Problem 100 - 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) the number of \(\mathrm{Na}^{+}\)ions per cubic centimeter of solution, (c) the mass of sodium per \(1000 \mathrm{~L}\) of solution?

Step-by-Step Solution

Answer

(a) The molarity of Na⁺ ions is 2.17 × \(10^{-9}\) M. (b) The number of Na⁺ ions per cubic centimeter of the solution is 1.31 × \(10^{9}\) ions/cm³. (c) The mass of sodium per 1000 L of solution is 5 × \(10^{-5}\) g.

Step 1: Determine the molarity of Na⁺ ions

: To find the molarity, we must convert the mass of sodium ions (50 pg/mL) into moles per liter. To do this, we need to know the molar mass of sodium ions (Na), which is 22.99 g/mol. First, we'll convert 50 pg/mL to g/L by considering the conversion factors: 1 g = \(10^{12}\) pg and 1 L = 1000 mL Molarity = moles/L

Step 2: Calculate Molarity

: First, we will use the given mass in pg/mL and convert it to g/L, and then use the molar mass of Na to find the moles of Na. (50 pg/mL) × (1 g/ \(10^{12}\) pg) × (1 L/1000 mL) = 5 × \(10^{-8}\) g/L Now, we'll use the molar mass of Na to find moles of Na. (5 × \(10^{-8}\) g/L) × (1 mol / 22.99 g) = 2.17 × \(10^{-9}\) mol/L So, the molarity of Na⁺ ions is 2.17 × \(10^{-9}\) M.

Step 3: Calculate the number of Na⁺ ions per cubic centimeter of solution

: To calculate the number of Na⁺ ions per cm³ of solution, we'll use Avogadro's number (6.022 × \(10^{23}\) ions/mol) to convert the molarity of Na⁺ ions into the number of Na⁺ ions and then convert it into cm³. Number of Na⁺ ions = (2.17 × \(10^{-9}\) mol/L) × (6.022 × \(10^{23}\) ions/mol) × (1 L/ \(10^3\) cm³) = 1.31 × \(10^{9}\) ions/cm³ Therefore, the number of Na⁺ ions per cubic centimeter of the solution is 1.31 × \(10^{9}\) ions/cm³.

Step 4: Calculate the mass of sodium per 1000 L of solution

: First, we will find the mass of sodium in 1 L of solution using its concentration in g/L. Mass of sodium in 1 L of solution = (5 × \(10^{-8}\) g/L) × (1 L) = 5 × \(10^{-8}\) g Now, we'll calculate the mass of sodium in 1000 L of solution. Mass of sodium in 1000 L of solution = (5 × \(10^{-8}\) g) × (1000 L) = 5 × \(10^{-5}\) g Therefore, the mass of sodium per 1000 L of solution is 5 × \(10^{-5}\) g. In conclusion: (a) The molarity of Na⁺ ions is 2.17 × \(10^{-9}\) M. (b) The number of Na⁺ ions per cubic centimeter of the solution is 1.31 × \(10^{9}\) ions/cm³. (c) The mass of sodium per 1000 L of solution is 5 × \(10^{-5}\) g.

Key Concepts

Understanding Molarity

Molarity is a term used in chemistry to describe the concentration of a substance within a solution. It expresses the amount of moles of a solute present in one liter of solution. Formally, it's represented by the symbol 'M' and calculated with the formula:
\( Molarity = \frac{{moles \ of \ solute}}{{volume \ of \ solution \ in \ liters}} \).

For instance, a molarity of 2.17 × \(10^{-9}\) M means that there are 2.17 × \(10^{-9}\) moles of the solute (in our exercise, sodium ions) in every liter of solution. This concept is particularly useful when dealing with reactions in solution because it allows for straightforward stoichiometric calculations.

Avogadro's Number and Its Significance

Avogadro's number is a constant that represents the number of particles found in one mole of a substance, named after the Italian scientist Amedeo Avogadro. It is approximately 6.022 × \(10^{23}\) particles per mole. This number helps us to link the macroscopic world, which we can measure, to the microscopic world of atoms and molecules.

By using Avogadro's number, chemists can determine the actual amount of particles in a given molarity of a solution. For instance, having a concentration of sodium ions expressed as a molarity can be used to calculate the exact number of ions in a particular volume of that solution. Understanding this connection is crucial for accurately describing chemical concentrations and carrying out various calculations in analytical chemistry.

Molar Mass: A Connecting Point Between Moles and Mass

Molar mass bridges the physical world's mass with the chemist’s mole concept. It defines the mass of one mole of a substance and is typically expressed in grams per mole (g/mol). Every element's molar mass is found on the periodic table as its atomic weight.

Each element has a unique molar mass—for sodium, it's 22.99 g/mol. This value is pivotal for converting between the mass of a substance and the amount of moles it comprises. To convert from a small mass like picograms (pg) to moles, especially in concentration calculations, the molar mass serves as an essential conversion factor.

Detecting Sodium Ions with Precision

Sodium ion detection is an important measure in many fields including medicine, environmental science, and food industry. Sodium ion concentration can be determined using various analytical techniques, one being the high sensitivity of modern instruments that permits the detection of minuscule amounts of sodium ions, down to picograms per milliliter levels.

Knowledge of molarity and Avogadro's number allows us to work out not just the concentration in mol/L, but also the actual quantity of sodium ions present in a given volume of solution. This capacity to determine low levels of sodium ions is essential for studies that require precise manipulation and measurement of sodium, such as understanding the balance of electrolytes in the human body or monitoring sodium levels in environmental samples.