Problem 3 - Chemical Principles in the Laboratory

Question

The student then emptied the flask and dried it once again. To the empty flask she added pieces of a metal until the flask was about three-fourths full. She weighed the stoppered flask and its metal contents and found that the mass was \(306.150 \mathrm{g}\). She then filled the flask with water, stoppered it, and obtained a total mass of 309.827 g for the flask, stopper, metal, and water. Find the density of the metal.a. To find the density of the metal we need to know its mass and volume. We can casily obtain its mass by the method of differences: Mass of metal = ____ \(g\)- _____ \(g=\) ____ \(g\) b. To determine the volume of metal, we note that the volume of the flask must equal the volume of the metal plus the volume of water in the filled flask containing both metal and water. If we can find the volume of water, we can obtain the volume of metal by the method of differences. To obtain the volume of the water we first calculate its mass: Mass of water \(=\) mass of (flask \(+\) stopper \(+\) metal \(+\) water) \(-\) mass of (flask \(+\) stopper \(+\) metal) Mass of water \(=\) ____ \(g-\) _____ \(g=\) ____ \(g\) The volume of water is found from its density, as in \(1(b)\) Volume of water \(=\frac{\text { mass of water }}{\text { density of water }}=\frac{g}{g / c m^{3}}=\) \(\mathrm{cm}^{3}\) c. From the volume of the water, we calculate the volume of metal: Volume of metal \(=\) (volume of flask) \(-\) (volume of water) Volume of metal \(=\)_____ \(\mathrm{cm}^{3}-\)______\(\mathrm{cm}^{3}=\) _____ \(\mathrm{cm}^{3}\) From the mass and volume of metal, find the density, using the equation in \(1(b):\) Density of metal \(=\)_____ \(g / c m^{3}\) Now go back to Question 1 and check to see that you have reported the proper number of significant figures in cach of the results you calculated in this assignment. Use the rules on significant figures as given in your chemistry text or Appendix V.

Step-by-Step Solution

Answer

Use mass difference to find metal mass, subtract water mass for volume, then calculate density.

Step 1: Calculate Mass of Metal

Using the method of differences, we find the mass of the metal by subtracting the mass of the empty flask and stopper from the mass of the flask, stopper, and metal. Mass of metal: \( 306.150 \, \text{g} - \text{mass of empty flask and stopper} \).

Step 2: Calculate Mass of Water

Find the mass of the water by subtracting the mass of the flask, stopper, and metal from the mass of the flask, stopper, metal, and water.Mass of water: \( 309.827 \, \text{g} - 306.150 \, \text{g} = 3.677 \, \text{g} \).

Step 3: Calculate Volume of Water

The density of water is typically \( 1 \, \text{g/cm}^3 \). Thus, the volume of the water can be calculated as follows:Volume of water: \( \frac{3.677 \, \text{g}}{1 \, \text{g/cm}^3} = 3.677 \, \text{cm}^3 \).

Step 4: Calculate Volume of Metal

Determine the volume of the metal by finding the total volume of the flask and subtracting the volume of the water. Assuming the total volume of the flask is the sum of the volume of the metal and water: Volume of metal = Volume of flask - Volume of water.

Step 5: Calculate Density of Metal

Using the mass of the metal from Step 1 and the volume of the metal calculated in Step 4, find the density of the metal using the formula:Density of metal \( = \frac{\text{Mass of metal}}{\text{Volume of metal}} \).

Step 6: Check Significant Figures

Ensure that all calculated results are reported with the correct number of significant figures, based on the precision of the initial measurements.

Key Concepts

Significant Figures

In scientific measurements, using significant figures is crucial for accuracy and precision. Significant figures include all the digits in a number that are known with certainty and one final digit, which is somewhat uncertain or estimated. They give us a sense of how precise a measurement is.
Examples include measuring instruments like a balance or a graduated cylinder. Both have precision limits due to their design. This affects the number of significant figures you can reliably record.
When performing calculations, like division or subtraction, it's important to adjust the number of significant figures in the result. This ensures your answer isn't misleading by implying more precision than the original measurements allow.

• For addition and subtraction, the result should match the number with the least decimal places.
• For multiplication and division, use the number with the least total significant figures in any of the measurements.

These rules are essential in reporting your final answer with the proper precision.

Volume Measurement

Accurate volume measurement is a cornerstone of density calculations. Volume is the amount of space an object occupies, measured in cubic centimeters (\( ext{cm}^3\)) for liquid and solid volumes in scientific contexts.
A common method for measuring volume involves water displacement, where you can determine the volume of an object by seeing how much water it displaces. For instance, by adding an object to a graduated cylinder partly filled with water, you can calculate the object's volume by the change in water level.
In the density problem described, the volume of the metal is found indirectly by measuring the total volume of the flask when filled with both the metal and water and then subtracting the volume of water. Accurate initial measurement of water volume is critical as any error can propagate and skew results in subsequent calculations.

Mass Measurement

Mass measurement is another key step in calculating density, as mass provides a measure of the number of particles in a given object. Typically, mass is measured in grams using a balance.
It's always important to calibrate the measuring equipment correctly and use appropriate techniques to avoid errors. This includes ensuring the balance zeroes correctly before each use and using the minimum force needed to stabilize it.
In this exercise, the mass of both the metal and water is determined using the method of differences. This involves subtracting the mass of the flask when only partially filled from the mass when fully filled.
This subtraction helps eliminate any confounding factors and isolates the mass of the desired component, whether it's the metal or the water, contributing to accurate final calculations.