Question

If you were to convert the measured length \(4.165 \mathrm{ft}\) to yards by multiplying by the conversion factor \((1 \mathrm{yd} / 3 \mathrm{ft})\), how many significant figures should the answer contain? Why?

Short answer

The answer should have four significant figures because the original measurement 4.165 ft had four significant figures and the conversion factor is exact.

Step-by-step solution

Identifying Significant Figures in the Given Measurement

Determine the number of significant figures in the original measurement. The measurement given is 4.165 ft, which contains four significant figures. All non-zero digits are considered significant, and any zeros between them or following a decimal point and a non-zero digit are also significant.

Applying the Conversion Factor

Multiply the given length by the conversion factor to convert feet to yards. The conversion factor (1 yd / 3 ft) does not limit the number of significant figures because it is a defined quantity with an infinite number of significant figures.

Matching Significant Figures in the Answer

The answer should be expressed with the same number of significant figures as the measurement with the smallest number of significant figures used in the calculation. Since the given value 4.165 ft has four significant figures and the conversion factor is exact, the answer should also have four significant figures.

Key concepts

Measurement Conversion

Understanding measurement conversion is crucial in scientific calculations. It's a process that allows us to convert a quantity expressed in one unit to another unit, maintaining the same physical quantity. For example, to convert a length measurement from feet to yards, you multiply the length in feet by the conversion factor \( (1 \mathrm{yd} / 3 \mathrm{ft}) \). It's important to note that conversion factors are defined values based on relationships between different units and carry an unlimited number of significant figures. They do not affect the precision of the calculated result.

When converting measurements, it’s essential to consider if the factor is an exact number or an estimate because this affects the final number of significant figures in the result. Exact numbers, often found in conversion factors, allow you to maintain the precision of your initial measurement. Always be mindful of the units involved in multi-step conversions to ensure accuracy and clarity in your calculations.

Scientific Notation

Scientific notation is a method of writing very large or very small numbers in a compact form. It is expressed as a product of a number between 1 and 10 and a power of 10. For instance, \( 4.165 \mathrm{ft} \) can be written in scientific notation as \( 4.165 \times 10^0 \mathrm{ft} \), which simplifies comparison and calculation with other numbers. This notation is particularly useful in chemistry and physics, where measurements can span across multiple orders of magnitude.

Using scientific notation helps maintain the precision of the numbers since it clearly indicates the significant figures. The digits before the exponential part are all significant, aiding in avoiding ambiguity during mathematical operations and when converting between units. It's a valuable tool for communicating precise measurements and ensuring consistent interpretation of data.

Precision in Chemistry Calculations

Precision in chemistry calculations is all about how closely individual measurements agree with each other. It's intrinsically linked to significant figures, which are the digits within a measurement that are known with certainty plus one that is estimated. When calculating with measurements, the rule of thumb is that the result should not be more precise than the least precise measurement involved in the calculation.

Precise measurements are vital in chemistry because they can impact the outcome and reproducibility of experiments. For example, when converting \(4.165 \mathrm{ft}\) to yards, the result must reflect the original precision of four significant figures, just as we've identified in our measurement. This ensures the validity of our calculations and the consistency of experimental results. Remember that precision is different from accuracy, which measures how close a measurement is to the true value. Both are crucial for high-quality scientific research and comprehension.