Question

Perform each conversion. (a) \(1.08 \mathrm{Mm}\) to kilometers (b) \(4.88\) fs to picoseconds (c) \(7.39 \times 10^{11} \mathrm{~m}\) to gigameters (d) \(1.15 \times 10^{-10} \mathrm{~m}\) to picometers

Short answer

The conversions are: (a) 1080 km, (b) 0.00488 ps, (c) 739 Gm, and (d) 115 pm.

Step-by-step solution

Convert Megameters to Kilometers

To convert from megameters (Mm) to kilometers (km), use the conversion factor that 1 Mm equals 1000 km. Multiply the given value by this factor: \(1.08 \mathrm{Mm} \times 1000 \frac{\mathrm{km}}{\mathrm{Mm}} = 1080 \, \mathrm{km}\).

Convert Femtoseconds to Picoseconds

To convert from femtoseconds (fs) to picoseconds (ps), use the conversion factor that 1 fs equals 0.001 ps. Multiply the given value by this factor: \(4.88 \, \mathrm{fs} \times 0.001 \frac{\mathrm{ps}}{\mathrm{fs}} = 0.00488 \, \mathrm{ps}\).

Convert Meters to Gigameters

To convert from meters (m) to gigameters (Gm), use the conversion factor that 1 Gm equals \(1 \times 10^{9}\) meters. Divide the given value by this factor: \(7.39 \times 10^{11} \, \mathrm{m} \times 1 \frac{\mathrm{Gm}}{10^{9} \, \mathrm{m}} = 739 \, \mathrm{Gm}\).

Convert Meters to Picometers

To convert from meters (m) to picometers (pm), use the conversion factor that 1 m equals \(1 \times 10^{12}\) pm. Multiply the given value by this factor: \(1.15 \times 10^{-10} \, \mathrm{m} \times 1 \times 10^{12} \frac{\mathrm{pm}}{\mathrm{m}} = 115 \, \mathrm{pm}\).

Key concepts

Understanding Metric System Conversions

The metric system is the international standard for measurement, widely used in science and industry. It is a decimal-based system, meaning measurements are scaled by powers of ten, making conversions straightforward once you know the prefixes and their corresponding powers of ten. For instance, prefixes like 'mega' (M) signify a million times (\times 10^6), and 'kilo' (k) indicates a thousand (\times 10^3).

To convert a measurement, such as megameters (Mm) to kilometers (km), we simply multiply by the relationship between their units. Since 1 Mm is equal to 1000 km, converting 1.08 Mm to kilometers requires multiplying by this fraction: \[1.08 \, \mathrm{Mm} \times 1000 \, \frac{\mathrm{km}}{\mathrm{Mm}} = 1080 \, \mathrm{km}\]. This is a clear example of using the prefix 'mega' to convert to 'kilo' simply by knowing the conversion factor.

Scientific Notation in Chemistry

Scientific notation is a method to express very large or very small numbers succinctly, which is crucial when dealing with the vast scale differences in chemistry. It's written as the product of a number between 1 and 10 and a power of ten. This notation tremendously simplifies calculations.

For example, in converting meters to gigameters, the number \(7.39 \times 10^{11} \mathrm{~m}\) can appear daunting. However, recognizing that 'giga' means a billion (\times 10^9), we can convert this easily by dividing: \[7.39 \times 10^{11} \, \mathrm{m} \times 1 \frac{\mathrm{Gm}}{10^9 \, \mathrm{m}} = 739 \, \mathrm{Gm}\]. Here, the power of ten in scientific notation makes the conversion process much less complex.

The Role of Dimensional Analysis

Dimensional analysis is a powerful tool used in chemistry to convert units and solve problems involving different measurement systems. It works by using conversion factors, which are fractions that represent the relationship between different units. This technique ensures that all the units cancel appropriately, leaving only the desired unit.

An example is converting meters to picometers. Knowing that 1 meter equals \(1 \times 10^{12}\) picometers, we multiply by a conversion factor: \[1.15 \times 10^{-10} \, \mathrm{m} \times 1 \times 10^{12} \frac{\mathrm{pm}}{\mathrm{m}} = 115 \, \mathrm{pm}\]. Dimensional analysis isn't just about simple multiplication or division; it's about ensuring the units you start with convert properly to the units you need to end with, making it an essential concept in chemistry conversions.