Question

Perform the following conversions: (a) 5.00 days to s, (b) \(0.0550 \mathrm{mi}\) to \(\mathrm{m}\) (c) \(\$ 1.89 /\) gal to dollars per liter, (d) 0.510 in. \(/ \mathrm{ms}\) to \(\mathrm{km} / \mathrm{hr}\), (e) \(22.50 \mathrm{gal} / \mathrm{min}\) to \(\mathrm{L} / \mathrm{s}\), (f) \(0.02500 \mathrm{ft}^{3} \mathrm{to} \mathrm{cm}^{3}\)

Short answer

(a) \(432000\,seconds\) (b) \(88.5137\,m\) (c) \(\$0.49897 / L\) (d) \(465.116\frac{km}{hr}\) (e) \(14.1749\frac{L}{s}\) (f) \(707.92\,cm^3\)

Step-by-step solution

(a) Conversion from days to seconds

For this conversion, we need to know how many seconds are in a day. There are 24 hours in a day, each hour contains 60 minutes and each minute contains 60 seconds. To convert 5.00 days to seconds, we will multiply by the number of seconds in a day: \( 5.00\,days \times 24\frac{hours}{day} \times 60\frac{minutes}{hour} \times 60\frac{seconds}{minute}= 432000\,seconds \)

(b) Conversion from miles to meters

To perform this conversion, we need to know the conversion factor between miles and meters. 1 mile = 1609.34 meters. To convert 0.0550 miles to meters, we will multiply by the conversion factor: \( 0.0550\,mi \times 1609.34\frac{m}{mi} = 88.5137\,m \)

(c) Conversion from dollars per gal to dollars per liter

To perform this conversion, we need to know the conversion factor between gallons and liters. 1 gal = 3.78541 liters. To convert $1.89/gal to dollars per liter, we will divide by the conversion factor: \( \frac{\$1.89}{gal} \times \frac{1\,gal}{3.78541\,L} = \$0.49897 / L \)

(d) Conversion from in/ms to km/hr

We can break this conversion into two steps. Firstly, we can convert inches to kilometers, and then convert milliseconds to hours. We know that 1 inch = 2.54 cm = 0.0000254 km, and 1 ms = 0.001 seconds = 0.000000277778 hours. To convert 0.510 in/ms to km/hr, we will multiply by the conversion factors: \( 0.510\frac{in}{ms} \times 0.0000254\frac{km}{in} \times \frac{1}{0.000000277778\frac{hr}{ms}} = 465.116\frac{km}{hr} \)

(e) Conversion from gal/min to L/s

To perform this conversion, we need to know the conversion factor between gallons and liters, and between minutes and seconds. 1 gal = 3.78541 L, and 1 min = 60 seconds. To convert 22.50 gal/min to L/s, we will multiply by the conversion factors: \( 22.50\frac{gal}{min} \times 3.78541\frac{L}{gal} \times \frac{1}{60\frac{s}{min}} = 14.1749\frac{L}{s} \)

(f) Conversion from ft³ to cm³

To perform this conversion, we need to know the conversion factor between cubic feet and cubic centimeters. 1 ft³ = 28316.8 cm³. To convert 0.02500 ft³ to cm³, we will multiply by the conversion factor: \( 0.02500\,ft^3 \times 28316.8\frac{cm^3}{ft^3} = 707.92\,cm^3 \)

Key concepts

Dimensional Analysis

Dimensional analysis is a method used to convert one set of units to another using the conversion factors associated with the dimensions involved. This process involves multiplying quantities by conversion factors, which are ratios representing equalities between different units. For instance, when converting between days and seconds, the ratio \(24 \frac{hours}{day}\) is used. Dimensional analysis ensures that units cancel each other out correctly, allowing for the correct conversion.

It involves carefully tracking units through a series of conversions, leading to the desired unit in the end. By doing so, this analysis not only helps in achieving accurate calculations but also ensures that the quantities are dimensionally consistent. This method is widely used in chemistry and physics to perform conversions effortlessly and accurately.

SI Units

SI Units, or the International System of Units, is the standardized system of measurements used around the world for scientific and technical purposes. This system includes units for length (meter), mass (kilogram), time (second), electric current (ampere), temperature (kelvin), amount of substance (mole), and luminous intensity (candela). Using a universal system, like SI Units, ensures consistency and clarity in scientific communication and calculations.

These units form the basis for defining derived units, such as the Newton for force or the Joule for energy, facilitating scientific exchange and commercial trade. When converting between units, it is essential to ensure compatibility with the SI Units to maintain accuracy and reliability in measurements and computations.

Metric System

The metric system is a decimal-based system of measurement that is used widely worldwide. It includes units such as meter for length, liter for volume, and gram for mass. The simplicity of the metric system lies in its base-10 structure, making it easy to convert between units by simply moving the decimal point. For example, converting centimeters to meters involves shifting the decimal point two places, as 1 meter equals 100 centimeters.

This system is not only practical for scientific calculations but also for everyday uses, as it is commonly used in most countries today. The Metric System's ease of conversion and wide acceptance make it a vital component of educational curricula and international commerce.

Conversion Factors

Conversion factors are ratios used in dimensional analysis to convert one unit of measurement into another. These factors are derived from known relationships between different units, such as 1 inch equals 2.54 centimeters or 1 gallon equals 3.78541 liters. A key feature of conversion factors is that they are equal to one, meaning they do not alter the actual numerical value, only the units.

For instance, when converting 0.0550 miles to meters, the conversion factor 1609.34 \(\frac{m}{mi}\) is applied, translating the distance into another unit without changing measurement precision. Using conversion factors consistently and effectively allows for the seamless transition of measurements between different unit systems, a critical skill in fields like chemistry, where precision and accuracy are paramount.