Problem 5 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

What is the wavelength of a helium atom with a velocity of \(1.00 \times 10^{3} \mathrm{ms}^{-1} ?(\text { Section } 3.4)\)

Step-by-Step Solution

Answer

The wavelength of a helium atom with a velocity of \(1.00 \times 10^{3} \mathrm{ms}^{-1}\) is approximately \(9.97 \times 10^{-11}\) meters.

Step 1: Identify the De Broglie Equation

The De Broglie wavelength \( \lambda \) for a particle is given by the equation \( \lambda = \frac{h}{mv} \), where \( h \) is Planck's constant, \( m \) is the mass of the particle, and \( v \) is the velocity of the particle.

Step 2: Gather Known Values

For a helium atom, approximate mass \( m \) is \( 4.00 \times 10^{-3} \text{ kg/mol} \) (mol represents the number of atoms). We use Avogadro's number \( 6.022 \times 10^{23} \text{ atoms/mol} \) to find the mass of a single atom. Planck's constant \( h = 6.626 \times 10^{-34} \text{ Js} \) and velocity \( v = 1.00 \times 10^{3} \text{ ms}^{-1} \).

Step 3: Calculate Mass of a Single Helium Atom

Convert the molar mass of helium atom to the mass of a single helium atom: \( m_{helium} = \frac{4.00 \times 10^{-3}}{6.022 \times 10^{23}} \approx 6.646 \times 10^{-27} \text{ kg} \).

Step 4: Substitute Values into the De Broglie Equation

Now substitute the values into the De Broglie equation: \[ \lambda = \frac{6.626 \times 10^{-34}}{6.646 \times 10^{-27} \times 1.00 \times 10^{3}} \].

Step 5: Calculate the Wavelength

Perform the calculation: \[ \lambda \approx 9.97 \times 10^{-11} \text{ m} \].

Key Concepts

Planck's constant

Planck's constant, denoted by the symbol \( h \), is a fundamental quantity in quantum mechanics. It is defined as \( h = 6.626 \times 10^{-34} \text{ Js} \). Understanding its role is crucial when you deal with phenomena at the atomic and subatomic levels.
Planck's constant serves as a bridge between the macroscopic and the quantum worlds. It is a tiny, precise number that describes the size of the quanta of energy. Each "blob" or "packet" of energy that particles like electrons interact with is underlined by this constant.
In the De Broglie equation, Planck's constant helps calculate the wavelength of a particle, such as a helium atom, by determining the ratio of energy to frequency. Without it, we wouldn't be able to relate seemingly wavy properties (like wavelengths) to tiny particles. This constant gave rise to the idea that matter exhibits wave-like and particle-like properties, showing how versatile atoms can be at their smallest scale.

Avogadro's number

Avogadro's number is a fundamental constant used to express the number of atoms, ions, or molecules in one mole of a substance. This number is an astounding \( 6.022 \times 10^{23} \). It acts as a crucial link between the macroscopic and molecular scales.
When dealing with the mass of atoms, Avogadro's number helps to transition us from "per mole" (a collective measurement) to "per atom" (an individual measurement). This number allows for the conversion of the molar mass of a substance to the actual mass of its constituent atoms or molecules.

• In the exercise, using Avogadro's number helps us find the mass of a single helium atom from its molar mass.
• This conversion is essential before applying it in calculations like finding the De Broglie wavelength of helium atoms as it gives specific atomic-level detail needed.

Mass of helium atom

Understanding the mass of a helium atom involves conversions based on molar mass, which is commonly expressed in grams per mole. Helium has a molar mass of approximately \( 4.00 \times 10^{-3} \text{ kg/mol} \).
To find the mass of a single helium atom, the molar mass is divided by Avogadro's number \( 6.022 \times 10^{23} \text{ atoms/mol} \).
This conversion results in the mass of a single helium atom being approximately \( 6.646 \times 10^{-27} \text{ kg} \). Emphasizing this step is critical because:

• It reflects the transition from bulk quantities (as perceived in chemistry) to individual particles (as necessary in quantum physics).
• The specific mass of a single atom is required when calculating properties like the De Broglie wavelength, to determine how quantum mechanics applies to small, moving atoms like helium.

Helium's atomic mass set against its quantum behaviors helps in illustrating how atomic particles are both predictable and yet wondrously complex.