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Chapter 19: Problem 43

A human body contains approximately \(5 \mathrm{~L}\) of blood. A bipolar disorder patient receives \(2.0 \mathrm{~g}\) lithium carbonate as a daily dose. Calculate the concentration of \(\mathrm{Li}^{+}(\mathrm{mmol} / \mathrm{L})\) that this dosage provides.

Short Answer

Expert verified
The concentration of \(\text{Li}^+\) ions is \(10.8 \, \text{mmol/L}\).

Step by step solution

01

Identify the Molecular Weight of Lithium Carbonate

Lithium carbonate has the chemical formula \(\text{Li}_2\text{CO}_3\). Calculate the molecular weight: \(\text{Li} = 7 \text{ g/mol}\), \(\text{C} = 12 \text{ g/mol}\), \(\text{O} = 16 \text{ g/mol}\). Therefore, the molecular weight is \(2 \times 7 + 12 + 3 \times 16 = 74 \text{ g/mol}\).
02

Determine Moles of Lithium Carbonate Administered

Find the number of moles of lithium carbonate in the daily dose: \(\text{mass} = 2.0 \text{ g}\), \(\text{molar mass} = 74 \text{ g/mol}\). Use the formula \(\text{moles} = \frac{\text{mass}}{\text{molar mass}}\). Hence, \(2.0 \text{ g} \div 74 \text{ g/mol} = 0.027 \, \text{mol}\).
03

Calculate Moles of Lithium Ions

Since lithium carbonate contains 2 lithium ions per molecule, the moles of \(\text{Li}^+\) ions is double the moles of \(\text{Li}_2\text{CO}_3\): \(0.027 \, \text{mol} \times 2 = 0.054 \, \text{mol of Li}^+\).
04

Convert Moles to Millimoles

Convert moles of \(\text{Li}^+\) to millimoles: \(0.054 \,\text{mol} = 54 \,\text{mmol of Li}^+\) (as 1 mol = 1000 mmol).
05

Calculate Concentration of Lithium Ions

Calculate the concentration of \(\text{Li}^+\) in the blood: \(\text{conc} = \frac{\text{mmol of Li}^+}{\text{volume of blood in L}} = \frac{54 \,\text{mmol}}{5 \,\text{L}} = 10.8 \,\text{mmol/L}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Molecular Weight Calculation
Molecular weight, also known as molar mass, is a calculation of the total mass of one mole of a molecule. It is crucial in converting between the mass of a substance and the number of moles. To determine the molecular weight of a compound, you must know the atomic weights of each element involved. These are usually listed on the periodic table.
  • Step 1: Identify the chemical formula: For lithium carbonate, it's \(\text{Li}_2\text{CO}_3\).
  • Step 2: Determine atomic weights: Lithium (\(\text{Li}\)) is 7 g/mol, Carbon (\(\text{C}\)) is 12 g/mol, Oxygen (\(\text{O}\)) is 16 g/mol.
  • Step 3: Calculate the molecular weight: Add up the atomic weights in accordance with the formula.
    • For \(\text{Li}_2\text{CO}_3\): \(2 \times 7 + 12 + 3 \times 16 = 74\) g/mol.
This calculation gives you the molecular weight which is essential to know the quantity of a given compound in a specific reaction or dose.
Molarity Calculation
Molarity is a way to express the concentration of a solute in a solution. It is particularly useful in chemistry to quantify the amount of a substance dissolved in a liquid.Molarity is calculated as the number of moles of solute divided by the liters of solution. The formula is:\[\text{Molarity (M)} = \frac{\text{moles of solute}}{\text{liters of solution}}\]Here鈥檚 how you can apply it for lithium carbonate:
  • Once the moles of lithium carbonate are known (in this case, 0.027 moles), you convert it to the moles of lithium ions.
  • Since each molecule of lithium carbonate provides two lithium ions, multiply the moles by 2.
  • Next, convert moles to millimoles (1 mole equals 1000 millimoles) for ease of calculation with blood volume.
  • The molarity is then the number of millimoles of \(\text{Li}^{+}\) divided by the total blood volume in liters.
    • Using 5 liters of blood, the concentration is \(\frac{54 \, \text{mmol}}{5 \, \text{L}} = 10.8 \, \text{mmol/L}\).
This represents the concentration of lithium ions in the blood after the given dosage.
Lithium Carbonate in Medicine
Lithium carbonate is commonly used in medicine, particularly in the treatment of mood disorders such as bipolar disorder. It functions to stabilize mood swings, thus playing a crucial role in mental health care. Some significant features of lithium carbonate in medical use:
  • Lithium works by affecting the flow of sodium through nerve and muscle cells in the body, influencing mood regulation.
  • Patients are prescribed specific dosages, often based on weight and blood volume, to ensure therapeutic efficiency while avoiding toxicity.
  • Regular monitoring of blood lithium levels is crucial to prevent side effects which can be caused by exceeding therapeutic levels.
  • It is essential for dosages to be accurate, harnessing concepts like molecular weight and molarity to guide medication administration.
Through its direct influence on neurotransmitters, lithium carbonate becomes a powerful tool in treating conditions that involve mood imbalance.

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