Problem 9 - Chemistry in Context

Question

Express each of these numbers in scientific notation. a. \(1500 \mathrm{~m}\), the distance of a foot race b. \(0.0000000000958 \mathrm{~m}\), the distance between \(\mathrm{O}\) and \(\mathrm{H}\) atoms in a water molecule c. \(0.0000075 \mathrm{~m}\), the diameter of a red blood cell d. \(150,000 \mathrm{mg}\) of \(\mathrm{CO}\), the approximate amount breathed daily

Step-by-Step Solution

Answer

a. \(1.5 \times 10^3\), b. \(9.58 \times 10^{-11}\), c. \(7.5 \times 10^{-6}\), d. \(1.5 \times 10^5\).

Step 1: Understand Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. The format is: \[ a \times 10^n \]where \(1 \leq |a| < 10\) and \(n\) is an integer.

Step 2: Convert 1500 m to Scientific Notation

The number 1500 can be expressed as:\[ 1.5 \times 10^3 \]In this expression, 1.5 is the base (where \(1 \leq 1.5 < 10\)) and 3 is the power of 10 that indicates the original decimal place shift.

Step 3: Convert 0.0000000000958 m to Scientific Notation

Move the decimal point 11 places to the right to get:\[ 9.58 \times 10^{-11} \]This is because the number is very small and the decimal shift right increases the negative exponent to ensure original value is retained.

Step 4: Convert 0.0000075 m to Scientific Notation

Shift the decimal point 6 places to the right to form:\[ 7.5 \times 10^{-6} \]This indicates a negative power due to the original small size of the number.

Step 5: Convert 150,000 mg to Scientific Notation

Move the decimal point 5 places to the left to form:\[ 1.5 \times 10^5 \]Here, '1.5' is used as the base and 5 is the positive exponent which shows the large magnitude of the original number.

Key Concepts

Decimal Point

The decimal point is a key concept when working with scientific notation. It acts as a marker that separates the whole part of a number from its fractional part. In scientific notation, we often manipulate the decimal point to express the number in a base value between 1 and 10. For instance, in the number 1500, moving the decimal point three places to the left transforms it into 1.5. This results in the scientific notation of \(1.5 \times 10^3\). By carefully positioning the decimal point, we express both very large and very small numbers in a consistent format.

• Decimal point placement is crucial for maintaining the accuracy of the original number.
• Scientific notation helps manage numbers with many trailing zeros or leading zeros effectively.

Negative Exponent

Negative exponents in scientific notation are used to express very small numbers. When the decimal point moves to the right, the exponent of 10 becomes negative to preserve the original value of the number. For example, the number 0.0000000000958 is written in scientific notation as \(9.58 \times 10^{-11}\). The negative exponent -11 indicates that the decimal point was shifted 11 places to the right.

• Negative exponents simplify the notation of small numbers.
• They help keep track of the decimal point's original position.

Understanding this concept is essential when handling measurements in scientific fields, where small quantities are frequent.

Positive Exponent

Positive exponents are used in scientific notation to express large numbers. When the decimal point moves to the left, the exponent of 10 becomes positive. This shift maintains the integrity of the original number. For instance, the number 150,000 can be expressed as \(1.5 \times 10^5\). The positive exponent 5 indicates that the decimal point moved five places to the left.

• Positive exponents make it easier to handle large numbers succinctly.
• They help in keeping the number format compact and manageable.

This method is widely applied in scientific and engineering disciplines for efficiency.

Base and Power

Base and power are fundamental elements of scientific notation. The base is the value that lies between 1 and 10, while the power, or exponent, shows how many places the decimal point has moved. In the expression \(1.5 \times 10^3\), the base is 1.5, and the power is 3.

• The base must be a number between 1 and 10, ensuring standardization.
• The power determines whether the decimal point has moved left or right.

Understanding these components is crucial for converting to and from scientific notation accurately. They ensure precision and consistency across computations.

Magnitude of Numbers

The magnitude of numbers in scientific notation helps compare and comprehend the size of different quantities. By expressing numbers with a standardized base and exponent, it's easier to assess their relative size. Scientific notation allows quick visual identification of a number's magnitude.

• A larger positive exponent signals a larger number.
• A larger negative exponent indicates a smaller number.

Using scientific notation, the magnitude becomes apparent and manageable, allowing for clearer analysis in fields like astronomy, chemistry, and physics. This method enables rapid assessment and comparison of vastly differing scales.