Problem 53 - Precalculus: Functions and Graphs

Question

The mass of a hydrogen atom is approximately $$0.0000000000000000000000017 gram.$$ Express this number in scientific form.

Step-by-Step Solution

Answer

The mass is approximately \(1.7 \times 10^{-24}\) grams.

Step 1: Identify the number

We are given the mass of a hydrogen atom, which is 0.0000000000000000000000017 grams.

Step 2: Count the decimal places

To express the given number in scientific notation, start by counting the number of places from the first non-zero digit to the decimal point. The number is 0.0000000000000000000000017, and there are 24 places from the decimal point to the digit 1.

Step 3: Write in the form of scientific notation

In scientific notation, a number is written as the product of a number (between 1 and 10) and a power of 10. Therefore, we can express 0.0000000000000000000000017 as 1.7 multiplied by 10 raised to the power of -24.

Key Concepts

Hydrogen Atom

Hydrogen is the simplest and most abundant element in the universe. A hydrogen atom consists of one proton and one electron. It doesn't have any neutrons in its most common form, which distinguishes it as the lightest atom on the periodic table.
The mass of a hydrogen atom is incredibly small, making it difficult to comprehend using regular numbers. This is where scientific notation comes in handy, enabling us to express such small quantities more easily. Understanding the basic structure of a hydrogen atom can help students appreciate why expressing its mass requires such precise notation.

Mass in Grams

When dealing with the mass of atoms, the unit mostly used is the gram. However, because atoms are extremely small, their masses in grams are tiny numbers with many decimal places.
For instance, the mass of a hydrogen atom is roughly 0.0000000000000000000000017 grams.
Such a small mass illustrates the minuscule scale of atoms, and highlights the importance of using a more manageable form, like scientific notation, to make sense of these values in chemistry and physics.

Exponential Form

Exponential form is a way to represent numbers as a base and an exponent. This makes it particularly useful for very large or very small numbers. In scientific notation, we'll write numbers in the form of \( a \times 10^{n} \), where \( a \) is a number between 1 and 10, and \( n \) is an integer.
For our hydrogen atom example, the mass in grams 0.0000000000000000000000017 can be converted to exponential form as \( 1.7 \times 10^{-24} \).
This format simplifies the way we communicate and calculate with extremely small or large numbers, ensuring accuracy and ease of understanding.

Decimal Places

Decimal places play a crucial role when we convert numbers to scientific notation. Each decimal place represents a factor of 10.
In our problem, the number of decimal places from the decimal point to the first non-zero digit "1" in 0.0000000000000000000000017 is 24. This counting helps us determine the correct power of 10 as \(-24\).
Understanding and counting decimal places correctly ensures that the converted scientific notation accurately represents the original number, preserving its value and allowing easier manipulation in calculations.