Problem 52 - Chemistry

Question

Carry out the following conversions: (a) 5 pm = _______ cm = _______ nm (b) \(8.5 \mathrm{~cm}^{3}=\) _______ \(\mathrm{m}^{3}=\) _______ \(\mathrm{mm}^{3}\) (c) \(65.2 \mathrm{mg}=\) _______ \(\mathrm{g}=\) _______ pg

Step-by-Step Solution

Answer

(a) 5 pm = 5×10^{-12} cm = 5×10^{-3} nm; (b) 8.5 cm³ = 8.5×10^{-6} m³ = 8500 mm³; (c) 65.2 mg = 0.0652 g = 6.52×10^{10} pg.

Step 1: Convert 5 pm to cm

To convert from picometres (pm) to centimetres (cm), we use the conversion factor: 1 pm = 1×10^{-12} cm.Multiply 5 pm by this factor: \[ 5 \text{ pm} \times 10^{-12} \text{ cm/pm} = 5 \times 10^{-12} \text{ cm} \]

Step 2: Convert 5 pm to nm

To convert from picometres (pm) to nanometres (nm), use the conversion factor: 1 pm = 1×10^{-3} nm. Multiply 5 pm by this factor: \[ 5 \text{ pm} \times 10^{-3} \text{ nm/pm} = 5 \times 10^{-3} \text{ nm} \]

Step 3: Convert 8.5 cm³ to m³

To convert from cubic centimetres (cm³) to cubic metres (m³), use the conversion factor: 1 cm³ = 1×10^{-6} m³. Multiply 8.5 cm³ by this factor: \[ 8.5 \text{ cm}^3 \times 10^{-6} \text{ m}^3/\text{cm}^3 = 8.5 \times 10^{-6} \text{ m}^3 \]

Step 4: Convert 8.5 cm³ to mm³

To convert from cubic centimetres (cm³) to cubic millimetres (mm³), we use the factor: 1 cm³ = 1000 mm³. Multiply 8.5 cm³ by this factor: \[ 8.5 \text{ cm}^3 \times 1000 \text{ mm}^3/\text{cm}^3 = 8500 \text{ mm}^3 \]

Step 5: Convert 65.2 mg to g

To convert from milligrams (mg) to grams (g), use the factor: 1 mg = 1×10^{-3} g. Multiply 65.2 mg by this factor: \[ 65.2 \text{ mg} \times 10^{-3} \text{ g/mg} = 0.0652 \text{ g} \]

Step 6: Convert 65.2 mg to pg

To convert from milligrams (mg) to picograms (pg), use the factor: 1 mg = 1×10^{9} pg. Multiply 65.2 mg by this factor: \[ 65.2 \text{ mg} \times 10^{9} \text{ pg/mg} = 6.52 \times 10^{10} \text{ pg} \]

Key Concepts

Picometres to Centimetres

Converting picometres (pm) to centimetres (cm) can sound tricky, but it's straightforward once you understand the scale involved. Picometres are part of the metric system, designed to measure very tiny lengths. They are often used on the atomic scale due to their small size. Here's a breakdown of how to do this conversion using the conversion factor:

• 1 picometre is equivalent to \(1 \times 10^{-12}\) centimetres. This factor means one picometre is one-trillionth of a centimetre.

To convert any length from picometres to centimetres:1. Take the number of picometres.2. Multiply it by \(1 \times 10^{-12}\).
For example, converting 5 pm into cm:\[5 \text{ pm} \times 10^{-12} \text{ cm/pm} = 5 \times 10^{-12} \text{ cm}\] This results in an extremely small number because picometres are very small as compared to centimetres. Understanding this conversion is crucial when dealing with atomic and molecular scales.

Cubic Centimetres to Cubic Metres

When dealing with volume, switching from cubic centimetres (cm³) to cubic metres (m³) can seem daunting, but the concept is simple. A cubic metre is a much larger unit than a cubic centimetre, which influences the conversion factor.

• 1 cubic centimetre equals \(1 \times 10^{-6}\) cubic metres.

To convert:1. Take your value in cubic centimetres.2. Multiply it by \(1 \times 10^{-6}\).
In practice, for 8.5 cm³, the conversion is:\[8.5 \text{ cm}^3 \times 10^{-6} \text{ m}^3/\text{cm}^3 = 8.5 \times 10^{-6} \text{ m}^3\] Understanding this conversion helps in fields like chemistry and physics, where materials are often measured in these units. Also, knowing that \(1 \text{ cm}^3 = 1000 \text{ mm}^3\) further aids conversions by helping bridge the gap between different measurements systems.

Milligrams to Grams

Milligrams (mg) to grams (g) is one of the most common conversions you'll encounter in science, particularly in chemistry and biology. This conversion is essential because it helps in accurately weighing substances and understanding their mass in a more practical unit.

• 1 milligram equals \(1 \times 10^{-3}\) grams. So, milligrams are a thousandth of a gram.

To convert:1. Start with your value in milligrams.2. Multiply it by \(1 \times 10^{-3}\).
Let's see how it works, using 65.2 mg as an example:\[65.2 \text{ mg} \times 10^{-3} \text{ g/mg} = 0.0652 \text{ g}\] This calculation shows how small amounts in milligrams are converted to a more prominent figure in grams. This makes it easier to handle and communicate in most scientific contexts.