Problem 65 - Chemistry: Atoms First

Question

The diameter of a red blood cell is about \(3 \times 10^{-4}\) in. What is its diameter in centimetres?

Step-by-Step Solution

Answer

The diameter of the red blood cell in centimeters is approximately \(7.62 \times 10^{-4}\) cm.

Step 1: Understand the Conversion Factor

To convert inches to centimeters, use the conversion factor that 1 inch is equal to 2.54 centimeters.

Step 2: Set Up the Conversion

Write the given diameter of the red blood cell in inches and multiply it by the conversion factor to convert to centimeters.

Step 3: Perform the Conversion

Multiply the diameter of the red blood cell by the conversion factor. \[ (3 \times 10^{-4} \, \text{in}) \times \frac{2.54 \, \text{cm}}{1 \, \text{in}} = (3 \times 2.54 \times 10^{-4}) \, \text{cm} \]

Key Concepts

Dimensional Analysis

Dimensional analysis, often referred to as unit factor method or unit conversion, is a critical skill in chemistry that involves the conversion of measurements from one unit to another. This technique uses the multiplication or division of conversion factors – ratios that express the same physical quantity in different units – to transform a given measurement into the desired unit.

When faced with a dimensional analysis problem, it's essential to understand the conversion factors at play. For example, the exercise asks for the conversion of the diameter of a red blood cell from inches to centimeters. Knowing that 1 inch equals 2.54 centimeters is a foundational step in applying dimensional analysis. By writing the given value and systematically applying the conversion factor, you are effectively 'canceling out' the original units and ending up with your desired unit of measurement.

Consider planning your steps before starting the calculations, ensuring each conversion factor is correctly positioned (numerator or denominator) to cancel out the previous units. With practice, dimensional analysis becomes an intuitive process that simplifies complex unit conversions in chemistry.

Metric Conversion

Metric conversion is a specific type of dimensional analysis that deals with the units of the metric system, such as meters, liters, and grams. The metric system is based on powers of ten, which makes conversions quite straightforward. For instance, a centimeter is one-hundredth of a meter, and a kilogram is one thousand grams.

In the given exercise, converting inches to centimeters is an example of metric conversion. As a rule of thumb, remember the key prefixes in the metric system: 'kilo-' for a thousandfold, 'centi-' for a hundredth, and 'milli-' for a thousandth, among others.

Students often benefit from reference charts or mnemonic devices to remember the order of metric prefixes. Also, using a consistent method such as writing down the conversion factors and cross-multiplying, as shown in the example, can reduce the chance of errors and increase the speed and accuracy of solving conversion problems in chemistry.

Scientific Notation

Scientific notation is a method of expressing very large or very small numbers in a concise form. In chemistry, it is frequently used for expressing measurements, concentration levels, or the number of atoms and molecules. The notation consists of a coefficient between 1 and 10 multiplied by ten raised to a power (exponent).

The diameter of a red blood cell in the exercise is given as \(3 \times 10^{-4}\) inches, a compact representation indicating a number significantly less than one inch. With scientific notation, you avoid writing out many zeros, removing the risk of miscounting them and making calculations more manageable.

When performing calculations with scientific notation, special care must be taken with the exponents during multiplication and division. To enhance understanding, try breaking down the steps: multiply the coefficients together, then add or subtract the exponents to calculate the power of ten. Recognizing how to manipulate scientific notation is instrumental in translating complex information into usable data.