Question

Few figures are expressed in scientific notation. Mark the incorrect one. (a) \(234000=2.34 \times 10^{5}\) (b) \(8008=8 \times 10^{8}\) (c) \(0.0048=4.8 \times 10^{-3}\) (d) \(500.0=5.00 \times 10^{2}\)

Short answer

The incorrect scientific notation is option (b): \(8008\) should be \(8.008 \times 10^3\), not \(8 \times 10^8\).

Step-by-step solution

Identify the correct form of scientific notation

Scientific notation is a way of expressing numbers that are too big or too small to be conveniently written in decimal form. It is typically written as the product of a number between 1 and 10 and a power of 10. The general form is \( a \times 10^n \) where \(1 \leq a < 10\) and \(n\) is an integer.

Evaluate Option (a)

Convert \(234000\) to scientific notation. Move the decimal point to the left until one digit is left before it. It moves 5 places to become \(2.34\), hence \(234000=2.34 \times 10^{5}\). This is correct.

Evaluate Option (b)

Convert \(8008\) to scientific notation. Moving the decimal after the first non-zero digit, we get \(8.008\). To express \(8008\) correctly, it should be \(8.008 \times 10^3\), not \(8 \times 10^{8}\). Option (b) is incorrect.

Evaluate Option (c)

Move the decimal point in \(0.0048\) to the right until we have a number between 1 and 10. After moving 3 places, it becomes \(4.8\), thus \(0.0048 = 4.8 \times 10^{-3}\) is correctly stated.

Evaluate Option (d)

To express \(500.0\) in scientific notation, we move the decimal 2 places to the left to get \(5.00\), therefore \(500.0 = 5.00 \times 10^2\) is correct.

Key concepts

Expressing Numbers in Scientific Notation

Scientific notation is a method often used in chemistry and other sciences to handle very large or very small numbers with ease. The central idea is to represent a number as the product of two components: a coefficient that is at least 1 but less than 10, and a power of 10. The general structure looks like this:
\[\begin{equation} a \times 10^n \end{equation}\]where \(a\) represents the coefficient and \(n\) is an integer indicating the number of places the decimal point has been moved. To convert a number into scientific notation:

• Identify the most significant digit in the number.
• Place the decimal point after this digit, ensuring the resulting number is between 1 and 10.
• Count the number of places the decimal point has moved. This count becomes the exponent \(n\) in your scientific notation. If the original number is greater than 1, \(n\) will be positive, and if it’s less than 1, \(n\) will be negative.

It's important to note that scientific notation is not only a compact way to write numbers, but it also simplifies mathematical operations like multiplication and division of very large or small quantities.

Scientific Notation Conversion

Converting between standard decimal notation and scientific notation is a critical skill in chemistry, as it allows for a more manageable representation of quantities. Here's a detailed look at the conversion process:
For a large number, like \[\begin{equation}234000\end{equation}\]we find the most significant digit (2 in this case), place a decimal to keep its value between 1 and 10 (\[\begin{equation}2.34\end{equation}\]), and then determine the power of 10 needed to return it to its original value, which is 5 in this case because we moved the decimal five places to the left. Hence, we express it as \[\begin{equation}2.34 \times 10^5\end{equation}\]Conversely, for a small number such as \[\begin{equation}0.0048\end{equation}\], we do the reverse. The decimal moves three places to the right, giving us 4.8, and since the number is smaller than 1, our power of 10 will be negative, resulting in \[\begin{equation}4.8 \times 10^{-3}\end{equation}\]This technique simplifies the understanding of chemical measurements and calculations, such as concentrations and mole calculations, which often involve very large or small numbers.

Identifying Correct Scientific Notation

Recognizing whether a number is expressed correctly in scientific notation is essential, as it ensures accuracy in scientific calculations. Here are key factors to identify correct scientific notation:

• The coefficient must be a number greater than or equal to 1, but less than 10.
• The exponent must accurately reflect the number of decimal places moved and be positive when the original number is greater than 1 and negative when it is less than 1.
• The resulting expression should streamline computations and accurately represent the original number’s magnitude.

Using our textbook example, option (b) \[\begin{equation}8008=8 \times 10^8\end{equation}\]is incorrect because the coefficient is not between 1 and 10. The correct scientific notation should reflect moving the decimal three places to the left, resulting in \[\begin{equation}8.008 \times 10^3\end{equation}\]This kind of attention to detail is vital when handling chemical data where precision is necessary, such as in calculating reaction yields or preparing solutions of precise concentrations. Keeping these guidelines in mind will help students accurately work with scientific notation in their assignments and exams.