Problem 13 - Chemistry Matter and Change

Question

Heating 235 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 \(7.06 \times 10^{4} \mathrm{J} ?\)

Step-by-Step Solution

Answer

Approximately \( 3.68 \times 10^{27} \) photons are needed.

Step 1: Calculate Energy per Photon

The energy of each photon can be calculated using the formula for energy of a photon: \( E = h u \), where \( h \) is Planck's constant \( (6.626 \times 10^{-34} \text{ J}\cdot\text{s}) \) and \( u \) is the frequency of the microwave \( (2.88 \times 10^{10} \text{ s}^{-1}) \). Substitute the values to find \( E \): \[ E = 6.626 \times 10^{-34} \text{ J}\cdot\text{s} \times 2.88 \times 10^{10} \text{ s}^{-1} \]. Calculate \( E \).

Step 2: Calculate the Total Number of Photons

To find out how many photons (quanta) are needed, divide the total energy \( 7.06 \times 10^{4} \text{ J} \) by the energy per photon calculated in Step 1. Use the formula: \[ \text{Number of photons} = \frac{7.06 \times 10^{4} \text{ J}}{E} \], where \( E \) is the result from Step 1. Perform this division to find the number of photons.

Key Concepts

Photon Energy

Photon energy is fundamental in understanding how different types of electromagnetic waves, including microwaves, interact with matter. A photon is a particle of light, and its energy can be calculated using the formula:

• \( E = h u \)

In this formula, \( E \) represents the energy of the photon, \( h \) is Planck's constant, and \( u \) is the frequency of the radiation.
Photon energy plays a critical role in processes like heating in a microwave oven. Here, microwaves provide the necessary energy to increase the temperature of the water molecules by interacting with them at the quantum level, which is essentially transferring energy from the photon to the molecules.

Planck's Constant

Planck's constant is a crucial element in the field of quantum mechanics.
It represents the proportionality constant that relates the energy of a photon to the frequency of its associated electromagnetic wave.
The value of Planck's constant is \( 6.626 \times 10^{-34} \text{ J} \cdot \text{s} \).

• It is a small number, indicating the energy of individual photons is tiny compared to everyday measurements.
• Named after Max Planck, a pioneer of quantum theory.
• Integral in calculations that predict how particles behave at atomic and subatomic levels.

Utilizing Planck's constant allows us to bridge the gap between macroscopic phenomena and the microscopic quantum world, helping us calculate the number of photons required for tasks like heating water in a microwave.

Microwave Frequency

Microwave frequency refers to the number of oscillations of the microwave waves per second. This frequency is crucial, as it determines the energy of the microwaves according to the formula:

• Photon Energy = \( h \times u \)

Microwaves typically have a high frequency, which means each photon carries a relatively low amount of energy compared to visible light but enough to cause molecular rotation. For the original exercise, the microwave frequency given was \( 2.88 \times 10^{10} \text{ s}^{-1} \).
This indicates the microwaves are at the right frequency to cause water molecules to vibrate and heat up, making them perfect for heating food efficiently.

Quantum Mechanics

Quantum mechanics is the branch of physics that studies particles at the atomic and subatomic levels. It provides the framework for understanding how particles like photons behave and interact.
One of the fundamental principles is that energy is quantized, meaning it comes in discrete packets called quanta.

• Encapsulates concepts such as wave-particle duality, where particles exhibit both wave and particle-like properties.
• Essential for calculating phenomena that classical mechanics cannot explain.
• Includes the use of Planck’s constant to quantify energy levels.

In quantum terms, all electromagnetic radiation, including microwaves, consists of packets of energy or photons. Understanding these interactions helps explain how energy from microwaves is absorbed by water in a microwave oven, leading to heating.