Problem 76 - Chemistry: The Central Science in SI Units, Global Edition

Question

(a) Determine the number of sodium ions in the chemical formula of albite, \(\mathrm{Na}_{x} \mathrm{AlSi}_{3} \mathrm{O}_{8} .(\mathbf{b})\) Determine the number of hydroxide ions in the chemical formula of tremolite, \(\mathrm{Ca}_{2} \mathrm{Mg}_{5}\left(\mathrm{Si}_{4} \mathrm{O}_{11}\right)_{2}(\mathrm{OH})_{x}\).

Step-by-Step Solution

Answer

(a) The number of sodium ions in the chemical formula of albite is 25, as we have \(\mathrm{Na}_{25} \mathrm{AlSi}_{3} \mathrm{O}_{8}\). (b) The number of hydroxide ions in the chemical formula of tremolite is 26, as we have \(\mathrm{Ca}_{2} \mathrm{Mg}_{5}\left(\mathrm{Si}_{4} \mathrm{O}_{11}\right)_{2}(\mathrm{OH})_{26}\).

Step 1: Identify the ions in the chemical formulas

In the first part of the exercise, we are given the chemical formula of albite: \(\mathrm{Na}_{x} \mathrm{AlSi}_{3} \mathrm{O}_{8}\). Here, the sodium ion is represented by Na and we are asked to find the value of x. In the second part of the exercise, we are given the chemical formula of tremolite: \(\mathrm{Ca}_{2} \mathrm{Mg}_{5}\left(\mathrm{Si}_{4}\mathrm{O}_{11}\right)_{2}(\mathrm{OH})_{x}\). Here, the hydroxide ion is represented by OH and we are asked to find the value of x.

Step 2: Determine the charge balance for the formulas

In order to determine the values of x, we must first understand that chemical formulas must be electrically neutral. This means that the total positive charge from the ions must equal the total negative charge from the ions. For albite: Na has a +1 charge, Al has a +3 charge, Si has a +4 charge while O has a -2 charge. For tremolite: Ca has a +2 charge, Mg has a +2 charge, Si has a +4 charge, O has a -2 charge, and OH has a -1 charge.

Step 3: Calculate the number of sodium ions in albite

In this step, we will calculate the positive and negative charges in the albite formula and find the number of sodium ions, keeping in mind that the formula must be electrically neutral. Positive charges: \(1x\) charge from \(\mathrm{Na}_{x}\) \(3\) charges from \(\mathrm{Al}\) Negative charges: \(12\) charges from \(\mathrm{Si}_{3}\) \(16\) charges from \(\mathrm{O}_{8}\) The total charge from positive ions must equal the total charge from negative ions: \(1x + 3 = 12 + 16\) Solving for x: \(x = 12 + 16 - 3 = 25\) Therefore, the number of sodium ions in the chemical formula of albite is 25.

Step 4: Calculate the number of hydroxide ions in tremolite

In this step, we will calculate the positive and negative charges in the tremolite formula and find the number of hydroxide ions, keeping in mind that the formula must be electrically neutral. Positive charges: \(4\) charges from \(\mathrm{Ca}_{2}\) \(10\) charges from \(\mathrm{Mg}_{5}\) \(16\) charges from \(\mathrm{Si}_{4}\times 2\) (because of the subscript 2 outside the brackets) Negative charges: \(22\times 2\) charges from \(\mathrm{O}_{11}\times 2\) (because of the subscript 2 outside the brackets) \(1x\) charge from \(\mathrm{(OH)}_{x}\) The total charge from positive ions must equal the total charge from negative ions: \(4 + 10 + 16 = 22\times 2 + 1x\) Solving for x: \(x = 4 + 10 + 16 - 22\times 2 = 30 - 44 = -14\) However, we have a negative value for x, which is not possible for the number of ions in the formula. Going back to the negative charge calculation, there is an error: the hydroxide ion (OH) has a -1 charge, not the oxygen atom. So the number of negative charges should be calculated as follows: Negative charges: \(2\times 2\) charges from \(\mathrm{O}_{2}\times 2\) (from the hydroxide ion, with subscript 2 outside the brackets) Now, the total charge from positive ions must equal the total charge from negative ions: \(4 + 10 + 16 = 2\times 2 + 1x\) Solving for x: \(x = 4 + 10 + 16 - 2\times 2 = 30 - 4 = 26\) Therefore, the number of hydroxide ions in the chemical formula of tremolite is 26.

Key Concepts

Ions

Ions are atoms or molecules that have gained or lost one or more electrons, which results in a net electrical charge. Understanding ions is crucial for grasping how chemical formulas represent compounds and how these compounds achieve charge neutrality.

There are two types of ions:

• Cations: Positively charged ions formed by losing electrons. For instance, the sodium ion (Na⁺) is a cation because it has lost one electron.
• Anions: Negatively charged ions formed by gaining electrons. Hydroxide ion (OH^-) is an example of an anion because it has gained an extra electron.

Balancing between cations and anions is key to understanding the overall electrical neutrality in chemical compounds.

Charge Balance

In a chemical compound, charge balance ensures the total positive charge equals the total negative charge. This equilibrium makes the compound electrically neutral.

To achieve charge balance:

• Add up all the positive charges from the cations.
• Add up all the negative charges from the anions.
• Ensure the sums are equal.

For example, in albite (\( \mathrm{Na}_{x} \mathrm{AlSi}_{3} \mathrm{O}_{8} \)), the charge balance is formulated as \(1x + 3 = 12 + 16\), where \(x\) represents the number of sodium ions. In this case, the combined positive charge must align with the negative charge to ensure neutrality, making \(x\) equal to 25.
Understanding and applying charge balance is fundamental when determining correct stoichiometric coefficients in chemical formulas.

Sodium Ions

Sodium ions (\(\mathrm{Na}^+\)) are crucial in many minerals, including albite. In the given chemical formula of albite, \(\mathrm{Na}_{x} \mathrm{AlSi}_{3} \mathrm{O}_{8}\), sodium ions carry a +1 charge, contributing to the compound's positive charge.

When determining the number of sodium ions needed:

• Identify sodium's charge as +1.
• Maintain charge neutrality by equating the total positive charge, including sodium, to the total negative charge.

By setting up and solving this equation (\(1x + 3 = 28\)), we find that there are 25 sodium ions, contributing a cumulative charge of +25.
This process highlights the need for strong understanding of ion charges when working with chemical formulas.

Hydroxide Ions

Hydroxide ions (\(\mathrm{OH}^-\)) are an essential component in many inorganic compounds like tremolite. These ions contain a single oxygen and hydrogen atom and possess a -1 charge. In the formula for tremolite, \(\mathrm{Ca}_{2} \mathrm{Mg}_{5}(\mathrm{Si}_{4} \mathrm{O}_{11})_{2}(\mathrm{OH})_{x}\), the hydroxide ion's negative charge plays a key role in balancing the positive charges from all other ions.

To find the number of hydroxide ions, facilitate charge neutrality by:

• Identifying the total negative charge contribution needed to balance positive charges.
• Using this to determine the count of hydroxide ions (x).

In the case provided, applying correct charge analysis results in \(x = 26\). This calculated value of 26 hydroxide ions ensures the compound's charge is completely balanced, highlighting the critical nature of ion charge understanding in such chemical equations.