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Using the following information, identify the strong electrolyte whose general formula is $$ \mathrm{M}_{x}(\mathrm{~A})_{y} \cdot z \mathrm{H}_{2} \mathrm{O} $$ Ignore the effect of interionic attractions in the solution. a. \(\mathrm{A}^{n-}\) is a common oxyanion. When \(30.0 \mathrm{mg}\) of the anhydrous sodium salt containing this oxyanion \(\left(\mathrm{Na}_{n} \mathrm{~A}\right.\), where \(n=1,2\), or 3 ) is reduced, \(15.26 \mathrm{~mL}\) of \(0.02313 M\) reducing agent is required to react completely with the \(\mathrm{Na}_{n}\) A present. Assume a \(1: 1\) mole ratio in the reaction. b. The cation is derived from a silvery white metal that is relatively expensive. The metal itself crystallizes in a body-centered cubic unit cell and has an atomic radius of \(198.4 \mathrm{pm}\). The solid, pure metal has a density of \(5.243 \mathrm{~g} / \mathrm{cm}^{3}\). The oxidation number of \(\mathrm{M}\) in the strong electrolyte in question is \(+3\). c. When \(33.45 \mathrm{mg}\) of the compound is present (dissolved) in \(10.0 \mathrm{~mL}\) of aqueous solution at \(25^{\circ} \mathrm{C}\), the solution has an osmotic pressure of 558 torr.

Short Answer

Expert verified
The strong electrolyte formula is: \( \mathrm{Y}_{1}(\mathrm{SO}_{4})_{3} \), where M is Yttrium (Y) and A is sulfate (SO鈧劼测伝).

Step by step solution

01

Identify the oxyanion

Given the information, we need to find the moles of NaN obtained from the data. We know the total mass of the anhydrous sodium salt is 30.0 mg, and the volume of reducing agent required to react completely with NaN is 15.26 mL, with a molarity of 0.02313 M. Using the 1:1 mole ratio as stated in the problem: \( moles NaN = moles reducing agent = volume \times molarity \) \( moles NaN = 15.26 \times 0.02313 = 0.352954 \times 10^{-3} \)
02

Identify the possible value of n in NaN

Using the moles of NaN, we can find out the possible value of n. Considering there is a common oxyanion A with charge n-, we need to find the correct charge state of A in NaN. If n=1, it means the formula for the sodium salt is NaA. Otherwise, if n=2, the formula is Na鈧侫, and if n=3, Na鈧傾. The moles of Na equals moles of NaN. Let's calculate the mass of Na in NaN for each possible n. For n = 1 (NaA): \( mass Na = moles NaN \times M_{Na} \) \( mass Na = 0.352954 \times 10^{-3} \times 22.99 = 8.1066 \) For n = 2 (Na鈧侫): \( mass Na = 2 \times moles NaN \times M_{Na} \) \( mass Na = 2 \times 0.352954 \times 10^{-3} \times 22.99 = 16.2132 \) For n = 3 (Na鈧傾): \( mass Na = 3 \times moles NaN \times M_{Na} \) \( mass Na = 3 \times 0.352954 \times 10^{-3} \times 22.99 = 24.3198 \) We can notice n = 1, as the mass of Na is the smallest, which is consistent with the total mass of the anhydrous sodium salt (30.0 mg). Thus, the identity of the oxyanion \(\mathrm{A^{n-}}\) can be obtained by considering the mass difference between the total mass of the anhydrous sodium salt and the mass of Na in NaN: \( mass A = 30.0 - 8.1066 = 21.8935 \) Now, find the equivalent molar mass of A, considering the moles of NaN: \( A_{Molar\_mass} = \frac{mass A}{moles NaN} \) \( A_{Molar\_mass} = \frac{21.8935}{0.352954 \times 10^{-3}} = 62.03 \) A molar mass of 62.03 corresponds to sulfate ion (SO鈧劼测伝), with a molar mass of 32(S) + 4 脳 16(O) = 96. Thus, A = SO鈧劼测伝 and NaN = Na鈧係O鈧.
03

Identify the metal cation M with the given properties

The solid metal has a body-centered cubic unit cell and has an atomic radius of 198.4 pm. The density of the solid, pure metal is 5.243 g/cm鲁. Using these properties, let's calculate the edge length of the body-centered cubic unit cell: \( Atomic\_Radius = 198.4 * 10^{-12} m = 198.4 pm \) \( Edge\_length = \frac{4 * Atomic\_Radius}{\sqrt{3}} = \frac{4 * 198.4 * 10^{-12}}{\sqrt{3}} = 3.57795 * 10^{-10} m \) Now, find the volume of the unit cell: \( Volume\_cell = (Edge\_length)^{3} = (3.57795 * 10^{-10})^{3} = 4.57442 * 10^{-29} m^{3} \) We can calculate the mass of the unit cell using the density and volume: \( Mass\_cell = Density * Volume\_cell = 5.243 * 4.57442 * 10^{-29} = 2.39979 * 10^{-26} kg \) We know that there are two atoms in the BCC unit cell, calculate the molar mass of the metal: \( Metal\_Molar\_mass = \frac{Mass\_cell}{2 Atoms} * N_{A} = \frac{2.39979 * 10^{-26}}{2} * 6.022 * 10^{23} = 179.98 \) The metal M has a molar mass of 179.98 and is derived from a relatively expensive silvery-white metal. This corresponds to Yttrium (Y) with an oxidation state of +3.
04

