Problem 59 - Algebra for College Students

Question

The mass of an electron is (9.11) \(\left(10^{-31}\right)\) kilogram, and the mass of a proton is \((1.67)\left(10^{-27}\right)\) kilogram. Approximately how many times more is the weight of a proton than the weight of an electron? Express the result in decimal form.

Step-by-Step Solution

Answer

A proton is approximately 1833 times heavier than an electron.

Step 1: Convert Scientific Notation to Decimal Multiplication

Convert the given masses of the electron and the proton into multiplication form. The mass of an electron is given as \( 9.11 \times 10^{-31} \) kg, while the mass of a proton is \( 1.67 \times 10^{-27} \) kg.

Step 2: Calculate the Ratio of Proton Mass to Electron Mass

Find the ratio of the proton's mass to the electron's mass by dividing the mass of the proton by the mass of the electron: \[ \text{Ratio} = \frac{1.67 \times 10^{-27}}{9.11 \times 10^{-31}} \].

Step 3: Simplify the Exponents

Simplify the division by dealing with the exponents separately: \[ \text{Ratio} = \frac{1.67}{9.11} \times 10^{(-27) - (-31)} = \frac{1.67}{9.11} \times 10^{4} \].

Step 4: Calculate the Decimal Value

Compute the value of \( \frac{1.67}{9.11} \), which is approximately \(0.1833\). Then multiply by \(10^4\) to get \(0.1833 \times 10^4 = 1833\).

Step 5: Conclude the Calculation

Therefore, the mass of a proton is approximately 1833 times the mass of an electron.

Key Concepts

Electron Mass

The mass of an electron is a fundamental aspect of atomic and particle physics. Electrons are one of the main subatomic particles, along with neutrons and protons, that make up an atom. They are extremely lightweight compared to protons and neutrons.

The mass of an electron is approximately 9.11 times 10 to the power of negative thirty-one kilograms. This means in scientific notation, it is written as \(9.11 \times 10^{-31}\) kg. Scientific notation is particularly helpful here because it simplifies the representation of very small or very large numbers, which are common in scientific calculations.

Understanding how minute the electron's mass is, allows us to appreciate the precision required to study and measure phenomena at the quantum level. Electrons have a negative electric charge, and they play a crucial role in electricity, chemistry, and thermal conductivity.

Proton Mass

Protons are positively charged particles found within the nucleus of an atom, and they play a significant role in defining the chemical element. Unlike the electron, protons have a much greater mass, which profoundly affects calculations in physics and chemistry.

The mass of a proton is approximately 1.67 times 10 to the power of negative twenty-seven kilograms. In scientific notation, this is expressed as \(1.67 \times 10^{-27}\) kg. By using scientific notation, it becomes easier to work with the values and conduct calculations without constantly dealing with an endless stream of zeros.

Comparing the mass of protons and electrons highlights how protons contribute a major portion of the atomic mass. This difference is crucial for understanding the structure of atoms and the behavior of matter under various conditions.

Decimal Form

Expressing numbers in decimal form is sometimes more useful for interpretation and communication, especially in contexts outside scientific research. Decimal representation helps to simplify the understanding of results for practical applications.

In the original exercise, we determined the ratio of proton mass to electron mass, resulting in a numerical value. After calculating, by dividing and handling the exponents, the result was 1833 when expressed in decimal form.

Decimals provide a straightforward way to show the magnitude of differences. Here, the decimal form clearly shows that a proton's mass is approximately 1833 times greater than that of an electron, giving a tangible sense of scale and aiding in both educational and professional contexts of science.