Problem 100 - Chemistry Matter and Change

Question

Express the following numbers in scientific notation. (Chapter 2) \(\begin{array}{ll}{\text { a. } 34,500} & {\text { d. } 789} \\ {\text { b. } 2665} & {\text { e. } 75,600} \\ {\text { c. } 0.9640} & {\text { f. } 0.002189}\end{array}\)

Step-by-Step Solution

Answer

a. \(3.45 × 10^4\) b. \(2.665 × 10^3\) c. \(9.640 × 10^{-1}\) d. \(7.89 × 10^2\) e. \(7.56 × 10^4\) f. \(2.189 × 10^{-3}\)

Step 1: a. Convert 34,500 to scientific notation.

To represent 34,500 in scientific notation, we need to find a and n such that \(34,500 = a × 10^n\). In this case, we have \(a=3.45\) and \(n=4\) because \(34,500=3.45 × 10^4\). So in scientific notation, 34,500 is \(3.45 × 10^4\).

Step 2: b. Convert 2,665 to scientific notation.

For 2,665, we have \(a=2.665\) and \(n=3\), since \(2,665=2.665 × 10^3\). So 2,665 in scientific notation is \(2.665 × 10^3\).

Step 3: c. Convert 0.9640 to scientific notation.

For 0.9640, we have \(a=9.640\) and \(n=-1\), since \(0.9640=9.640 × 10^{-1}\). So 0.9640 in scientific notation is \(9.640 × 10^{-1}\).

Step 4: d. Convert 789 to scientific notation.

For 789, we have \(a=7.89\) and \(n=2\), since \(789=7.89 × 10^2\). So 789 in scientific notation is \(7.89 × 10^2\).

Step 5: e. Convert 75,600 to scientific notation.

For 75,600, we have \(a=7.56\) and \(n=4\), since \(75,600=7.56 × 10^4\). So 75,600 in scientific notation is \(7.56 × 10^4\).

Step 6: f. Convert 0.002189 to scientific notation.

For 0.002189, we have \(a=2.189\) and \(n=-3\), since \(0.002189=2.189 × 10^{-3}\). So 0.002189 in scientific notation is \(2.189 × 10^{-3}\).

Key Concepts

Exponential Notation

Exponential notation is a way to express numbers that are too large or too small to write conveniently in decimal form. It involves two main components: a base (usually 10) and an exponent. The base tells us what number we are repeatedly multiplying, and the exponent tells how many times to multiply the base. For example, in math, the expression \(10^4\) means 10 multiplied by itself four times: \(10 \times 10 \times 10 \times 10\). The number 35,000 can be written in exponential notation as \(3.5 \times 10^4\), where 3.5 is the coefficient and 4 is the exponent.

Using exponential notation simplifies calculations and helps when dealing with very large numbers such as the speed of light (\(3 \times 10^8\) meters per second) or tiny values like the size of a hydrogen atom.

• The coefficient is a number greater than or equal to 1 and less than 10.
• The exponent is an integer which shifts the decimal point.

Exponential notation is also the foundation for scientific notation, making it an essential tool in both mathematics and science.

Mathematics

Mathematics is composed of various concepts that help solve problems by counting, measuring, and analyzing shapes and structures. Among its many branches, algebra often uses scientific notation to simplify calculations and express extremely large or small numbers easily. This is particularly useful in mathematical operations like multiplication or division of numbers with many zeros.

Sometimes in mathematics, scientific notation allows us to perform arithmetic operations without directly dealing with large decimal numbers. For example, multiplying \(3.45 \times 10^4\) by \(2 \times 10^2\) can be done by multiplying the coefficients (3.45 and 2) and adding their exponents (4 and 2), resulting in \(6.9 \times 10^6\).

• It provides precision by maintaining significant figures.
• Simplifies mathematical equations and error comparisons.

Through these applications, mathematics uses scientific notation as a tool to express quantities clearly and manage computations effectively.

Chemistry

Chemistry often deals with vast ranges of measurements, from enormous quantities like the number of molecules in a substance to microscopic measurements of atomic structures. Scientific notation is incredibly practical in this field because it allows chemists to handle and communicate these very large or extremely small numbers concisely.

For instance, Avogadro's number, which is approximately \(6.022 \times 10^{23}\), represents the number of atoms or molecules in one mole of a substance. Without scientific notation, this number would be cumbersome to write and calculate with.

• Facilitates accurate communication of large quantities.
• Ensures precision when measuring reactant quantities or atomic masses.

Understanding scientific notation is crucial in chemistry because it provides a consistent format for reporting and comparing chemical data, aiding in accurate calculation and analysis.