Problem 71 - Introductory Chemistry Atoms First

Question

Convert the following measured values from scientific notation to standard notation. For each one, indicate the number of significant figures. (a) \(5.60 \times 10^{1} \mathrm{~kg}\) (b) \(2.5 \times 10^{-4} \mathrm{~m}\) (c) \(5.600 \times 10^{6}\) miles (d) \(0.02 \times 10^{2} \mathrm{ft}\)

Step-by-Step Solution

Answer

(a) \(56.0 \mathrm{~kg}\) (3 significant figures) (b) \(0.00025 \mathrm{~m}\) (2 significant figures) (c) \(5,600,000 \thinspace miles\) (4 significant figures) (d) \(2 \mathrm{~ft}\) (1 significant figure)

Step 1: (a) Case 1: \(5.60 \times 10^{1} \mathrm{~kg}\)

To convert from scientific notation to standard notation, we simply need to multiply the coefficient (5.60) by the corresponding power of 10 (\(10^{1}\)). And also count the significant figures in the coefficient: \(5.60 = 5.6 \) \( 5.60 \times 10^{1} = 5.60 \times 10 = 56.0 \mathrm{~kg} \) Number of significant figures = 3

Step 2: (b) Case 2: \(2.5 \times 10^{-4}\mathrm{~m}\)

Again, multiply the coefficient (2.5) by the corresponding power of 10 (\(10^{-4}\)) to find the standard notation. Count the significant figures in the coefficient: \(2.5\) \( 2.5 \times 10^{-4} = 0.00025 \mathrm{~m} \) Number of significant figures = 2

Step 3: (c) Case 3: \(5.600 \times 10^{6}\) miles

Multiply the coefficient (5.600) by the corresponding power of 10 (\(10^{6}\)) to convert it into standard notation. Count the significant figures in the coefficient: \(5.600\) \( 5.600 \times 10^{6} = 5,600,000 \thinspace miles \) Number of significant figures = 4

Step 4: (d) Case 4: \(0.02 \times 10^{2} \mathrm{~ft}\)

Multiply the coefficient (0.02) by the corresponding power of 10 (\(10^{2}\)) to find the standard notation. In determining significant figures, the leading zero should not be counted in the coefficient: \(0.02 = 2\) \( 0.02 \times 10^{2} = 0.02 \times 100 = 2 \mathrm{~ft} \) Number of significant figures = 1

Key Concepts

Significant Figures

Understanding significant figures is crucial when reporting measurements and results in science. Significance in a number indicates the amount of reliable digits that are known, usually from some form of measurement. For instance, in the number 5.60, the 5 is certain and the two decimal places (6 and 0) are considered significant because they tell how precisely the quantity has been measured.

Significant figures also play an important role when converting from scientific notation to standard notation because they determine the exactness of the final number. As we can see from the exercise solutions, the number 5.60 has three significant figures. Even the trailing zero after the 6 is significant here, because it affects the precision of the measurement. When 5.60 is multiplied by the power of 10, the resulting number in standard notation, 56.0 kg, reflects the same level of precision.

Power of 10

The 'power of 10' in scientific notation represents how many times the base number 10 is multiplied by itself. In scientific notation, a number is written as a coefficient (a number typically between 1 and 10) multiplied by 10 raised to an exponent. This exponent can be positive or negative, showing whether we are dealing with very large or very small numbers respectively.

For example, the expression \(2.5 \times 10^{-4}\) in scientific notation indicates the coefficient 2.5 is being multiplied by 10 raised to the power of -4. The negative exponent tells us that we are working with a small number, which means the decimal point in the standard notation has to move 4 places to the left, resulting in the standard notation 0.00025 m. When changing such expressions from scientific to standard notation, understanding the power of 10 is essential.

Standard Notation Conversion

Converting from scientific to standard notation involves moving the decimal point of the coefficient to the right for positive exponents or to the left for negative exponents of 10. This is akin to multiplying or dividing the coefficient by ten the number of times indicated by the exponent.

As seen in the example \(0.02 \times 10^{2}\) ft, the coefficient is 0.02 and the power of 10 is 2. To convert this to standard notation, we move the decimal two places to the right (since our exponent is +2), which equalizes to multiplying 0.02 by 100. Hence, we achieve the standard notation of 2 ft. Notably, leading zeros present before the first non-zero digit aren't significant, which is why our significant figure count for the given value is 1.