Answered

Ionic compounds can be made from a metallic ion and a nonmetallic ion. Metallic ions (M) are positive (cations). Nonmetallic ions (N) are negative (anions). $x$ stands for the number of metallic ions, and $y$ stands for the number of nonmetallic ions. For each equation given, find the values that correctly balance the charge.


  1. $x(+1 M) + (-2 N) = 0$

What does $x$ have to be?


  1. $(+2 M) + y(-1 N) = 0$

What does $y$ have to be?


  1. $x(+2 M) + y(-3 N) = 0$

What do $x$ and $y$ have to be? $x =$ [ ] and $y =$ [ ]

Verified Answer :

  • For equation 1, \(x(+1) + 1(-2) = 0\) simplifies to \(x = 2\).
  • For equation 2, 1(+2) + y(-1) = 0 simplifies to \(y = 2\).
  • For equation 3, \(x(+2) + y(-3) = 0\) simplifies to \(2x = 3y\), giving the smallest positive integer solutions \(x = 3\) and \(y = 2\).
  • The final results are \(x = \boxed{2}\) for the first, \(y = \boxed{2}\) for the second, and \(x = \boxed{3}, y = \boxed{2}\) for the third.

Explanation

  1. Problem Analysis and Data

The goal is to determine the number of metallic ions (\(x\)) and nonmetallic ions (\(y\)) needed to ensure that ionic compounds are electrically neutral. In each case, the sum of all charges contributed by the cations (positive metallic ions) and anions (negative nonmetallic ions) must equal zero. We are given three distinct scenarios representing different charge balances.

  1. Solving for Equation 1

The first scenario is given by the equation \(x(+1) + 1(-2) = 0\). This represents a metallic ion with a charge of \(+1\) and a nonmetallic ion with a charge of \(-2\). Expanding the equation, we have \(x - 2 = 0\). Adding 2 to both sides gives \(x = 2\). Thus, 2 metallic ions are required to balance the charge of one nonmetallic ion.

  1. Solving for Equation 2

The second scenario is \(1(+2) + y(-1) = 0\). Here, we have one metallic ion with a charge of \(+2\) and \(y\) nonmetallic ions, each with a charge of \(-1\). Expanding this, we get \(2 - y = 0\). Adding \(y\) to both sides, we find \(y = 2\). Therefore, 2 nonmetallic ions are required to balance one metallic ion.

  1. Solving for Equation 3

The third scenario is \(x(+2) + y(-3) = 0\). This involves metallic ions with a charge of \(+2\) and nonmetallic ions with a charge of \(-3\). The equation simplifies to \(2x - 3y = 0\), which implies \(2x = 3y\). To find the simplest positive integer ratio for \(x\) and \(y\), we observe that \(\frac{x}{y} = \frac{3}{2}\). The smallest positive integer solutions that satisfy this ratio are \(x = 3\) and \(y = 2\). We can verify this: \(3(+2) + 2(-3) = 6 - 6 = 0\), which satisfies the condition of electrical neutrality.

Examples

In chemistry, understanding ion stoichiometry is fundamental to predicting the formulas of ionic compounds. For instance, when forming common table salt (sodium chloride, NaCl), the sodium ion (\(Na^+\)) and chloride ion (\(Cl^-\)) combine in a 1:1 ratio. Conversely, for magnesium chloride (\(MgCl_2\)), the magnesium ion (\(Mg^{2+}\)) requires two chloride ions (\(Cl^-\)) to achieve neutrality. Calculating these ratios, as done here with \(x\) and \(y\), is exactly how chemists determine the chemical formulas of stable compounds found in nature.