Question

A student calculates the density of iron as 6.80 \(\mathrm{g} / \mathrm{cm}^{3}\) by using lab data for mass and volume. A handbook reveals that the correct value is 7.86 \(\mathrm{g} / \mathrm{cm}^{3} .\) What is the percentage error?

Short answer

The percentage error is 13.49%.

Step-by-step solution

- Identify given values

Note the experimental value and the true value for the density of iron. The experimental value is 6.80 \(\text{g/cm}^3\) and the true value is 7.86 \(\text{g/cm}^3\).

- Understand the formula for percentage error

The formula for percentage error is: \[ \text{Percentage Error} = \frac{|\text{Experimental Value} - \text{True Value}|}{\text{True Value}} \times 100 \]

- Apply the values to the formula

Substitute the given values into the formula: \[ \text{Percentage Error} = \frac{|6.80 - 7.86|}{7.86} \times 100 \]

- Calculate the absolute difference

Find the absolute difference between the experimental value and the true value: \[ |6.80 - 7.86| = 1.06 \]

- Complete the calculation

Substitute this difference back into the formula and solve: \[ \text{Percentage Error} = \frac{1.06}{7.86} \times 100 = 13.49\text{%} \]

Key concepts

density

Density is a measure of how much mass is contained in a given volume. It is calculated by dividing the mass of an object by its volume. The formula for density is: \( \text{Density} = \frac{ \text{Mass} }{ \text{Volume} } \). In this exercise, the student measured the density of iron using lab data.
Understanding density is crucial in many fields, including materials science, engineering, and chemistry. It helps to identify substances and predict how they will behave.

• The student found a density of 6.80 \( \text{g/cm}^3 \), but the true density of iron is 7.86 \( \text{g/cm}^3 \).

experimental error

Experimental error refers to the differences between a measured value and the true value. It can arise from various factors, including measurement precision, human error, and environmental conditions. Recognizing and accounting for experimental error is essential for accurate scientific work.
The step-by-step calculation demonstrates how to identify and correct for experimental errors. This helps improve the reliability of the student's results and their understanding of lab procedures.

• In this case, the experimental value for the density was 6.80 \( \text{g/cm}^3 \).
• The true value was 7.86 \( \text{g/cm}^3 \).
• Identifying this error helps in refining the measurement methods and accuracy.

absolute difference

The absolute difference is the positive difference between two numbers, disregarding which one is larger. It is calculated using the formula: \( |a - b| \), where \(|x|\) denotes the absolute value of \(x\).
In our example, the absolute difference between the experimental density and the true density of iron is calculated by:
\(|6.80 - 7.86| = 1.06\). This shows how much the student's measured value deviates from the correct value.
Understanding absolute difference is essential in calculating errors and assessing measurement accuracy.

• It tells us how far off a measurement is from the true value.
• It simplifies the error calculation process.

error analysis

Error analysis involves studying the types and sources of errors in measurements. It helps to understand how much uncertainty is present in experimental results.
There are two main types of experimental errors:

• Systematic Errors: Consistent, repeatable errors linked to faulty equipment or flawed methods.
• Random Errors: Irregular fluctuations that arise from unpredictable factors, such as environmental conditions.

In this exercise, calculating the percentage error highlights the deviation quantitatively.
The percentage error is given by:
\( \text{Percentage Error} = \frac{|\text{Experimental Value} - \text{True Value}|}{\text{True Value}} \times 100 \)
This helps students recognize the significance of precision in experimental work and the need for careful data analysis.