Question

What is the molarity of \(\mathrm{Na}^{+}\) in a solution of \(\mathrm{NaCl}\) whose salinity is \(5.6\) if the solution has a density of \(1.03 \mathrm{~g} / \mathrm{mL}\) ?

Short answer

The molarity of \(\mathrm{Na}^+\) ions in a solution of \(\mathrm{NaCl}\) with a salinity of \(5.6\) and a density of \(1.03 \ \mathrm{g/mL}\) is \(98.83 \ \mathrm{M}\).

Step-by-step solution

1. Determine the mass of the solution

Since we have the density and volume of the solution, we can calculate the mass of the solution. We know that density is equal to mass divided by volume. Therefore, mass (\(m_{solution}\)) can be calculated as: \[m_{solution} = \text{density} \times \text{volume}\] We have the density, but we need to assume a volume for the solution in order to calculate its mass. We can choose any volume, but for simplicity let's choose a \(1 \ \mathrm{L}\) solution. This means the volume of the solution is \(1 \ \mathrm{L} = 1000 \ \mathrm{mL}\). Now, we can find the mass of the \(1 \ \mathrm{L}\) solution: \[m_{solution} = (1.03 \ \mathrm{g/mL}) \times (1000 \ \mathrm{mL}) = 1030 \ \mathrm{g}\]

2. Determine the mass of NaCl in the solution

Now that we have the mass of the solution, we can calculate the mass of \(\mathrm{NaCl}\) in it, using salinity. We know that salinity is the mass of solute (\(\mathrm{NaCl}\)) divided by the mass of the solution: \(salinity = \frac{m_{\mathrm{NaCl}}}{m_{solution}}\) So, we can find the mass of \(\mathrm{NaCl}\) in the solution by rearranging the equation: \[m_{\mathrm{NaCl}} = \text{salinity} \times m_{solution}\] Now, we can find the mass of \(\mathrm{NaCl}\) in the \(1\ \mathrm{L}\) solution: \[m_{\mathrm{NaCl}} = (5.6) \times (1030 \ \mathrm{g}) = 5768 \ \mathrm{g}\]

3. Convert mass of NaCl to moles

To calculate the molarity of \(\mathrm{Na}^+\) ions, we need to know the number of moles of \(\mathrm{NaCl}\) in the solution. We can do this using the molar mass of \(\mathrm{NaCl}\): \(M_{\mathrm{NaCl}} = 22.99\, \mathrm{g/mol} + 35.45\, \mathrm{g/mol} = 58.44\, \mathrm{g/mol}\). Now, we will convert the mass of \(\mathrm{NaCl}\) into moles using the molar mass: \[\text{moles of } \mathrm{NaCl} = \frac{m_{\mathrm{NaCl}}}{M_{\mathrm{NaCl}}} = \frac{5768 \ \mathrm{g}}{58.44 \ \mathrm{g/mol}} = 98.83 \ \mathrm{moles}\]

4. Calculate the molarity of Na+ ions

Now that we know the number of moles of \(\mathrm{NaCl}\), we can find the molarity of \(\mathrm{Na}^+\) ions. Note that in a \(\mathrm{NaCl}\) solution, one mole of \(\mathrm{Nacl}\) will dissociate into one mole of \(\mathrm{Na}^+\) ions and one mole of \(\mathrm{Cl}^-\) ions. To find the molarity of \(\mathrm{Na}^+\) ions, we divide the number of moles of \(\mathrm{NaCl}\) by the volume of the solution in liters: \[\text{Molarity of }\mathrm{Na}^+ = \frac{\text{moles of } \mathrm{NaCl}}{\text{volume of solution (in L)}} = \frac{98.83 \ \mathrm{moles}}{1 \ \mathrm{L}} = 98.83 \ \mathrm{M}\] So, the molarity of \(\mathrm{Na}^+\) ions in the given solution is \(98.83 \ \mathrm{M}\).

Key concepts

Density

Density is a measure of how much mass is contained in a given volume. It helps us understand how compact or concentrated a substance is. The formula for density is given by:

• \[\text{Density} = \frac{\text{mass}}{\text{volume}}\]

In the context of solutions, knowing the density allows us to determine the mass if we know the volume. This is crucial for calculations involving concentrations and reactions. To find the mass of a solution with a given density, simply multiply the density by the volume:

• \[m_{\text{solution}} = \text{Density} \times \text{Volume}\]

For example, given a density of \(1.03\ g/mL\) and a volume of \(1000\ mL\), the mass is \(1030\ g\). This serves as a foundational step in calculating properties like molarity.

Salinity

Salinity refers to the concentration of salt in a solution. It's often expressed as the ratio of the mass of the solute (such as NaCl) to the mass of the entire solution. Salinity is crucial for determining how much salt is present and can be calculated using:

• \[\text{Salinity} = \frac{m_{\text{NaCl}}}{m_{\text{solution}}}\]

To find the mass of NaCl in a solution, rearrange this formula:

• \[m_{\text{NaCl}} = \text{Salinity} \times m_{\text{solution}}\]

For our exercise, with a salinity of 5.6 and the solution mass of \(1030\ g\), the NaCl mass amounts to \(5768\ g\). Understanding salinity helps determine the exact amount of salt, which is vital for further conversions into chemical quantities like moles.

Moles of NaCl

The concept of moles is essential in chemistry for expressing amounts of a substance. It links the mass of a compound to the number of particles it contains through the molar mass. The molar mass of NaCl, a common table salt, is \(58.44\ g/mol\), calculated as the sum of the atomic masses of sodium (22.99 g/mol) and chlorine (35.45 g/mol).

• To convert the mass of NaCl to moles, use:
• \[\text{Moles of } \mathrm{NaCl} = \frac{m_{\mathrm{NaCl}}}{M_{\mathrm{NaCl}}}\]

For instance, with a mass of \(5768\ g\) of NaCl, the number of moles calculated is approximately \(98.83\). This measurement is crucial because it allows us to calculate other important properties, such as molarity.

Dissociation of NaCl

When NaCl is dissolved in water, it dissociates into ions. Each molecule separates into one sodium ion \((\mathrm{Na}^+)\) and one chloride ion \((\mathrm{Cl}^-)\). This dissociation is vital for understanding solutions and reactions, especially when calculating ion concentrations.

• The dissociation equation is:
• \[\mathrm{NaCl} \rightarrow \mathrm{Na}^+ + \mathrm{Cl}^-\]

For every mole of NaCl dissolved, you get one mole of \(\mathrm{Na}^+\). Therefore, knowing the amount of NaCl directly gives you the molarity of \(\mathrm{Na}^+\) ions.

• Molarity of \(\mathrm{Na}^+\) is calculated as:
• \[\text{Molarity of } \mathrm{Na}^+ = \frac{\text{moles of } \mathrm{NaCl}}{\text{volume of solution (in L)}}\]

Thus, with \(98.83\) moles of NaCl in 1 liter, the molarity is \(98.83\ M\), indicating the concentration of sodium ions in the solution.