Question

Calculate Heating 235 \(\mathrm{g}\) of water from \(22.6^{\circ} \mathrm{C}\) to \(94.4^{\circ} \mathrm{C}\) in a microwave oven requires \(7.06 \times 10^{4} \mathrm{J}\) of energy. If the microwave frequency is \(2.88 \times 10^{10} \mathrm{s}^{-1}\) , how many quanta are required to supply the \(7.06 \times 10^{4} \mathrm{J} ?\)

Short answer

Approximately \(3.70 \times 10^{27}\) quanta are required to heat the water to the desired temperature.

Step-by-step solution

Recall Planck's Equation

Planck's equation relates the energy of a single quanta (photon) to its frequency. The equation is: \(E = h \cdot \nu\) where \(E\) = energy of a photon, \(h\) = Planck's constant, approximately \(6.63 \times 10^{-34} \, \mathrm{Js}\), and \(\nu\) = frequency of the photon.

Calculate the Energy of a Single Quanta

We are given the microwave's frequency, so we can use Planck's equation to calculate the energy of a single quanta.\\ Frequency: \(\nu = 2.88 \times 10^{10} \, \mathrm{s^{-1}}\)\\ Energy of a single quanta:\\ \(E = h \cdot \nu\)\\ \(E = (6.63 \times 10^{-34} \, \mathrm{Js}) \cdot (2.88 \times 10^{10} \, \mathrm{s^{-1}})\)\\ \(E = 1.91 \times 10^{-23} \, \mathrm{J}\)

Calculate the Number of Quanta Required

Now that we have the energy of a single quanta and the total energy required to heat the water, we can calculate the number of quanta needed to supply that energy. We can do this by dividing the total energy by the energy of a single quanta: Number of quanta (\(N\)): \(N = \frac{E_{\text{total}}}{E_{\text{single}}}\)\\ Total energy: \(E_{\text{total}} = 7.06 \times 10^{4} \, \mathrm{J}\)\\ Energy of a single quanta: \(E_{\text{single}} = 1.91 \times 10^{-23} \, \mathrm{J}\)\\ Number of quanta:\\ \(N = \frac{7.06 \times 10^{4}}{1.91 \times 10^{-23}}\)\\ \(N = 3.70 \times 10^{27}\) So, approximately \(3.70 \times 10^{27}\) quanta are required to heat the water to the desired temperature.

Key concepts

Energy Calculation

In this exercise, the calculation of energy involves determining the total energy required to heat a specific amount of water using microwave radiation. The process is guided by understanding that energy can be transferred in quantized packets, specifically through photons in the context of electromagnetic radiation.
To find out how much energy is necessary, we rely on the given value of energy needed, which is 70,600 Joules. This tells us how much energy corresponds to increasing the temperature of 235 grams of water from 22.6°C to 94.4°C.
In physics, energy calculation is often utilized to determine how much energy is needed for processes like heating or powering devices. The energy here is encapsulated in microwaves, which are part of the electromagnetic spectrum. Understanding how to calculate the energy in this case also bridges into the realm of quantum mechanics, where energy quantization becomes significant.

Quantum Mechanics

Quantum Mechanics is the area of physics that deals with phenomena at the smallest scales, like atomic and subatomic particles. It introduces the idea that energy levels are not infinite but rather quantized, meaning energy changes occur in discrete values.
In the context of this exercise, Planck's Equation (E = h \cdot u) is a vital tool. This equation links the energy of a photon to its frequency, emphasizing the concept of energy quantization. The constant \(h\), known as Planck's constant (6.63 \times 10^{-34} \, \text{Js}), represents the proportionality factor between energy and frequency.
In simpler terms, this expression means that a photon with a higher frequency will have more energy. When solving the exercise, we calculated the energy of a single quantum, reflecting a foundational quantum mechanics principle. Understanding these calculations helps us appreciate how energy packets behave at micro levels.

Microwave Frequency

Microwaves represent a specific frequency range of electromagnetic waves, often used in cooking appliances and communication technologies. In this exercise, we consider the frequency of microwaves as \(2.88 \times 10^{10} \, \text{s}^{-1}\).
This frequency is crucial as it directly affects the energy of the photons emitted by the microwave's radiation according to Planck's Equation. The higher the frequency, the more energy each quantum (photon) carries.

• Microwave frequency is vital because it determines how quickly molecules, particularly water molecules, rotate or vibrate, generating heat efficiently.
• The microwaves energize water molecules, causing them to heat up and increase the temperature of the water.

Understanding microwave frequency helps explain the effectiveness of microwaves in heating food. It's also a prime example of how electromagnetic waves interact with matter, which is key in various technologies and scientific fields.