Problem 64 - Chemistry: The Central Science in SI Units, Global Edition

Question

A person suffering from hyponatremia has a sodium ion concentration in the blood of \(0.118 \mathrm{M}\) and a total blood volume of \(4.6 \mathrm{~L}\). What mass of sodium chloride would need to be added to the blood to bring the sodium ion concentration up to \(0.138 \mathrm{M}\), assuming no change in blood volume?

Step-by-Step Solution

Answer

Approximately \(5.375 \, \mathrm{g}\) of sodium chloride should be added to the blood to increase the sodium ion concentration to \(0.138 \, \mathrm{M}\) without any change in blood volume.

Step 1: Calculate the initial amount of sodium ions in the blood

To calculate the initial amount of sodium ions in the blood, we can use the formula: Initial amount of sodium ions = Initial concentration × Blood volume Let's plug in the given values: Initial concentration = 0.118 M Blood volume = 4.6 L Initial amount of sodium ions = \(0.118 \mathrm{M} \times 4.6 \mathrm{L}\) = \(0.5428 \,\text{moles of sodium ions}\)

Step 2: Calculate the final amount of sodium ions required for the desired concentration

To calculate the final amount of sodium ions for a concentration of 0.138 M, we can use the same formula as in step 1: Final amount of sodium ions = Final concentration × Blood volume We have: Final concentration = 0.138 M Blood volume = 4.6 L Final amount of sodium ions = \(0.138 \,\mathrm{M} \times 4.6 \,\mathrm{L}\) = \(0.6348 \,\text{moles of sodium ions}\)

Step 3: Calculate the net amount of sodium ions needed

Now, let's calculate the difference between the initial and final amount of sodium ions: Net amount of sodium ions needed = Final amount of sodium ions - Initial amount of sodium ions Net amount of sodium ions needed = \(0.6348 \,\text{moles} - 0.5428 \,\text{moles}\) = \(0.092 \,\text{moles of sodium ions}\)

Step 4: Calculate the mass of sodium chloride needed

Since sodium chloride (NaCl) dissociates into sodium ions (Na+) and chloride ions (Cl-) in a 1:1 ratio, the moles of sodium ions required will be equal to the moles of NaCl. To determine the mass of sodium chloride needed, we can use the formula: Mass of NaCl = Moles of NaCl × Molar mass of NaCl The molar mass of NaCl = 58.44 g/mol (22.99 g/mol for Na + 35.45 g/mol for Cl) Now let's calculate the mass of NaCl: Mass of NaCl = \(0.092 \,\mathrm{moles} \times 58.44 \,\frac{\mathrm{g}}{\mathrm{mol}}\) = \(5.375 \,\mathrm{g}\) So, approximately 5.375 grams of sodium chloride should be added to the blood to increase the sodium ion concentration to 0.138 M without any change in blood volume.

Key Concepts

Sodium Ion Concentration

Sodium ion concentration in the blood is an essential factor in maintaining proper bodily functions. Our body needs a precise balance of sodium ions to ensure that nerve and muscle cells function correctly. Having too little sodium in the blood can lead to a condition called hyponatremia, which can cause various symptoms such as headaches, confusion, and even seizures.
To measure sodium ion concentration, we use molarity (M), which is the moles of solute per liter of solution. In the exercise, a person suffering from hyponatremia has a reduced sodium ion concentration at 0.118 M compared to what's considered normal or necessary. To rectify this, the concentration needs to be increased to 0.138 M.
By understanding sodium ion concentration, we can appreciate how our body maintains balance and how external intervention, like adding sodium chloride, might be necessary to correct imbalances.

Blood Volume

Blood volume is the amount of blood circulating in your body, and it's crucial for ensuring that all cells receive the necessary nutrients and oxygen. For adults, the average blood volume is usually around 4.5 to 5.5 liters.
In this exercise, the blood volume is set at 4.6 liters, which is needed for calculating the amount of sodium ions currently present and required in the system. The blood volume remains constant throughout the problem, making it a fixed variable that simplifies calculations.
By keeping the blood volume constant (in this case 4.6 L), it allows us to focus on changing the concentration of sodium ions rather than worrying about changes in the overall volume. This is a typical approach when calculating concentration changes in chemistry.

Molar Mass of Sodium Chloride

The molar mass of a compound, like sodium chloride (NaCl), is the mass of one mole of its molecules and is expressed in grams per mole (g/mol). For NaCl, this molar mass is the sum of the atomic masses of sodium (22.99 g/mol) and chlorine (35.45 g/mol), resulting in 58.44 g/mol.

• This molar mass helps in determining how much of a substance is needed for chemical reactions, such as increasing sodium ion concentration.
• The calculations in the exercise show us that 0.092 moles of sodium ions need to be added.

Since the dissolution of NaCl into sodium and chloride ions is a 1:1 ratio, the same amount of moles of NaCl are needed to supply 0.092 moles of sodium ions. Knowing the molar mass also allows us to calculate the exact mass of NaCl required, which in this problem is approximately 5.375 grams. This mass is crucial for practitioners who need to correct hyponatremia effectively.