energy translates into the heating process in our kitchen appliances.

Specific Heat Formula

In physics, the specific heat capacity is a substance's property that indicates the amount of heat required to change its temperature by a certain amount. The formula to calculate this energy is \( Q = mc\Delta T \), where \(Q\) stands for the energy in joules, \(m\) for mass in grams, \(c\) for the specific heat capacity, and \(\Delta T\) for the change in temperature in Celsius.

Specific heat capacity is crucial in our problem for understanding how much energy is needed to heat our cup of coffee. With water's specific heat capacity being approximately 4.18 J/g°C, we can calculate the precise amount of energy it takes to bring our drink to a cozy temperature. This very concept plays a fundamental role in designing heating and cooling systems in various applications, from culinary arts to industrial processes.

Power and Energy Relationship

Power and energy are closely related yet distinct concepts in physics. Power is the rate at which energy is used, usually measured in watts, and one watt is defined as one joule per second. The relationship between power (P), energy (E), and time (t) is given by the equation \(E = P \times t\).

In the context of heating our coffee, knowing the microwave's power allows us to calculate the time needed to reach the desired temperature. If a microwave operates at 900 watts, and we have already determined the amount of energy required to heat our coffee, we can easily work out the time by rearranging our equation to \(t = E / P\). This formula is not only useful for food preparation but also broadly applicable in various energy-related industries such as electricity generation and consumption, highlighting the universal importance of understanding these foundational relationships in physics.