Question

The following water measurements are made: \(18 \mathrm{~mL}\) of water measured with a beaker, \(128.7 \mathrm{~mL}\) of water measured with a graduated cylinder, and \(23.45 \mathrm{~mL}\) of water measured with a buret. If all of these water samples are then poured together into one container, what total volume of water should be reported? Support your answer.

Short answer

The total volume of water that should be reported is \(170 \mathrm{~mL}\), as it is rounded to the nearest whole number based on the least precise measurement from the beaker.

Step-by-step solution

Analyze the measuring tools

In this step, we will look at the given reflecting tools. The beaker is the least accurate measuring tool, the graduated cylinder provides an intermediate level of precision, and the buret is the most accurate.

Find the total volume measured

Add all the given volumes of water measured: \[Total\: volume = Volume_{beaker} + Volume_{graduated\:cylinder} + Volume_{buret}\] Plug in the given volumes: \[Total\: volume = 18 \, \mathrm{mL} + 128.7 \, \mathrm{mL} + 23.45 \, \mathrm{mL}\] Calculate the total volume: \[Total\: volume = 170.15 \, \mathrm{mL}\]

Determine the precision of the total volume

For this step, we need to determine the precision of the reported value based on the least precise measuring tool, which is the beaker. As the beaker has a lower limit of accuracy than the other instruments, we should use its precision as our final answer. The beaker's given volume is \(18 \mathrm{~mL}\). It has no decimal places, so our final reported value also should not have any decimal places.

Round the final value

To comply with the precision from the beaker, round the total volume value calculated in step 2 to the nearest whole number: \[Reported\: volume = 170 \, \mathrm{mL}\] The total volume of water that should be reported is 170 mL.

Key concepts

Significant Figures

Significant figures are crucial when working with measurements to ensure that results are reported accurately based on the precision of the tools used. These figures include all known digits plus one estimated digit. Significant figures help communicate not just the size but also the accuracy of the measurement.

Whenever you measure something, the number of significant figures you use tells others how precise your measurement is. For instance, a measurement of 18 mL from a beaker has just two significant figures. In contrast, 23.45 mL from a buret has four significant figures. Each digit in these figures adds to how trustworthy or reliable the measurement is.

• Non-zero numbers are always significant.
• Any zeros between significant numbers are significant.
• Leading zeros (zeros before the first non-zero digit) are not significant.
• Trailing zeros (zeros after the last non-zero digit) in a decimal number are significant.

Understanding significant figures helps in ensuring you do not overstate the accuracy and mislead about the precision of a measurement.

Precision in Measurements

Precision refers to the closeness of multiple measurements to each other. When different laboratory equipment is used, understanding the precision of each tool is vital.

For example, a beaker offers minimal precision as it might only provide a rough estimate, being more suited to holding and mixing than measuring with high accuracy. On the other hand, devices like a buret or a graduated cylinder can give finer measurements due to smaller scales and precise markings. A buret is ranked higher in precision, allowing for readings with more significant figures.

The precision of measurements affects the reliability of data. When you combine measurements, always refer to the least precise one to determine the precision of your total or final calculation. This principle ensures that your final result does not overstate the accuracy implied by the least precise measurement.

Volume Calculation

Calculating the total volume requires summing up all individual measurements. It might seem straightforward, but it's essential to consider the precision and significant figures from each measurement.

When adding quantities with different precision levels, the least precise measurement determines the precision of the result. For example, if you sum 18 mL, 128.7 mL, and 23.45 mL, the sum equals 170.15 mL. However, since 18 mL (from the beaker) is the least precise (no decimal places), the sum is rounded to 170 mL.

This practice is crucial in science, where reporting overly precise results from imprecise data can lead to errors. Always adjust the final result based on the lowest precision level, ensuring the integrity of your calculations.

Laboratory Equipment

The choice of laboratory equipment significantly impacts the precision and accuracy of measurements. Different tools serve different purposes, offering varying levels of precision.

• Beakers: Ideal for mixing or holding reactions rather than precise measurements. They have large margins of error.
• Graduated Cylinders: Provide better accuracy than beakers for measuring liquid volumes, often used for preparing solutions.
• Burets: Offer the highest precision among these tools. Used for titration, they measure volumes as small as tenths of a milliliter.

Choosing the correct equipment for a specific task depends on the required precision and accuracy of the measurement. It's vital to understand these differences to ensure that the data collected is as reliable and precise as possible, adhering to scientific standards.