Problem 47 - Chemistry Matter and Change

Question

If you had a balance that could determine the mass of a proton, how many clectrons would you need to weigh on the same balance to measure the same mass as that of a single proton?

Step-by-Step Solution

Answer

1836 electrons are needed to match the mass of one proton.

Step 1: Understand the Mass of a Proton

The mass of a proton is approximately equal to \(1.67 \times 10^{-27}\) kilograms. This is a standardized value used in scientific calculations.

Step 2: Understand the Mass of an Electron

The mass of an electron is much smaller than that of a proton, approximately \(9.11 \times 10^{-31}\) kilograms.

Step 3: Set Up the Equation

To find out how many electrons would be needed to equal the mass of one proton, we set up the equation: \[ n \times (9.11 \times 10^{-31}) = 1.67 \times 10^{-27} \]where \(n\) is the number of electrons.

Step 4: Solve for n

Solve the equation for \(n\) by dividing the mass of the proton by the mass of an electron:\[ n = \frac{1.67 \times 10^{-27}}{9.11 \times 10^{-31}} \] Calculating this gives approximately \(1836\).

Step 5: Final Interpretation

Therefore, it would take about 1836 electrons to have a mass equivalent to that of a single proton.

Key Concepts

Proton Mass

The proton is a subatomic particle that resides in the nucleus of an atom. Protons have a positive electric charge and are significantly more massive than electrons. In scientific studies, the mass of a proton is standardized and known to be approximately \(1.67 \times 10^{-27}\) kilograms. This value is crucial for calculations involving atomic structure and helps in understanding how atomic forces operate within the nucleus. The proton's mass is measured with high precision, allowing scientists to conduct various experiments in particle physics and chemistry effectively.

Electron Mass

An electron is another fundamental subatomic particle but, unlike the proton, an electron carries a negative charge and is found outside the nucleus in electron clouds. Electrons are much lighter and faster than protons and neutrons. The mass of an electron is approximately \(9.11 \times 10^{-31}\) kilograms. These tiny particles are essential for the chemical properties of atoms. The significantly smaller mass of the electron compared to the proton plays a vital role in atomic structure and arrangement, affecting how atoms form molecules and interact with each other.

Scientific Calculation

Scientific calculations are essential for solving problems in physics and other sciences, requiring precision and accuracy. In this exercise, knowing the masses of the proton and electron allows us to perform a calculation to determine how many electrons equal the mass of one proton.
The set-up of the equation is straightforward:

• The mass of a proton = \(1.67 \times 10^{-27}\) kg
• The mass of an electron = \(9.11 \times 10^{-31}\) kg

We set up the equation \[ n \times (9.11 \times 10^{-31}) = 1.67 \times 10^{-27} \]where \( n \) is the number of electrons equivalent to one proton's mass. To solve for \( n \), divide the proton's mass by the electron's mass.
This yields:\[ n = \frac{1.67 \times 10^{-27}}{9.11 \times 10^{-31}} \approx 1836 \]Such calculations help scientists understand fundamental principles of atomic physics.

Physics Problem Solving

In physics, problem solving often involves breaking down complex concepts into manageable steps. This exercise is a classic illustration of physics problem-solving methodology:

• Identify known values: Use standard measurements for subatomic particle masses.
• Set up equations: Ensure equations are logical and reflect the relationships between known values.
• Perform calculations: Apply mathematical operations with care to maintain accuracy.
• Interpret results: Consider what the results mean practically, such as understanding that it takes 1836 electrons to equate a proton's mass.

By systematically applying these techniques, physicists can explore various scientific phenomena and derive reliable results, contributing to our comprehensive understanding of the universe.