Question

There are 269,000,000,000,000,000,000,000 molecules in 10 liters of oxygen gas at standard temperature and pressure. Express this number in scientific notation.

Short answer

The number in scientific notation is \( 2.69 \times 10^{25} \).

Step-by-step solution

Identify the Number

The given number is 269,000,000,000,000,000,000,000.

Determine the Significant Figures

Count the nonzero digits starting from the left in the number 269,000,000,000,000,000,000,000. These digits are '269'.

Write as a Product of a Decimal and a Power of Ten

Rewrite the number 269,000,000,000,000,000,000,000 in the form \( 2.69 \times 10^x \), where \( x \) is the exponent to be determined.

Count the Number of Zeros

Since there are 23 zeros following the '269', the power of ten is 23 plus the 2 additional digits before the first zero, yielding an exponent of 23 + 2 = 25.

Write the Scientific Notation

Combine the determined decimal and power of ten: The scientific notation of 269,000,000,000,000,000,000,000 is \( 2.69 \times 10^{25} \).

Key concepts

Significant Figures

Significant figures are crucial in scientific calculations as they indicate the precision of a measured or calculated quantity. When expressing numbers, recognizing which digits are significant helps convey the accuracy of measurements.
Here’s what you need to know:

• All non-zero digits (1-9) are considered significant. These digits indicate the measured value.
• Any zeros between significant digits are also significant. For example, 105 has three significant figures: 1, 0, and 5.
• Zeros at the beginning of a number are not significant because they don’t affect the precision of the measurement. For example, 0.0042 has two significant figures: 4 and 2.
• Zeros at the end of a number that includes a decimal point are significant because they help to clarify the precision of the measurement. For example, 5.00 has three significant figures.

In the exercise example, the significant figures of 269,000,000,000,000,000,000,000 are the digits '269' because they determine the precision before any trailing zeros added by the magnitude of the number.

Powers of Ten

The concept of powers of ten is central in scientific notation because it allows for concise representation of extremely large or small numbers through exponents. A power of ten is an expression like 10, 100, or 1,000, which can be written as 10 raised to a certain exponent:

• 10 is written as \(10^1\)
• 100 is \(10^2\)
• 1,000 is \(10^3\), and so on.

In scientific notation, a number is scaled by a power of ten to place the decimal after the first non-zero digit. This standard position simplifies comparing and computing vast numbers.
In the oxygen molecules exercise, converting 269,000,000,000,000,000,000,000 to \(2.69 \times 10^{25}\), powers of ten show the large number as a compact expression, distinctly separated by a coefficient \(2.69\) and the exponent \(25\). Each count of zero in the given value increases the power of ten.

Exponents

Exponents, or powers, are a shorthand notation for expressing how many times a number, the base, is used in a multiplication.
For example, in the expression \(2^3\), 2 is the base and 3 is the exponent, indicating that 2 is multiplied by itself three times: \(2 \times 2 \times 2 = 8\).
This system is integral to scientific notation, where it succinctly scales numbers.

• Positive exponents show how many times to multiply by ten: \(10^3 = 1,000\)
• Negative exponents indicate division by ten: \(10^{-3} = 0.001\)

In the molecular count exercise, the exponent 25 in \(2.69 \times 10^{25}\) shows that the original number has 25 places to the right of the decimal, demonstrating how scientific notation leverages exponents to easily manage large quantities.