Problem 79 - Chemistry and Chemical Reactivity

Question

A Most standard analytical balances can measure accurately to the nearest \(0.0001 \mathrm{g}\). Assume you have weighed out a \(2.0000-\mathrm{g}\) sample of carbon. How many atoms are in this sample? Assuming the indicated accuracy of the measurement, what is the largest number of atoms that can be present in the sample?

Step-by-Step Solution

Answer

Approx. 1.002 x 10^23 atoms; maximum is 1.0022 x 10^23 atoms.

Step 1: Determine molar mass of carbon

The element carbon has a molar mass of approximately 12.01 g/mol. This value represents the mass of one mole (or Avogadro's number of) carbon atoms.

Step 2: Calculate moles of carbon in the sample

Using the given sample mass of 2.0000 g, we can calculate the number of moles of carbon using the formula: \[ ext{moles of carbon} = \frac{2.0000 \, \text{g}}{12.01 \, \text{g/mol}}\approx 0.1665 \, \text{mol}\]

Step 3: Convert moles to atoms

To find the number of atoms, multiply the moles of carbon by Avogadro's number, which is approximately \(6.022 \times 10^{23}\) atoms/mol:\[\text{Number of atoms} = 0.1665 \, \text{mol} \times 6.022 \times 10^{23} \, \text{atoms/mol} \approx 1.002 \times 10^{23} \, \text{atoms}\]

Step 4: Determine the range for the largest number of atoms

Given the measurement accuracy \(\pm 0.0001 \, \text{g}\), the maximum mass would be 2.0001 g. Calculate the maximum moles:\[\text{Max moles} = \frac{2.0001 \, \text{g}}{12.01 \, \text{g/mol}} \approx 0.166511 \text{ mol}\]Multiply the maximum moles by Avogadro's number:\[\text{Max number of atoms} = 0.166511 \, \text{mol} \times 6.022 \times 10^{23} \, \text{atoms/mol} \approx 1.0022 \times 10^{23} \, \text{atoms}\]

Key Concepts

Understanding the Role of Analytical Balance

An analytical balance is an essential piece of laboratory equipment used to measure mass with high precision. It's specifically designed to measure small sample masses and can provide measurements accurate to within a tiny fraction of a gram, often to four decimal places. For example, if you measure a 2.0000 g carbon sample, it means the balance can accurately distinguish that mass within \(\pm 0.0001 \, \text{g}\). This high degree of precision is critical across disciplines like chemistry and pharmacology.

Some important features of an analytical balance include:

• High sensitivity: It detects even minute changes in mass.
• Environmental controls: They are often enclosed to minimize air currents and static electricity that can affect weighing.
• Calibration: Balances should regularly be calibrated with standard weights to ensure accuracy.

When dealing with chemical samples, achieving precise measurements ensures the success of experiments or the accurate preparation of solutions. Without such precision, the margins for error increase significantly.

Explaining Avogadro's Number

Avogadro's number is a central concept in chemistry, providing a bridge between the atomic scale and the macroscopic scale that we can observe. This number, approximately \(6.022 \times 10^{23}\), represents the quantity of entities, usually atoms or molecules, present in one mole of a substance. In the context of our exercise, it helps convert the moles of carbon into a comprehensible count of atoms.

Why is Avogadro's number important?

• It allows chemists to count particles at the atomic level. Instead of saying 'a few atoms,' they can precisely define how many atom-sized units are present.
• A mole as a unit embraces Avogadro's number, bringing the subatomic world to a human scale.
• It facilitates stoichiometry, which ensures accurate chemical equations and reactions.

When we say that a carbon sample contains about 1.002 \times 10^{23}\ atoms, Avogadro's number is the transformative figure that makes this possible. By working with this constant, chemists simplify the calculation of molar masses and chemical formulas.

Importance of Measurement Accuracy

Measurement accuracy is crucial in scientific experiments and calculations as it determines the reliability of the results. Accuracy refers to how close a measured value is to the actual value, while precision pertains to the repeatability of measurements. The exercise highlights the importance of precise weighing, which affects further calculations and the final outcome.

In our exercise, considering the measurement accuracy \(\pm 0.0001 \, \text{g}\), this uncertainty impacts both the computed moles of carbon and the resulting number of atoms. Here’s why measurement accuracy is vital:

• Reliable results: Accurate measurements ensure that calculations are representative of reality.
• Consistent quality: Especially in manufacturing or pharmaceuticals, precise measurements ensure product consistency.
• Error minimization: Identifying measurement inaccuracies can reduce systematic error in experiments.

For the carbon sample, small differences in mass (even as tiny as 0.0001 g) can reflect significant changes in the count of atoms, showing why precision and accuracy are indispensable in science.