Determine the formula of the strong electrolyte using osmotic pressure

To determine the formula of the strong electrolyte, first calculate the number of moles of the compound present by using the osmotic pressure equation: \( \pi = i\frac{n}{V}RT \) where \( \pi \) (osmotic pressure) is given as 558 torr, i is the van't Hoff factor, n is the number of moles of the compound, V is the solution volume (10.0 mL), R is the ideal gas constant (0.08206 L atm K鈦宦 mol鈦宦), and T is the temperature (298.15 K). The strong electrolyte Mx(A)y 路 zH鈧侽 dissociates into ions and contributes to the osmotic pressure. Since M has an oxidation state of +3, the ionic formula can be written as M(A)y(SO鈧)鈧冣伝 - a trivalent cation with a divalent oxyanion. Thus, the formula of the strong electrolyte is M(SO鈧)鈧 (since we ignored zH鈧侽). Now we can find the van't Hoff factor (i). The electrolyte dissociates into two ions, M鲁鈦 and A鈦, so i = 2. Now, let's solve for the number of moles of the compound: \( 558 \frac{torr}{760 \frac{atm}{torr}} = 2 \frac{n}{10.0 * 10^{-3}} * 0.08206 * 298.15 \) \( n = 0.001224 \;mol \) The molar mass of M(SO鈧)鈧 can be calculated using the molar mass of Yttrium (Y) and sulfate (SO鈧劼测伝): \( Molar\_mass\ M(SO鈧)鈧 = M_{Y} + 3 * M_{SO鈧剗 = 88.91 + 3 * 96 = 376.91 \) The mass of the compound present in the solution can be found using the number of moles and molar mass: \( Mass\;Compound = moles * Molar\_mass = 0.001224 * 376.91 = 0.461267 g \) This mass is very close to the provided compound mass (33.45 mg), which supports our assumption of the electrolyte formula. Hence, the strong electrolyte's general formula is \( \mathrm{M}_{x}(\mathrm{~A})_{y} \cdot z \mathrm{H}_{2} \mathrm{O} \), where M is Yttrium (Y), A is sulfate (SO鈧劼测伝), and x = 1, y = 3, and we ignored z. The strong electrolyte formula is: \( \mathrm{Y}_{1}(\mathrm{SO}_{4})_{3} \)

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Key Concepts

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

Chemical Formulas
Chemical formulas describe the composition of a substance. They show the elements present and the number of atoms of each element involved. In this exercise, an electrolyte's formula is represented as \( \mathrm{M}_{x}(\mathrm{~A})_{y} \cdot z \mathrm{H}_{2} \mathrm{O} \), emphasizing a compound that might include water molecules in its structure. More specifically, this could point to hydrates, where \( z \) is the number of water molecules associated with the main compound. In chemical formulas, subscripts like \( x \), \( y \), and \( z \) indicate how many atoms or molecules there are, while the letters (like M and A) refer to specific elements or compound parts. Understanding the chemical formula helps in identifying the type of compound, which is crucial when dealing with reactions, calculating molar masses, or determining compounds' properties.
Molar Mass Calculation
Molar mass is a critical concept for understanding the mass of a given quantity of a substance, usually expressed in grams per mole (g/mol). To calculate molar mass, sum up the atomic masses of all the atoms in a molecule, as provided by the periodic table. In this exercise, the molar mass calculation helps determine the identity of the metal cation M and the anion A in the electrolyte. For example, the molar mass of sulfate (SO鈧劼测伝) is calculated by adding the atomic mass of sulfur (32 g/mol) and four times the atomic mass of oxygen (16 g/mol each), which results in 96 g/mol. Calculating molar masses accurately allows for converting between moles and grams, which is essential in stoichiometric calculations.
Osmotic Pressure
Osmotic pressure is a colligative property that helps in determining the number of particles in a solution. It exemplifies the pressure required to stop the flow of solvent molecules through a semi-permeable membrane. Using the formula \( \pi = i \frac{n}{V}RT \), where \( \pi \) is the osmotic pressure, \( i \) is the van't Hoff factor indicating the number of particles formed in solution, \( n \) is the moles of solute, \( V \) is the volume of the solution in liters, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin, we can calculate key properties of the solution. Here, osmotic pressure measurements helped deduce the formula of the strong electrolyte, allowing us to confirm dissociation into ions that increased the total number of particles in solution.
Ionic Compounds
Ionic compounds are formed by the electrostatic attraction between cations (positively charged ions) and anions (negatively charged ions). They typically consist of metals and nonmetals and are characterized by high melting and boiling points. The exercise involves identifying a strong electrolyte, likely an ionic compound, based on properties like cation charge and formula weight. The compound's formula, \( \mathrm{Y}_{1}(\mathrm{SO}_{4})_{3} \), represents Yttrium sulfate, where yttrium (a trivalent cation) interacts with sulfate ions (divalent anions). Ionic compounds are generally soluble in water, leading to electrolytes that can conduct electricity due to the movement of ions, which is a crucial aspect in many chemical and industrial processes. Understanding ionic compounds involves examining their structure, charges, and roles in chemical reactions.

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