Problem 129 - Chemistry The Molecular Science

Question

A high-quality analytical balance can weigh accurately to the nearest \(1.0 \times 10^{-4} \mathrm{~g}\). A sample of carbon weighed on this balance has a mass of \(1.000 \mathrm{mg}\). Calculate the number of carbon atoms in the sample. Given the precision of the balance, determine the maximum and minimum number of carbon atoms that could be in the sample.

Step-by-Step Solution

Answer

Between \(4.52 \times 10^{19}\) and \(5.51 \times 10^{19}\) carbon atoms.

Step 1: Convert Mass to Grams

First, convert the mass of the sample from milligrams to grams. We have a sample weighing \(1.000 \text{ mg}\). Since there are \(1000 \text{ mg}\) in \(1 \text{ g}\), the weight in grams is \(0.001000 \text{ g}\).

Step 2: Calculate Moles of Carbon

Use the molar mass of carbon (which is approximately \(12.01 \text{ g/mol}\)) to determine the number of moles. This is done by dividing the mass of the sample in grams by the molar mass.\[\text{Moles of carbon} = \frac{0.001000 \text{ g}}{12.01 \text{ g/mol}} \approx 8.33 \times 10^{-5} \text{ mol}\]

Step 3: Calculate Number of Atoms in the Sample

Use Avogadro's number, \(6.022 \times 10^{23} \text{ atoms/mol}\), to find the number of atoms in the sample. Multiply the moles of carbon by Avogadro's number:\[\text{Number of atoms} = 8.33 \times 10^{-5} \text{ mol} \times 6.022 \times 10^{23} \text{ atoms/mol} \approx 5.01 \times 10^{19} \text{ atoms}\]

Step 4: Determine Maximum and Minimum Mass

Given the precision of the balance (\(1.0 \times 10^{-4} \text{ g}\)), calculate the possible range of masses: the maximum mass is \(0.0010 + 0.0001 = 0.0011 \text{ g}\) and the minimum mass is \(0.0010 - 0.0001 = 0.0009 \text{ g}\).

Step 5: Calculate Maximum and Minimum Number of Atoms

Calculate the number of atoms for both the maximum and minimum mass scenarios: - For \(0.0011 \text{ g}\):\[\text{Moles of carbon} = \frac{0.0011 \text{ g}}{12.01 \text{ g/mol}} \approx 9.16 \times 10^{-5} \text{ mol}\]\[\text{Number of atoms} = 9.16 \times 10^{-5} \text{ mol} \times 6.022 \times 10^{23} \text{ atoms/mol} \approx 5.51 \times 10^{19} \text{ atoms}\]- For \(0.0009 \text{ g}\):\[\text{Moles of carbon} = \frac{0.0009 \text{ g}}{12.01 \text{ g/mol}} \approx 7.50 \times 10^{-5} \text{ mol}\]\[\text{Number of atoms} = 7.50 \times 10^{-5} \text{ mol} \times 6.022 \times 10^{23} \text{ atoms/mol} \approx 4.52 \times 10^{19} \text{ atoms}\]

Key Concepts

Analytical Balance

An analytical balance is an essential tool in the field of analytical chemistry. It allows for precise and accurate measurement of small masses, which is crucial in experiments where accuracy is significant.
The balance described in the exercise can measure accurately to the nearest 0.0001 grams. This level of precision is vital when calculating the number of atoms or molecules in a sample, as even tiny errors could lead to significant discrepancies in the results.
Analytical balances usually feature a draft shield to protect the sample from environmental factors, such as air currents, which could affect the balance reading. This makes them highly suitable for chemical laboratories where small quantities are frequently measured. They are also calibrated regularly to ensure accuracy.

• Key Feature: High precision to 0.0001 g.
• Usage: Ideal for measuring small sample masses.
• Important: Regular calibration and protection from air currents.

Molar Mass

Molar mass is a fundamental concept in chemistry that refers to the mass of one mole of a given substance, typically expressed in grams per mole (g/mol). For carbon, the commonly used molar mass is approximately 12.01 g/mol.
This value is crucial for converting between mass and moles, which is a step seen in the solution. By knowing the molar mass, chemists can determine the number of moles in a sample when given its mass.
Understanding molar mass is essential for stoichiometric calculations in chemical reactions, as it allows for the conversion of mass to moles, facilitating the calculation of reactants or products.

• Key Concept: Mass of one mole of a substance.
• Unit: Grams per mole (g/mol).
• Example: Molar Mass of Carbon = 12.01 g/mol.

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry, defined as the number of constituent particles, usually atoms or molecules, that are contained in one mole of a substance. The accepted value is approximately \(6.022 \times 10^{23}\) particles per mole.
This constant is crucial because it bridges the gap between macroscopic measurements and atomic scale measurements. For example, it allows scientists to estimate the number of atoms in a given sample of matter, as demonstrated in the exercise solution.
Using Avogadro's number, chemists can convert moles into particles, such as atoms or molecules, providing a tangible understanding of chemical quantities.

• Concept: Indicates particles per mole.
• Value: \(6.022 \times 10^{23}\) particles/mol.
• Application: Converts moles to atoms/molecules.

Precision and Accuracy

Precision and accuracy are two critical characteristics of measurements in analytical chemistry.
Precision refers to the consistency of repeated measurements, indicating how close the measurements are to each other. It does not necessarily imply accuracy but rather, reliability of the instrument or method.
Accuracy, on the other hand, measures how close a result is to the true or accepted value. A perfectly accurate measurement hits the exact or near-exact target.
In the given exercise, the precision of the analytical balance ensures the weight measurement is consistent (to the nearest 0.0001 g), while accuracy determines the closeness of this measurement to the actual mass of the carbon sample.

• Precision: Consistency in measurement.
• Accuracy: Closeness to true value.
• Importance: Both are critical for reliable data in experiments.