Question

A \(10^{6}\) liter tank of seawater contains \(16,600 \mathrm{~kg}\) of chlorine (Cl^ ), \(9200 \mathrm{~kg}\) of sodium \(\left(\mathrm{Na}^{+}\right)\) and \(1180 \mathrm{~kg}\) of magnesium \(\left(\mathrm{Mg}^{++}\right) .\) Calculate the molarity of each. Is all the charge accounted for?

Short answer

The molarity of Chlorine (Cl^-) is \( \frac{16600\, kg}{35.453\, g/mol} * \frac{1000\,g}{1\, kg} * \frac{1}{10^6\,L} \approx 0.469\,M \). The molarity of Sodium (Na^+) is \( \frac{9200\, kg}{22.990\, g/mol} * \frac{1000\,g}{1\, kg} * \frac{1}{10^6\,L} \approx 0.400\,M \). The molarity of Magnesium (Mg^2+) is \( \frac{1180\, kg}{24.305\, g/mol} * \frac{1000\,g}{1\, kg} * \frac{1}{10^6\,L} \approx 0.049\,M \). The total positive charge is equal to the total negative charge, so all the charge is accounted for.

Step-by-step solution

Convert mass to moles

First, we need to convert the mass (in kg) of each element to moles using the molar masses of each element (in g/mol). The molar masses are as follows: Cl^- = 35.453 g/mol, Na^+ = 22.990 g/mol, Mg^2+ = 24.305 g/mol. Number of moles for Chlorine (Cl^-): \( \frac{16600\, kg}{35.453\, g/mol} * \frac{1000\,g}{1\, kg} \) Number of moles for Sodium (Na^+): \( \frac{9200\, kg}{22.990\, g/mol} * \frac{1000\,g}{1\, kg} \) Number of moles for Magnesium (Mg^2+): \( \frac{1180\, kg}{24.305\, g/mol} * \frac{1000\,g}{1\, kg} \)

Calculate the molarity of each element

Now, we will calculate the molarity of each element. Molarity (M) is defined as the number of moles of solute per liter of solution. Molarity of Chlorine (Cl^-): \( \frac{moles\, of\, Cl^-}{10^6\,L} \) Molarity of Sodium (Na^+): \( \frac{moles\, of\, Na^+}{10^6\,L} \) Molarity of Magnesium (Mg^2+): \( \frac{moles\, of\, Mg^2+}{10^6\,L} \)

Check if all the charge is accounted for

To determine whether all the charge is accounted for, we will calculate the total positive charge (from Na^+ and Mg^2+) and the total negative charge (from Cl^-). Total positive charge: (Number of moles of Na^+ x +1) + (Number of moles of Mg^2+ x +2) Total negative charge: (Number of moles of Cl^- x -1) If the total positive charge is equal to the total negative charge, then all the charge is accounted for.

Key concepts

Seawater Chemistry

Seawater is a complex solution that contains a mixture of various dissolved salts. These salts are composed primarily of ions such as chloride \(Cl^-\), sodium \(Na^+\), and magnesium \(Mg^{2+}\). The chemistry of seawater is essential for understanding natural processes and environmental conditions in marine ecosystems. Chloride and sodium are the dominant ions, while magnesium also plays a significant role. Understanding the concentration of these ions is crucial for maintaining the ionic balance in marine life. In this context, the concentration is often measured in terms of molarity, which represents the number of moles of solute per liter of seawater. This measurement helps in understanding the reactivity and interactions of seawater with other substances, vital for marine biology and industrial applications.

Molar Mass

Molar mass is a critical concept in chemistry that refers to the mass of one mole of a given substance. It is expressed in grams per mole (g/mol) and is used to convert the mass of a substance to the number of moles, which is a fundamental quantity for stoichiometric calculations.

• For chlorine \(Cl^-\), the molar mass is 35.453 g/mol.
• Sodium \(Na^+\) has a molar mass of 22.990 g/mol.
• Magnesium \(Mg^{2+}\) has a molar mass of 24.305 g/mol.

Knowing the molar masses of these ions allows us to convert their given masses from kilograms to moles, using the formula:\[ \text{Number of moles} = \frac{\text{mass (kg)} \times 1000}{\text{molar mass (g/mol)}} \]This conversion is necessary to calculate the molarity of each ion in seawater precisely.

Ionic Charge Balance

Ionic charge balance is essential for the stability of any solution, including seawater. The balance involves ensuring that the total positive and negative charges in the solution are equal. This equilibrium is governed by the principle of electroneutrality, which states that in any chemical species, the sum of positive charges must equal the sum of negative charges.
To verify the charge balance in the exercise:

• Total positive charge from \(Na^+\) and \(Mg^{2+}\) is calculated by multiplying the moles of \(Na^+\) by +1 and the moles of \(Mg^{2+}\) by +2.
• Total negative charge is obtained by multiplying the moles of \(Cl^-\) by -1.

If the sum of the total positive charge equals the total negative charge, the ionic balance is maintained, indicating that all charges are accounted for in the solution.

Stoichiometry

Stoichiometry is the branch of chemistry that deals with the quantitative relationships between reactants and products in a chemical reaction. In the context of seawater chemistry, stoichiometry helps us understand how to calculate the amounts of different ions in a solution. This involves using moles and molarity, which allows us to predict the quantities of elements needed or produced during chemical reactions.
The exercise required us to convert the given masses of chloride, sodium, and magnesium into moles and then calculate their molarities. This process is a classic example of stoichiometry in action. By doing this, we ensure that we can accurately determine the chemical makeup and interactions within seawater, which is essential for processes such as desalination and marine ecosystem management.