Question

A sample of Ringer's solution contains the following concentrations (mEq/L) of cations: \(\mathrm{Na}^{+} 147, \mathrm{~K}^{+} 4\), and \(\mathrm{Ca}^{2+} 4\). If \(\mathrm{Cl}^{-}\) is the only anion in the solution, what is the \(\mathrm{Cl}^{-}\) concentration, in milliequivalents per liter?

Short answer

The \(\text{Cl}^-\) concentration is 159 mEq/L.

Step-by-step solution

Write down the given cation concentrations

List the concentrations of the cations given in the exercise: \(\text{Na}^+ 147 \text{ mEq/L}, \text{K}^+ 4 \text{ mEq/L}, \text{Ca}^{2+} 4 \text{ mEq/L}.\)

Calculate the total charge contributed by each cation

To maintain electrical neutrality, the total positive charge must equal the total negative charge. Calculate the total mEq/L for each cation: \(\text{Na}^+: 147 \text{ mEq/L}, \text{K}^+: 4 \text{ mEq/L}, \text{Ca}^{2+}: 2 \times 4 \text{ mEq/L} = 8 \text{ mEq/L}.\)

Sum the total positive charge

Add the mEq/L of each cation to find the total positive charge: \(\text{Total positive charge} = 147 \text{ mEq/L} + 4 \text{ mEq/L} + 8 \text{ mEq/L} = 159 \text{ mEq/L}.\)

Determine the Cl- concentration

Since the solution must be electrically neutral and \(\text{Cl}^-\) is the only anion, the concentration of \(\text{Cl}^-\) must equal the total positive charge: \(\text{Cl}^- \text{ concentration} = 159 \text{ mEq/L}.\)

Key concepts

Cation Concentration

Cations are positively charged ions. In the given problem, Ringer's solution contains cations like \(\text{Na}^+\), \(\text{K}^+\), and \(\text{Ca}^{2+}\). The concentration of each is provided using the unit milliequivalent per liter (mEq/L), which measures the amount of charge contributed by each ion in a liter of solution. Specifically, \(\text{Na}^+\) has a concentration of 147 mEq/L, \(\text{K}^+\) has 4 mEq/L, and \(\text{Ca}^{2+}\) has 4 mEq/L. To understand how these concentrations are used, it's important to remember:

• The concentration in mEq/L indicates the number of charges an ion contributes to a solution.
• A higher mEq/L means a higher charge contribution.

So, by knowing these concentrations, we can calculate the total positive charge from all cations in the solution.

Anion Concentration

Anions are negatively charged ions. In our problem, \(\text{Cl}^-\) is the sole anion present in Ringer's solution. Determining the concentration of \(\text{Cl}^-\) is crucial for maintaining the balance of charges in the solution. The concentration of \(\text{Cl}^-\) will be such that it exactly balances the total positive charge from the cations. Here's how the process works:

• We calculate the concentration of \(\text{Cl}^-\) by ensuring that the solution remains electrically neutral.
• Since \(\text{Cl}^-\) is the only anion, its total charge must match the total positive charge from all cations.

This step is crucial for verifying the electrical neutrality of the solution.

Electrical Neutrality

Electrical neutrality means the total charge in the solution must be zero. For this to happen in our Ringer's solution, the total positive charge from all cations \(\text{Na}^+\), \(\text{K}^+\), and \(\text{Ca}^{2+}\) must equal the total negative charge from the anion \(\text{Cl}^-\). To achieve this balance:

• Calculate the charge contributed by each ion.
  \text{\(\text{Na}^+: 147 \text{ mEq/L}, \text{K}^+: 4 \text{ mEq/L}, \text{Ca}^{2+}: 4 \text{ mEq/L}\)}
• Note that \(\text{Ca}^{2+}\) contributes double the charge compared to a single equivalent, so its total charge is \(4 \text{\times}2=8 \text{ mEq/L}\)
• The total positive charge is obtained by summing these values: \(147 \text{ mEq/L}+ 4 \text{ mEq/L}+ 8 \text{ mEq/L}=159 \text{ mEq/L}\)

Finally, the concentration of \(\text{Cl}^-\) must equal this total positive charge to keep the solution electrically neutral.

Milliequivalent Calculation

A milliequivalent (mEq) is a unit used to express the amount of a substance that will combine with or displace 1 milligram of hydrogen. It is particularly useful in chemistry and medicine for dealing with electrolytes. To calculate milliequivalents:

• Identify the charge of the ion and the amount present in the solution.
• For monovalent ions like \(\text{Na}^+\) and \(\text{K}^+\), their milliequivalent value is simply equal to their molar concentration multiplied by the charge.
• For divalent ions like \(\text{Ca}^{2+}\), multiply their molar concentration by 2 (the charge).

Hence, in our example:
\( \text{mEq/L for Ca^{2+} }= 4 \text{ mEq/L} \text{\times} 2 = 8 \text {mEq/L} \)

Summing up the contributions gives the total positive charge healthily maintained in the solution.