Problem 102 - General Chemistry: Principles and Modern Applications

Question

Of the following masses, two are expressed to the nearest milligram. The two are (a) \(32.7 \mathrm{g}\); (b) \(0.03271 \mathrm{kg} ;(\mathrm{c}) 32.7068 \mathrm{g} ;(\mathrm{d}) 32.707 \mathrm{g} ;(\mathrm{e}) 30.7 \mathrm{mg};\) (f) \(3 \times 10^{3} \mu g.\)

Step-by-Step Solution

Answer

The two masses expressed to the nearest milligram are (a) \(32.7 \, \mathrm{g}\) and (f) \(3 \times 10^{3} \, \mu g.\)

Step 1: Convert all masses to milligrams

First, all mass values are converted from grams (g), kilograms (kg) and micrograms (μg) to milligrams (mg). One gram is equivalent to 1000 milligrams, one kilogram is equivalent to 1,000,000 milligrams, and one microgram is equivalent to 0.001 milligrams. Therefore:\n(a) \(32.7 \, \mathrm{g}\) = 32,700\, \mathrm{mg}\n(b) \(0.03271 \, \mathrm{kg}\) = 32,710 \, \mathrm{mg}\n(c) \(32.7068 \, \mathrm{g}\) = 32,706.8 \, \mathrm{mg}\n(d) \(32.707 \, \mathrm{g}\) = 32,707 \, \mathrm{mg}\n(e) \(30.7 \, \mathrm{mg}\) = 30.7 \, \mathrm{mg}\n(f) \(3 \times 10^{3} \, \mu g.\) = 3 \, \mathrm{mg}

Step 2: Identify masses expressed to the nearest milligram

Next, take note of which masses do not have decimals in their milligram form, as they are expressed to the nearest milligram.\nSo, from the calculations in the previous step, the masses which do not contain decimals are (a) 32,700 \, \mathrm{mg} and (f) 3 \, \mathrm{mg}.

Key Concepts

Unit Conversion

Unit conversion is an essential aspect of scientific calculations. It allows us to switch between different measurement systems while maintaining the quantity's value. For mass, common conversions include grams (g), kilograms (kg), milligrams (mg), and micrograms (μg). A fundamental understanding of these conversions helps you switch seamlessly:

• 1 gram (g) = 1,000 milligrams (mg)
• 1 kilogram (kg) = 1,000 grams (g) = 1,000,000 milligrams (mg)
• 1 microgram (μg) = 0.001 milligrams (mg)

In practical scenarios, you'll encounter different units based on precision requirements or industry standards. The ability to convert units enables comparing values effectively, especially when working with data presented in varying units. For instance, in the given exercise, converting all values to milligrams helps unify the comparison process. Always check your conversion factors and ensure calculations are precise to avoid errors.

Precision in Measurements

Precision in measurements refers to the detail in the quantity expressed. It concerns how small a change can be detected or the exactness in measuring the quantity. When working with measurements, like in our exercise, it is crucial to say to what degree the number is precise:

• Significant figures help control this precision.
• Expressing a value "to the nearest milligram" means there shouldn't be any fractional milligrams included.

In the given problem, we aim to find which values are expressed at this precise level. Once converted to milligrams, those without decimals fit the criterion. This principle ensures consistency and reliability, especially in scientific research, where measurement precision can significantly impact results. A measurement without decimals after conversion implies rounding to the nearest whole unit, reflecting its precision.

Scientific Notation

Scientific notation is a method to express very large or very small numbers in a concise form. This helps in keeping calculations manageable and clear, especially in scientific contexts. It represents numbers as a product of a coefficient (between 1 and 10) and a power of ten. For instance, the number 3,000 may be represented as:

• 3,000 = \(3 \times 10^3\)

In our context, scientific notation helps simplify how we deal with small units like micrograms or express large quantities. For example, 3,000 micrograms can be represented in scientific notation to manage calculations better and make it easier to compare with other values. Using scientific notation not only saves space but also clarifies the scale at which you are measuring, aiding in precision and accuracy in reporting results.