Problem 120 - College Algebra Essentials

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 approximately \(1.34 \times 10^{-19}\) gram.

Step 1: Understand the Problem

The problem states that the mass of one hydrogen atom is \(1.67 \times 10^{-24}\) gram. We need to find the combined mass of \(80,000\) hydrogen atoms.

Step 2: Multiply the Mass of One Atom with the Total Number of Atoms

To find the total mass, just multiply the given mass of the single hydrogen atom by the total number of atoms. So, \(1.67 \times 10^{-24} \text{gram} \times 80,000 = 1.336 \times 10^{-20} \text{gram}\). Here, the number 80,000 is converted to scientific notation \(8.0 \times 10^{4}\) before multiplying with the mass of the atom. This allows for easier multiplication and handling of the scientific notation.

Step 3: Standardize the Answer in Scientific Notation

Scientific notation asks for a number between 1 and 10, multiplied by a power of 10. With \(1.336 \times 10^{-20}\), the number before the multiplication is larger than 10, which is not standard. To convert it, we count how many places we must move the decimal point to get a number between 1 and 10, which turns out to be 1 place to the left, giving us \(1.336\). To balance this, we increase the exponent by 1, giving us \(1.34 \times 10^{-19}\) which is now in proper scientific notation.