Problem 120 - College Algebra

Question

The mass of one hydrogen atom is \(1.67 \times 10^{-24}\) gram. Find the mass of \(80,000\) hydrogen atoms. Express the answer in scientific notation.

Step-by-Step Solution

Answer

The mass of \(80,000\) hydrogen atoms is \(1.336 \times 10^{-19}\) gram.

Step 1: Identify the Given Numbers

The given numbers are the mass of one hydrogen atom which is \(1.67 \times 10^{-24}\) gram and the number of hydrogen atoms which is \(80,000\).

Step 2: Multiply the Mass of One Atom by the Number of Atoms

Multiply the mass of one hydrogen atom by the number of hydrogen atoms, i.e. \(1.67 \times 10^{-24} \) gram \times \(80,000\). This can be written in scientific notation as \(1.67 \times 10^{-24}\) gram \times \(8.0 \times 10^{4}\).

Step 3: Perform the Multiplication

When multiplying numbers in scientific notation, multiply the coefficients and then add the exponents, i.e. \(1.67 \times 8.0\) and \(-24 + 4\). This results in \(13.36 \times 10^{-20}\) gram.

Step 4: Adjust the Final Answer to Scientific Notation

The scientific notation should have a coefficient between \(1\) and \(10\), so we rewrite our answer as \(1.336 \times 10^{-19}\) gram by moving the decimal point one position to the left and adding one to the exponent. This is the mass of \(80,000\) hydrogen atoms.

Key Concepts

Multiplying Powers of Ten

When you multiply numbers in scientific notation, you deal with two crucial parts: the coefficients and the powers of ten. Let's start with the miracle of multiplying powers of ten. The powers of ten refer to the exponents attached to the base 10 in scientific notation, such as in the example \(10^{-24}\) and \(10^4\).
To multiply these, simply add the exponents. Here's why that's neat:

• Multiplying \(10^{-24}\) and \(10^{4}\) is like adding the numbers \(-24\) and \(4\).
• This results in a new power of ten: \(10^{-20}\).

So, when dealing with these tiny or huge numbers, it's less about the size and more about simply managing the exponents. This makes operations straightforward and efficient! Once you handle the multiplication of the powers of ten, you're all set to adjust the coefficient later if needed.

Coefficients in Scientific Notation

Now, let's talk about coefficients in scientific notation. The coefficient is the number in front of the power of ten in scientific notation, like \(1.67\) or \(8.0\) in our example. Here’s what you do with coefficients:

• Multiply the coefficients as you would with any ordinary numbers.
• For example, \(1.67\) multiplied by \(8.0\) equals \(13.36\).

However, there's a rule to remember: in scientific notation, the coefficient should be between \(1\) and \(10\). In the example, \(13.36\) doesn't fit this range, so we adjust it by moving the decimal point one position to the left, resulting in \(1.336\).
When you move the decimal point left, increase the exponent by one to maintain the correct scale of the number. Consequently, the exponent increases from \(-20\) to \(-19\). This keeps your scientific notation true and precise.

Mass and Weight in Chemistry

In chemistry, understanding mass and weight is fundamental, especially when dealing with tiny particles like atoms. Mass refers to the amount of matter in an object, which remains constant regardless of location, while weight is mass affected by gravity.
When calculating the mass of a multitude of atoms, like hydrogen atoms, scientific notation becomes handy because:

• It allows us to express and compute extremely small masses, such as \(1.67 \times 10^{-24}\) grams for a single hydrogen atom.
• By using scientific notation, we can easily work with and comprehend very large numbers, for instance, the collective mass of many atoms.

This method not only helps in managing calculations but also in communicating these quantities clearly and concisely. The precision and clarity provided by scientific notation are invaluable, especially in chemistry, where accurate measurements are critical.