Question

A handbook gives the density of calcium as 1.54 \(\mathrm{g} / \mathrm{cm}^{3}\) . Based on lab measurements, what is the percentage error of a density calculation of 1.25 \(\mathrm{g} / \mathrm{cm}^{3} ?\)

Short answer

The percentage error is 18.83%.

Step-by-step solution

Understand the Formula for Percentage Error

The percentage error is given by the formula: \[ \text{Percentage Error} = \frac{|\text{Experimental Value} - \text{Theoretical Value}|}{\text{Theoretical Value}} \times 100 \]In this exercise, the theoretical value is the handbook density (1.54 \, \text{g/cm}^3) and the experimental value is the lab measurement (1.25 \, \text{g/cm}^3).

Subtract the Experimental Value from the Theoretical Value

Calculate the absolute difference between the experimental value and the theoretical value:\[ |1.25 \, \text{g/cm}^3 - 1.54 \, \text{g/cm}^3| = 0.29 \, \text{g/cm}^3 \]

Divide by the Theoretical Value

Divide the absolute difference by the theoretical value (handbook density):\[ \frac{0.29 \, \text{g/cm}^3}{1.54 \, \text{g/cm}^3} \]

Multiply by 100 to Get the Percentage Error

Multiply the result from Step 3 by 100 to find the percentage error: \[ \frac{0.29}{1.54} \times 100 = 18.83\text{%} \]

Key concepts

percentage error

Understanding percentage error can be quite beneficial when evaluating scientific measurements and experiments. It allows you to measure how far off an experimental value is from a known or theoretical value, expressed as a percentage. In our case, the percentage error formula is \[\text{Percentage Error} = \frac{|\text{Experimental Value} - \text{Theoretical Value}|}{\text{Theoretical Value}} \times 100\].
This calculation helps us determine how accurate our experimental results are compared to the accepted standard or theoretical value.
Start by subtracting the experimental value from the theoretical value to get the absolute difference.
Then, divide this difference by the theoretical value, and finally, multiply by 100 to convert it into a percentage.
By following these steps systematically, you can easily compute the percentage error and understand the reliability of your experimental data.
It's a great tool for anyone interested in ensuring their measurements are close to the accepted value.

experimental value

The experimental value is the measurement or result obtained during laboratory experiments.
It often comes from direct observation or measurement using various instruments and can be influenced by human error or equipment limitations.
In our example, the experimental value is the density of calcium measured in the lab (1.25 \(\text{g/cm}^3\)).
Experimentation results can vary, and understanding the experimental value helps us analyze these variances.
When calculating percentage error, the experimental value is used to determine how close or far it is from the theoretical value.

theoretical value

The theoretical value is a known or accepted value based on scientific research and authoritative sources.
It is considered the 'true' value that experiments aim to replicate.
For this exercise, the theoretical value is the handbook density of calcium, which is 1.54 \(\text{g/cm}^3\).
The theoretical value serves as the benchmark against which the experimental value is measured.
In calculations like percentage error, the theoretical value is crucial as it provides the standard for comparison.
Hence, understanding the theoretical value is key in validating experimental results.

absolute difference

The absolute difference represents the magnitude of the numerical difference between the experimental and theoretical values, irrespective of the sign.
It's calculated by taking the absolute value of the subtraction of the two values involved.
In this exercise, it is computed as \[|1.25 \, \text{g/cm}^3 - 1.54 \, \text{g/cm}^3| = 0.29 \, \text{g/cm}^3\].
The absolute difference is used to ensure the difference is always a positive number, which simplifies communication and comparison.
This value is then divided by the theoretical value to find the proportion of how far off the experimental value is when compared to the theoretical value.
Understanding absolute difference is essential in evaluating the degree of deviation in experimental measurements.