Question

Perform the following conversions: (a) \(0.076 \mathrm{~L}\) to \(\mathrm{mL}\), (b) \(5.0 \times 10^{-8} \mathrm{~m}\) to \(\mathrm{nm},(\mathrm{c}) 6.88 \times 10^{5} \mathrm{~ns}\) to \(\mathrm{s}\), (d) \(0.50 \mathrm{lb}\) to \(\mathrm{g}\), \((\mathrm{e}) 1.55 \mathrm{~kg} / \mathrm{m}^{3}\) to \(\mathrm{g} / \mathrm{L}\), (f) \(5.850 \mathrm{gal} / \mathrm{hr}\) to \(\mathrm{L} / \mathrm{s}\).

Short answer

Short Answer: (a) \(76 \: mL\), (b) \(50 \: nm\), (c) \(6.88 \times 10^{-4} \: s\), (d) \(226.796 \: g\), (e) \(1.55 \: \frac{g}{L}\), (f) \(\approx 0.000798 \: \frac{L}{s}\)

Step-by-step solution

Identify the conversion factor

The conversion factor between liters and milliliters is 1 L = 1000 mL.

Perform the conversion

To convert 0.076 L to mL, multiply the given value by the conversion factor: \[0.076 \; L \times \frac{1000 \:mL}{1 \; L} = 76 \: mL\] (b) Convert \(5.0 \times 10^{-8} \; m\) to nm:

Identify the conversion factor

The conversion factor between meters and nanometers is 1 m = \(10^9\) nm.

Perform the conversion

To convert \(5.0 \times 10^{-8} \; m\) to nm, multiply the given value by the conversion factor: \[5.0 \times 10^{-8} \; m \times \frac{10^9 \: nm}{1 \; m} = 50 \: nm\] (c) Convert \(6.88 \times 10^5 \; ns\) to s:

Identify the conversion factor

The conversion factor between nanoseconds and seconds is 1 s = \(10^9\) ns.

Perform the conversion

To convert \(6.88 \times 10^5 \; ns\) to s, multiply the given value by the conversion factor: \[6.88 \times 10^5 \; ns \times \frac{1 \; s}{10^9 \: ns} = 6.88 \times 10^{-4} \: s\] (d) Convert 0.50 lb to g:

Identify the conversion factor

The conversion factor between pounds and grams is 1 lb = 453.592 g.

Perform the conversion

To convert 0.50 lb to g, multiply the given value by the conversion factor: \[0.50 \; lb \times \frac{453.592 \: g}{1 \; lb} = 226.796 \: g\] (e) Convert 1.55 kg/m³ to g/L:

Identify the conversion factor

The conversion factor between kilograms and grams is 1 kg = 1000 g, and the conversion factor between cubic meters and liters is 1 m³ = 1000 L.

Perform the conversion

To convert 1.55 kg/m³ to g/L, multiply the given value by the conversion factors: \[1.55 \; \frac{kg}{m^3} \times \frac{1000 \; g}{1 \; kg} \times \frac{1 \; m^3}{1000 \; L} = 1.55 \: \frac{g}{L}\] (f) Convert 5.850 gal/hr to L/s:

Identify the conversion factor

The conversion factor between gallons and liters is 1 gal = 3.78541 L, and the conversion factor between hours and seconds is 1 hour = 3600 s.

Perform the conversion

To convert 5.850 gal/hr to L/s, multiply the given value by the conversion factors: \[5.850 \; \frac{gal}{hr} \times \frac{3.78541 \; L}{1 \; gal} \times \frac{1 \; hr}{3600 \; s} \approx 0.000798 \: \frac{L}{s}\]

Key concepts

Conversion Factors

A conversion factor is a number used to change one set of units to another, by multiplying or dividing. It's like a bridge between different units. They are essential when changing from one unit to another. For example, if you know that 1 liter equals 1000 milliliters, this acts as your conversion factor. You can multiply or divide by this number to perform conversions.

• To convert from a larger to a smaller unit, multiply by the conversion factor.
• To convert from a smaller to a larger unit, divide by the conversion factor.

Using conversion factors accurately is key to solving conversion problems efficiently. Always ensure the units cancel appropriately to reach the correct answer.

Metric System

The metric system is a system of measurement based on powers of ten. It's the most commonly used measurement system worldwide.

• It uses units such as meters, liters, and grams for measurements of length, volume, and mass, respectively.
• Prefixes like milli-, centi-, and kilo- are used to denote multiples or fractions of base units. For example, 1 millimeter is 1/1000th of a meter.
• The system's simplicity makes it easy to convert between units by simply moving the decimal point.

Since the metric system is decimal-based, conversions are consistent and straightforward, making it ideal for scientific and everyday measurements.

Dimensional Analysis

Dimensional analysis, also known as the factor-label method, is a problem-solving technique used to convert one unit to another. It ensures that calculations are correct by checking the consistency of units throughout a calculation.

• Write down the quantity to be converted with its unit.
• Multiply by conversion factors to cancel out the original units and bring in the new units.
• Ensure all units except the desired result cancel out properly.

Dimensional analysis not only helps with conversions but also checks the plausibility of calculations by verifying that the final units match the expected result.

Volume Conversion

Volume conversions often occur in cooking, chemistry, and other fields where knowing how much space something occupies is essential.

• Common units of volume include liters, milliliters, cubic meters, and gallons.
• Conversion between units like liters to milliliters, or gallons to liters, often requires the use of conversion factors.
• When converting, it's important to understand the magnitude of each unit to ensure accuracy in your calculations.

The key to mastering volume conversions is familiarity with commonly used units and practice using the appropriate conversion factors.