Problem 101 - Chemistry: The Central Science

Question

Using modern analytical techniques, it is possible to detect sodium ions in concentrations as low as \(50 \mathrm{pg} / \mathrm{mL}\). What is this detection limit expressed in (a) molarity of \(\mathrm{Na}^{+}\), (b) \(\mathrm{Na}^{+}\) ions per cubic centimeter?

Step-by-Step Solution

Answer

The detection limit of sodium ions expressed in (a) molarity of \(\mathrm{Na}^{+}\) is \(2.17 \times 10^{-9}\,\mathrm{M}\), and (b) \(\mathrm{Na}^{+}\) ions per cubic centimeter is \(1.31 \times 10^{12}\,\mathrm{ions/cm^{3}}\).

Step 1: Convert the given concentration to grams per liter

We are given the concentration of sodium ions as \(50\,\mathrm{pg/mL}\). Let's convert it to \(\mathrm{g/L}\): $$ 50\,\mathrm{pg/mL} \times \frac{1\,\mathrm{g}}{10^{12}\,\mathrm{pg}} \times \frac{1000\,\mathrm{mL}}{1\,\mathrm{L}} = 5.0 \times 10^{-8}\,\mathrm{g/L} $$

Step 2: Calculate the molar mass of sodium ions

The molar mass of \(\mathrm{Na}^{+}\) is equal to the atomic mass of sodium (Na), which is approximately \(22.99\,\mathrm{g/mol}\).

Step 3: Convert the concentration to molarity

To convert the concentration to molarity, we will divide the mass by the molar mass of sodium ions and the volume in liters: $$ \mathrm{Molarity} = \frac{5.0 \times 10^{-8}\,\mathrm{g/L}}{22.99\,\mathrm{g/mol}} = 2.17 \times 10^{-9}\,\mathrm{mol/L} $$ Hence, the detection limit expressed in molarity of \(\mathrm{Na}^{+}\) is \(2.17 \times 10^{-9}\,\mathrm{M}\).

Step 4: Convert molarity to ions per cubic centimeter

To find the number of \(\mathrm{Na}^{+}\) ions per cubic centimeter, we will first use Avogadro's number \((6.022 \times 10^{23}\,\mathrm{ions/mol})\) to convert moles to ions: $$ \mathrm{Ions\,per\,Liter} = 2.17 \times 10^{-9}\,\mathrm{mol/L} \times 6.022 \times 10^{23}\,\mathrm{ions/mol} = 1.31 \times 10^{15}\,\mathrm{ions/L} $$ Next, we will convert the volume from liters to cubic centimeters: $$ \mathrm{Ions\,per\,cubic\,centimeter} = \frac{1.31 \times 10^{15}\,\mathrm{ions/L}}{1000\,\mathrm{cm^{3}/L}} = 1.31 \times 10^{12}\,\mathrm{ions/cm^{3}} $$ In conclusion, the detection limit of sodium ions expressed in (a) molarity of \(\mathrm{Na}^{+}\) is \(2.17 \times 10^{-9}\,\mathrm{M}\), and (b) \(\mathrm{Na}^{+}\) ions per cubic centimeter is \(1.31 \times 10^{12}\,\mathrm{ions/cm^{3}}\).

Key Concepts

Analytical Chemistry Techniques

Analytical chemistry techniques are essential for scientists to identify, quantify, and study chemical components within samples. Modern analytical tools can detect extremely low concentrations of substances, down to parts per trillion. This kind of sensitivity is crucial in fields like environmental monitoring, food safety, and pharmacology.

For example, spectroscopic methods such as mass spectrometry or atomic absorption can be used to determine the presence of sodium ions in a solution. These techniques often employ principles of light absorption, emission, and mass-to-charge ratios, providing great specificity and sensitivity in analysis. The ability to detect minute quantities, such as the 50 picograms per milliliter for sodium ions mentioned in the exercise, exemplifies the potency of these modern analytical techniques.

Molarity Calculation

The concept of molarity plays a central role in chemistry. It is defined as the number of moles of solute per liter of solution, given by the formula: \[\text{Molarity} = \frac{\text{Moles of solute}}{\text{Liters of solution}}\]. Calculating molarity is a fundamental skill that relates the molecular scale to the macroscopic properties of a chemical solution.

The process usually involves converting a given mass of solute to moles, using the molar mass of the substance. As exemplified in the solution to the exercise, the mass of sodium ions was first converted from picograms per milliliter to grams per liter, and then to moles using the molar mass of sodium. This demonstrates how molarity calculation can translate between different scales of measurement, from the mass of individual atoms to the concentration in a laboratory-sized sample.

Avogadro's Number

Avogadro's number is a cornerstone of chemistry that sets the link between the macroscopic and microscopic worlds. It is defined as the number of constituent particles, usually atoms or molecules, that are contained in one mole of a substance. Avogadro's number is approximately \(6.022 \times 10^{23}\) particles per mole.

This constant allows chemists to count particles by weighing them. When the exercise mentions converting molarity to ions per cubic centimeter, Avogadro's number is used to translate moles into actual numbers of sodium ions. It bridges the gap between a mole, which is an abstract concept used to count molecules in a bulk sample, and the more intuitive notion of individual particles -- in this case, sodium ions in a volume of one cubic centimeter. Understanding and manipulating Avogadro's number is essential when dealing with reactions and concentrations at the molecular level.