Question

Carry out the following conversions: (a) \(185 \mathrm{nm}\) to meters (b) 4.5 billion years (roughly the age of Earth) to seconds (assume 365 days in a year), (c) \(71.2 \mathrm{~cm}^{3}\) to cubic meters, (d) \(88.6 \mathrm{~m}^{3}\) to liters.

Short answer

(a) \(1.85 \times 10^{-7} \) meters, (b) \(1.419 \times 10^{17}\) seconds, (c) \(7.12 \times 10^{-5} \) m³, (d) 88600 liters.

Step-by-step solution

Convert Nanometers to Meters

For part (a), you need to convert nanometers (nm) to meters. Use the conversion factor that 1 meter equals \(10^9\) nanometers. Therefore, the conversion is done as follows:\[185 \text{ nm} = \frac{185}{10^9} \text{ meters} = 1.85 \times 10^{-7} \text{ meters}\]

Convert Billion Years to Seconds

For part (b), you need to convert 4.5 billion years to seconds. Begin with converting years to days, then days to hours, hours to minutes, and finally minutes to seconds. Assume there are 365 days in a year, 24 hours in a day, 60 minutes in an hour, and 60 seconds in a minute.\[4.5 \times 10^9 \text{ years} \times 365 \text{ days/year} \times 24 \text{ hours/day} \times 60 \text{ minutes/hour} \times 60 \text{ seconds/minute} = 1.419 \times 10^{17} \text{ seconds}\]

Convert Cubic Centimeters to Cubic Meters

For part (c), you need to convert cubic centimeters (cm³) to cubic meters (m³). Note that 1 meter is 100 centimeters, so 1 cubic meter is \(100^3\) (or \(10^6\)) cubic centimeters:\[71.2 \text{ cm}^3 = \frac{71.2}{10^6} \text{ m}^3 = 7.12 \times 10^{-5} \text{ m}^3\]

Convert Cubic Meters to Liters

For part (d), you need to convert cubic meters to liters. Knowing that 1 cubic meter equals 1000 liters, the conversion is straightforward:\[88.6 \text{ m}^3 \times 1000 \text{ liters/m}^3 = 88600 \text{ liters}\]

Key concepts

Nanometers to Meters Conversion

When you're working in the vast world of chemistry, you'll often encounter the need to convert between different units of measurement. One common conversion is from nanometers to meters. Nanometers are incredibly tiny, which is why this unit is fundamental in fields like chemistry and nanotechnology. Did you know that there are 1 billion nanometers in a single meter? Yes, 1 meter equals \(10^9\) nanometers. This is why when converting nanometers to meters, you divide by \(10^9\). For instance, converting 185 nanometers gives you:

• Using the formula: \( \frac{185}{10^9} \)
• You get: \(1.85 \times 10^{-7}\) meters

Just think of it as compressing a large number of nanometers into a much shorter measurement unit.

Years to Seconds Conversion

Converting time from years to seconds is a multi-step process, but it’s straightforward once you know the right path. This type of conversion requires multiplying through the units of time: days, hours, minutes, and then seconds. The Earth is about 4.5 billion years old. Let’s convert that to seconds:

• Start with 4.5 billion years.
• There are 365 days in a year, so multiply: \(4.5 \times 10^9 \times 365\).
• Each day has 24 hours, so the next step is multiplying by 24.
• With 60 minutes in an hour and 60 seconds in a minute, multiply by 60 twice.
• The result is: \(1.419 \times 10^{17}\) seconds

Understanding time in such small units gives insights into the grand scales at which the universe operates.

Cubic Centimeters to Cubic Meters Conversion

Volume conversions from cubic centimeters to cubic meters are common in chemistry when dealing with substances at different scales. A cubic meter is a significantly larger unit compared to a cubic centimeter. Knowing that 1 meter equals 100 centimeters, you understand that:

• 1 cubic meter equals \(10^6\) cubic centimeters (since \((100)^3 = 10^6\)).
• To convert 71.2 cubic centimeters to cubic meters, divide by \(10^6\):
• \( \frac{71.2}{10^6} \)
• This gives: \(7.12 \times 10^{-5}\) cubic meters

Understanding this conversion is crucial for accurate measurements, especially when dealing with chemical reactions and solutions.

Cubic Meters to Liters Conversion

The metric system makes volume conversions straightforward because of its decimal-based structure. When converting from cubic meters to liters, you only need to remember that 1 cubic meter equals 1000 liters. So, for any conversion:

• Simply multiply the volume in cubic meters by 1000.
• If you have 88.6 cubic meters, like in this problem:
• Multiply: \(88.6 \times 1000\)
• And you get: 88600 liters

This conversion is often encountered in engineering and chemistry for predicting containers and volumes.