Problem 94 - Chemistry Matter and Change

Question

The accepted density for copper is 8.96 \(\mathrm{g} / \mathrm{mL} .\) Calculate the percent error for each of these measurements. a. 8.86 \(\mathrm{g} / \mathrm{mL} \quad\) c. 9.00 \(\mathrm{g} / \mathrm{mL}\) b. 8.92 \(\mathrm{g} / \mathrm{mL} \quad\) d. 8.98 \(\mathrm{g} / \mathrm{mL}\)

Step-by-Step Solution

Answer

Percent errors: a. 1.12%, b. 0.45%, c. 0.45%, d. 0.22%.

Step 1: Understand the Percent Error Formula

Percent error is a measure of how inaccurate a measurement is, compared to the true or accepted value. It is calculated using the formula: \[\text{Percent Error} = \left( \frac{|\text{Experimental Value} - \text{Accepted Value}|}{|\text{Accepted Value}|} \right) \times 100\%\] where the 'Experimental Value' is the measured density, and the 'Accepted Value' is the known density of copper, which is 8.96 g/mL.

Step 2: Calculate Percent Error for Measurement a

For measurement a, the experimental value is 8.86 g/mL. Substituting into the percent error formula: \[\text{Percent Error for a} = \left( \frac{|8.86 - 8.96|}{8.96} \right) \times 100\% = \left( \frac{0.10}{8.96} \right) \times 100\% \approx 1.12\%\]

Step 3: Calculate Percent Error for Measurement b

For measurement b, the experimental value is 8.92 g/mL. Substituting into the percent error formula: \[\text{Percent Error for b} = \left( \frac{|8.92 - 8.96|}{8.96} \right) \times 100\% = \left( \frac{0.04}{8.96} \right) \times 100\% \approx 0.45\%\]

Step 4: Calculate Percent Error for Measurement c

For measurement c, the experimental value is 9.00 g/mL. Substituting into the percent error formula: \[\text{Percent Error for c} = \left( \frac{|9.00 - 8.96|}{8.96} \right) \times 100\% = \left( \frac{0.04}{8.96} \right) \times 100\% \approx 0.45\%\]

Step 5: Calculate Percent Error for Measurement d

For measurement d, the experimental value is 8.98 g/mL. Substituting into the percent error formula: \[\text{Percent Error for d} = \left( \frac{|8.98 - 8.96|}{8.96} \right) \times 100\% = \left( \frac{0.02}{8.96} \right) \times 100\% \approx 0.22\%\]

Key Concepts

Density of Copper

The density of a substance tells us how much mass is contained in a given volume. For copper, a commonly used metal, the accepted density is 8.96 grams per milliliter (g/mL). This value is determined through numerous precise measurements and is considered universally accurate for pure copper.
The density can be expressed simply as:

• Mass per volume = 8.96 g/mL for copper.

Density plays a critical role in applications involving copper, such as in wiring and electronics, as it affects the material's weight and strength. A change in density might indicate impurities or changes in temperature or pressure.

Experimental Value

An experimental value is a measurement obtained through experiments. It represents the value you arrive at using your methods, equipment, and materials.
In a laboratory setting, various factors can affect the experimental value obtained for a substance's property, such as density.
For instance, when you measure the density of copper, your experimental values might differ from the accepted value due to:

• Instrument precision: The tools used may not measure with complete accuracy.
• Human error: Mistakes can occur during the experiment.
• Environmental conditions: Factors like temperature can affect measurements.

Understanding the experimental value helps compare and improve measurement techniques.

Accepted Value

The accepted value is a widely acknowledged benchmark that serves as the standard reference for a particular measurement. In scientific contexts, it is often derived from multiple experiments and careful analysis by experts.
For copper, the accepted density value used here, 8.96 g/mL, allows researchers and students to compare their findings straightforwardly.
When working in experiments, always keep in mind:

• The accepted value is considered highly reliable.
• It serves as a baseline to determine the accuracy of your experimental results.
• Deviations from the accepted value can inform you about potential measurement errors or experimental limitations.

Using accepted values is critical for verifying the credibility of your findings.

Measurement Error

Measurement error is the difference between the experimental value and the accepted value. It's a natural part of any experimental process but must be minimized to ensure accuracy.
To calculate the measurement error, the percent error formula is employed, which quantifies the size of the error:

• Percent Error = \( \left( \frac{| \text{Experimental Value} - \text{Accepted Value} |}{ | \text{Accepted Value} |} \right) \times 100\% \)

Measurement errors can result from various sources, such as:

• Faulty equipment giving inaccurate readings.
• Misreading scales or recordings.
• Uncontrolled variables affecting conditions throughout the experiment.

Understanding measurement error is crucial for improving experimental reliability and ensuring that results are as close to the truth as possible.