Problem 22 - General Chemistry

Question

1.22 Which of the following represents the smallest mass? a \(23 \mathrm{cg}\) b \(2.3 \times 10^{3} \mu \mathrm{g}\) c \(0.23 \mathrm{mg}\) d \(0.23 \mathrm{~g}\) e \(2.3 \times 10^{-2} \mathrm{~kg}\)

Step-by-Step Solution

Answer

c \(0.23 \mathrm{mg}\) is the smallest mass.

Step 1: Convert centigrams to grams

The given value is \(23\, \text{cg}\). To convert centigrams to grams, use the conversion: \(1\, \text{cg} = 0.01\, \text{g}\). Therefore, \(23\, \text{cg} = 23 \times 0.01\, \text{g} = 0.23\, \text{g}\).

Step 2: Convert micrograms to grams

The given value is \(2.3 \times 10^{3}\, \mu\text{g}\). To convert micrograms to grams, use the conversion: \(1\, \mu\text{g} = 10^{-6}\, \text{g}\). Thus, \(2.3 \times 10^{3}\, \mu\text{g} = 2.3 \times 10^{3} \times 10^{-6}\, \text{g} = 2.3 \times 10^{-3}\, \text{g}\).

Step 3: Convert milligrams to grams

The given value is \(0.23\, \text{mg}\). To convert milligrams to grams, use the conversion: \(1\, \text{mg} = 0.001\, \text{g}\). Therefore, \(0.23\, \text{mg} = 0.23 \times 0.001\, \text{g} = 0.00023\, \text{g}\).

Step 4: Express grams in grams (already in grams)

The given value is \(0.23\, \text{g}\). This value is already in grams, so no conversion is needed.

Step 5: Convert kilograms to grams

The given value is \(2.3 \times 10^{-2}\, \text{kg}\). To convert kilograms to grams, use the conversion: \(1\, \text{kg} = 1000\, \text{g}\). Thus, \(2.3 \times 10^{-2}\, \text{kg} = 2.3 \times 10^{-2} \times 1000\, \text{g} = 23\, \text{g}\).

Step 6: Compare all values in grams

Now, compare all the masses in grams:- \(23\, \text{cg} = 0.23\, \text{g}\)- \(2.3 \times 10^{3}\, \mu\text{g} = 0.0023\, \text{g}\)- \(0.23\, \text{mg} = 0.00023\, \text{g}\)- \(0.23\, \text{g}\ = 0.23\, \text{g}\)- \(2.3 \times 10^{-2}\, \text{kg} = 23\, \text{g}\)The smallest mass is \(0.23\, \text{mg}\) which equates to \(0.00023\, \text{g}\).

Key Concepts

Metric Units

In science and everyday life, we use metric units to measure different quantities. Metric units are based on the decimal system, making them easy to convert by adding or removing zeros.
The most common metric unit for mass is the gram (g). One thousand grams become a kilogram (kg), while one-thousandth of a gram is a milligram (mg).
Similarly, a centigram (cg) is one-hundredth of a gram, and a microgram (μg) is one-millionth of a gram. Understanding these units is crucial in calculating and comparing weights easily. This is particularly handy when performing unit conversion tasks.

Unit Conversion

Converting between different metric units might seem daunting at first, but it's all about moving the decimal point. This task becomes much simpler with a clear understanding of the conversion factors:

• 1 kilogram (kg) = 1,000 grams (g)
• 1 gram (g) = 100 centigrams (cg)
• 1 gram (g) = 1,000 milligrams (mg)
• 1 gram (g) = 1,000,000 micrograms (μg)

So, if you want to convert centigrams to grams, multiply by 0.01. Need to turn micrograms to grams? Multiply by 0.000001. Remember, these conversion factors help you seamlessly move between different units, ultimately simplifying complex problems.

Mass Comparison

Comparing masses requires them to be in the same unit. Once converted, you can easily rank them from smallest to largest.
For example, in the exercise, you were comparing the following masses: 23 cg, 2.3 x 10^3 μg, 0.23 mg, 0.23 g, and 2.3 x 10^-2 kg.
Converting them all to grams gives:

• 23 cg = 0.23 g
• 2.3 x 10^3 μg = 0.0023 g
• 0.23 mg = 0.00023 g
• 0.23 g = 0.23 g
• 2.3 x 10^-2 kg = 23 g

Once all values are in grams, comparing becomes straightforward. As seen, 0.23 mg is the smallest, equating to 0.00023 g. Emphasizing conversion accuracy enhances your understanding of how each mass relates.