Question

Perform each of the following conversions, being sure to set up clearly the appropriate conversion factor in each case. a. \(8.43 \mathrm{cm}\) to millimeters b. \(2.41 \times 10^{2} \mathrm{cm}\) to meters c. \(294.5 \mathrm{nm}\) to centimeters d. \(404.5 \mathrm{m}\) to kilometers e. \(1.445 \times 10^{4} \mathrm{m}\) to kilometers f. \(42.2 \mathrm{mm}\) to centimeters g. \(235.3 \mathrm{m}\) to millimeters h. \(903.3 \mathrm{nm}\) to micrometers

Short answer

a. 84.3 mm b. \(2.41 \times 10 \, \mathrm{m}\) c. \(2.945\times 10^{-5} \, \mathrm{cm}\) d. 0.4045 km e. \(1.445 \times 10 \, \mathrm{km}\) f. 4.22 cm g. \(2.353 \times 10^{5} \, \mathrm{mm}\) h. \(0.9033 \, \mathrm{\mu m}\)

Step-by-step solution

a. Convert 8.43 cm to millimeters

To convert centimeters to millimeters, we use the conversion factor: 1 cm = 10 mm So 8.43 cm x (10 mm/1 cm) = \(8.43 \times 10 \, \mathrm{mm}\) = 84.3 mm

b. Convert \(2.41 \times 10^{2}\) cm to meters

To convert centimeters to meters, we use the conversion factor: 1 m = 100 cm So \(2.41 \times 10^{2}\, \mathrm{cm} \, \times 10^{-2} \, \mathrm{m}\) = \(2.41 \times 10 \, \mathrm{m}\)

c. Convert 294.5 nm to centimeters

To convert nanometers to centimeters, we use the conversion factor: 1 cm = \(10^{7}\) nm So \(294.5 \, \mathrm{nm} \, \times 10^{-7} \, \mathrm{cm}\) = \(2.945\times 10^{-5} \, \mathrm{cm}\)

d. Convert 404.5 m to kilometers

To convert meters to kilometers, we use the conversion factor: 1 km = 1000 m So \(404.5 \, \mathrm{m} \, \times 10^{-3} \, \mathrm{km}\) = 0.4045 km

e. Convert \(1.445 \times 10^{4}\) m to kilometers

To convert meters to kilometers, we use the conversion factor: 1 km = 1000 m So \(1.445 \times 10^{4}\, \mathrm{m} \, \times 10^{-3} \, \mathrm{km}\) = \(1.445 \times 10 \, \mathrm{km}\)

f. Convert 42.2 mm to centimeters

To convert millimeters to centimeters, we use the conversion factor: 1 cm = 10 mm So 42.2 mm x (1 cm/10 mm) = 4.22 cm

g. Convert 235.3 m to millimeters

To convert meters to millimeters, we use the conversion factor: 1 m = 1000 mm So \(235.3 \, \mathrm{m} \, \times 10^{3} \, \mathrm{mm}\) = \(2.353 \times 10^{5} \, \mathrm{mm}\)

h. Convert 903.3 nm to micrometers

To convert nanometers to micrometers, we use the conversion factor: 1 μm = \(10^{3}\) nm So \(903.3 \, \mathrm{nm} \, \times 10^{-3} \, \mathrm{\mu m}\) = \(0.9033 \, \mathrm{\mu m}\)

Key concepts

Metric System Conversion

Understanding unit conversion within the metric system is vital for interpreting and communicating scientific measurements. The metric system is an internationally agreed decimal system of measurement which is based on multiples of ten. Most commonly in chemistry, you will see lengths measured in meters (m), weights in grams (g), and volumes in liters (l).

When converting from one metric unit to another, the key is to use conversion factors, which are fractions representing the relationship between two units. For example, to convert centimeters to millimeters, as shown in the exercise with 8.43 cm, you multiply by the conversion factor where 1 cm equals 10 mm. Similarly, to go from a larger to a smaller unit, multiply, and to go from a smaller to a larger unit, divide.

Sub-headings for this section on metric system conversion might include:

• Common Metric Units in Chemistry
• Multiples and Submultiples
• Conversion Factors

Each of these plays a crucial role in ensuring accurate and precise measurements, a fundamental aspect of successful chemical experiments and calculations.

Measurement in Chemistry

Measurement is a cornerstone of all scientific disciplines, enabling chemists to acquire quantitative data about substances and their reactions. It includes various physical quantities such as length, mass, volume, temperature, and concentration.

In chemistry, precision and accuracy in measurement are imperative. The tools and techniques used to measure substances must be calibrated correctly, and the units of measurement must be communicated unambiguously. For instance, scientists use precise instruments like analytical balances for mass and spectrophotometers for measuring light absorbance. Consistency in units allows for a common understanding and the ability to compare results across different experiments.

Using appropriate significant figures reflects the precision of the measurement and ensures that no artificial precision is implied. These significant figures play an important role in scientific notation, which we use when dealing with very large or very small numbers, as often encountered in chemical quantities.

Scientific Notation

Scientific notation is a convenient way to express very large or very small numbers, which are common in chemistry, such as the size of molecules or the number of atoms in a sample. It is written as the product of a number between 1 and 10 and a power of ten. This is especially useful when converting between units of different scales.

To convert a number into scientific notation, you must move the decimal point to the position after the first non-zero digit, counting the number of places moved as the exponent. If the original number is less than one, the exponent is negative; if more than one, the exponent is positive. For example, when converting 294.5 nanometers to centimeters in the exercise, we use scientific notation to express the final result as 2.945 x 10^-5 cm, indicating that the decimal moved 7 places to the left.

When multiplying or dividing numbers in scientific notation, you can simply add or subtract the exponents respectively. This technique greatly simplifies calculations with very large or very small numbers in chemistry.