Problem 111 - General Chemistry

Question

For each of the following, write the measurement in terms of an appropriate prefix and base unit. A The mass of calcium per milliliter in a sample of blood serum is \(0.0912 \mathrm{~g}\). B The radius of an oxygen atom is about \(0.000000000066 \mathrm{~m}\). C A particular red blood cell measures \(0.0000071 \mathrm{~m}\). D The wavelength of a certain ultraviolet radiation is \(0.000000056 \mathrm{~m} .\)

Step-by-Step Solution

Answer

A: 91.2 mg, B: 66 pm, C: 7.1 μm, D: 56 nm

Step 1: Convert Mass of Calcium to Milligrams

The mass of calcium per milliliter is given as \(0.0912 \mathrm{~g}\). To convert grams to milligrams, note that there are 1000 milligrams in a gram. Thus, \(0.0912 \mathrm{~g} = 0.0912 \times 1000 = 91.2 \mathrm{~mg}\).

Step 2: Convert Radius of Oxygen Atom to Picometers

The radius of an oxygen atom is given as \(0.000000000066 \mathrm{~m}\). To convert meters to picometers, recall that one meter is equal to \(10^{12}\) picometers. Thus, \(0.000000000066 \mathrm{~m} = 0.000000000066 \times 10^{12} = 66 \mathrm{~pm}\).

Step 3: Convert Size of Red Blood Cell to Micrometers

A red blood cell measures \(0.0000071 \mathrm{~m}\). To convert meters to micrometers, note that there are \(10^6\) micrometers in a meter. Therefore, \(0.0000071 \mathrm{~m} = 0.0000071 \times 10^6 = 7.1 \mathrm{~\mu m}\).

Step 4: Convert Wavelength of Ultraviolet Radiation to Nanometers

The wavelength of the ultraviolet radiation is \(0.000000056 \mathrm{~m}\). To express this in nanometers, remember there are \(10^9\) nanometers in a meter. Hence, \(0.000000056 \mathrm{~m} = 0.000000056 \times 10^9 = 56 \mathrm{~nm}\).

Key Concepts

Metric System

The metric system is a decimal-based system of measurement that is used all around the world. It is especially common in scientific fields because of its consistency and ease of conversion between units.
Almost all measurements in the metric system are based on powers of ten. This makes it straightforward to convert from one unit to another by simply multiplying or dividing by 10, 100, 1000, and so on, which is incredibly convenient in calculations.
In the metric system, you can find a base unit like the meter (for length) or the gram (for mass), and these units can be modified with prefixes such as kilo-, centi-, milli-, etc., to represent multiples or fractions of the unit.

• 1 kilometer (km) = 1000 meters (m)
• 1 meter = 100 centimeters (cm)
• 1 gram (g) = 1000 milligrams (mg)
• 1 liter (L) = 1000 milliliters (mL)

The metric system's logical structure and standardization simplify conversion and calculations across different fields.

Scientific Notation

Scientific notation is a method of writing very large or very small numbers in a more compact form. This approach is crucial in fields like chemistry and physics where the numbers can be extremely tiny or enormously large.
Scientific notation expresses numbers as a product of a number between 1 and 10, and a power of ten. It makes calculations easier and reduces the potential for errors with cumbersome numbers.
For instance, the number 0.000000000066 meters, which is the radius of an oxygen atom, can be expressed as 6.6 x 10^-11 meters in scientific notation.

• To convert a number to scientific notation:
  • Move the decimal point so that you have a number between 1 and 10.
  • Count how many places you moved the decimal point; this number becomes your exponent of 10.
  • If you move the decimal point to the right, the exponent is negative. If you move it to the left, the exponent is positive.

Scientific notation is incredibly useful in making these large and small numbers manageable and comparable at a glance.

Mass and Length Measurements

Mass and length are fundamental properties we regularly measure in science and everyday life. In the metric system, mass is typically measured in grams and kilograms, while length is measured in meters and its derivatives.
For example, the mass of calcium in blood serum can be converted from grams to a smaller unit, milligrams, making it easier to relate to daily nutritional values in micrograms or milligrams rather than grams.
Length measurements such as the size of red blood cells or atoms frequently require smaller units like micrometers or picometers, because they are incredibly small.

• To convert mass:
  • Recall that 1 gram = 1000 milligrams which means to convert from grams to milligrams, multiply by 1000.
• To convert length measurements:
  • 1 meter = 10⁹ nanometers or 10¹² picometers, thus, knowing the right multiplier helps in precision.

Chemistry Calculations

Chemistry calculations involve converting and comparing different quantities and units, which is a routine task to perform analyses, understand reactions, and derive conclusions.
This includes converting units of mass and length to appropriate ones that match the scale of atoms, molecules, or reactions. Understanding how to effectively work across different units is essential in chemistry calculations.
For instance:

• In calculating the concentration of a substance in a solution, converting grams to milligrams can allow for more nuanced and precise formulations.
• When considering the wavelengths of radiation, converting meters to nanometers helps fit the scale and makes it practical to relate to other known values.

Mastering these conversions enhances understanding and accuracy in chemistry studies and ensures that different measurements align with their respective scales. This way, the chemist can focus on interpreting data rather than grappling with unit confusion.