Density
Density is a measure of how much mass is contained in a given volume. It's calculated using the formula: \[Density = \frac{Mass}{Volume}\] In this exercise, the density of the solution is given as \ 1.025 \mathrm{~g/cm^3}, with a mass of \ 1.00 \mathrm{~kg} \ or \ 1000 \mathrm{~g}. To find the volume of the solution, we divide the mass by the density: \[Volume = \frac{1000 \mathrm{~g}}{1.025 \mathrm{~g/cm^3}} = 975.61 \mathrm{~cm^3} \approx 0.9756 \mathrm{~L}\]. This volume is crucial for subsequent calculations, including determining ion molarity.
Molar Mass
Molar mass refers to the mass of one mole of a given substance, expressed in grams per mole (g/mol). It is a critical component in converting the mass of a compound to moles. For instance, in the exercise, the molar mass of \ NaCl is \frac{58.44 \text{g/mol}} \ and the mass is \ 26.5 \mathrm{~g} \. To find moles of \ NaCl, we use: \[ \text{Moles of NaCl} = \frac{26.5 \mathrm{~g}}{58.44 \mathrm{~g/mol}} = 0.4539 \mathrm{~mol} \]. This conversion is performed for each compound in the solution, setting up critical data for determining ion concentrations.
Ion Concentration
Ion concentration, or molarity, measures the number of moles of an ion in a solution per liter of solution. It's determined using the moles of each ion and the solution's volume. For example, the moles of \ \text{Na}^+ \ from \ NaCl, NaHCO_{3} \ and \ NaBr \ are summed and divided by the volume: \[ \text{[Na}^{+}] = \frac{0.4539 + 0.00375 + 0.00095}{0.9756} = 0.468 \mathrm{~M} \]. This concept is applied to calculate the molarity of each ion in the solution, including \ Cl^{-}, Mg^{2+}, Ca^{2+}, K^{+}, SO_{4}^{2-}, HCO_{3}^{-}, and Br^{-}.
Alkali Metal Ions
Alkali metals are elements in group 1 of the periodic table, which include \ Li, Na, K, Rb, Cs, \ and \ Fr \. In this exercise, \ Na \ and \ K \ are the alkali metal ions present. Their total molarity is the sum of their individual molarities: \[ \text{[Total~Alkali~Metal~Ions]} = 0.468 + 0.0145 = 0.4825 \mathrm{~M} \]. This tells us how concentrated these alkali metal ions are in the solution.
Alkaline Earth Metal Ions
Alkaline earth metals are found in group 2 of the periodic table, and include \ Be, Mg, Ca, Sr, Ba, \ and \ Ra \. Here, \ \text{Mg}^{2+} \ and \ \text{Ca}^{2+} \ ions are present. Their total molarity is calculated by summing their individual molarities: \[ \text{[Total~Alkaline~Earth~Metal~Ions]} = 0.0546 + 0.0111 = 0.0657 \mathrm{~M} \]. Understanding this helps in determining the concentration of these metal ions in the solution.
Anions
Anions are negatively charged ions. In this context, the key anions are \ Cl^{-}, SO_{4}^{2-}, HCO_{3}^{-}, \ and \ Br^{-} \. Each anion's concentration is calculated by finding the moles from each source compound and dividing by the solution volume. The total molarity of anions is the sum of their concentrations: \[ \text{[Total~Anions]} = 0.587 + 0.0285 + 0.00384 + 0.000974 = 0.6203 \mathrm{~M} \]. This figure represents the overall concentration of anions in the solution.
Solution Volume
The solution volume is an essential factor for calculating molarity. It's derived from the mass and density of the solution as: \[ \text{Volume} = \frac{1000 \mathrm{~g}}{1.025 \mathrm{~g/cm^3}} = 0.9756 \mathrm{~L} \]. This volume is used in all subsequent molarity calculations, forming the denominator for converting moles of each ion into molarity. Ensuring accurate volume measurements is crucial for precise chemical calculations.
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