Question

Lanthanum was reacted with hydrogen in a given experiment to produce the nonstoichiometric compound ${\mathrm{LaH}}_{2.90}$ Assuming that the compound contains ${\mathrm{H}}^{-},{\mathrm{La}}^{2+},$ and ${\mathrm{La}}^{3+},$ calculate the fractions of ${\mathrm{La}}^{2+}$ and ${\mathrm{La}}^{3+}$ present.

Short answer

The fractions of ${\mathrm{La}}^{2+}$ and ${\mathrm{La}}^{3+}$ ions present in the ${\mathrm{LaH}}_{2.90}$ compound are: Fraction of ${\mathrm{La}}^{2+}$: x = 0.10 Fraction of ${\mathrm{La}}^{3+}$: y = 0.90 This means that 10% of the $\mathrm{La}$ ions in the compound are in the ${\mathrm{La}}^{2+}$ form and 90% are in the ${\mathrm{La}}^{3+}$ form.

Step-by-step solution

1. Write the general formula for the compound

Considering that the compound contains ${\mathrm{H}}^{-}$, ${\mathrm{La}}^{2+}$, and ${\mathrm{La}}^{3+}$, the general formula for the compound can be represented as follows: $${\mathrm{La}}_x^{2+}{\mathrm{La}}_y^{3+}{\mathrm{H}}_z^{-}$$

2. Write the charge balance equation

The charge balance equation states that the total positive charge must be equal to the total negative charge. In this case, the charges from ${\mathrm{La}}^{2+}$ and ${\mathrm{La}}^{3+}$ ions must be equal to the charge from the ${\mathrm{H}}^{-}$ ions. So, we can write: $$2x+3y=z$$

3. Write the composition constraint equation

The composition constraint equation relates the stoichiometry of the compound to the number of atoms of each ion in the whole compound. In this problem, the stoichiometry of the compound is ${\mathrm{LaH}}_{2.90}$; therefore, we can write: $$x+y=1$$

4. Solve the system of equations

Now let's solve the system of equations composed of the charge balance equation and the composition constraint equation to find the fractions of ${\mathrm{La}}^{2+}$ and ${\mathrm{La}}^{3+}$ ions. First, let's express the second equation in terms of x and substitute it into the first equation: From the second equation, we get: $$x=1-y$$ Substituting in the first equation, we get: $$2(1-y)+3y=2.90$$ Solving this equation, we find y: $$3y-2y=2.90-2$$ $$y=0.90$$ Now, substituting the value of y in the expression for x, we get: $$x=1-y=1-0.90$$ $$x=0.10$$

5. Interpret the results

The fractions of ${\mathrm{La}}^{2+}$ and ${\mathrm{La}}^{3+}$ ions present in the ${\mathrm{LaH}}_{2.90}$ compound are: Fraction of ${\mathrm{La}}^{2+}$: x = 0.10 Fraction of ${\mathrm{La}}^{3+}$: y = 0.90 This means that 10% of the $\mathrm{La}$ ions in the compound are in the ${\mathrm{La}}^{2+}$ form and 90% are in the ${\mathrm{La}}^{3+}$ form.

Key concepts

Lanthanide Chemistry

The lanthanide series consists of 15 elements, from lanthanum ($La$) to lutetium ($Lu$), which sit at the bottom of the periodic table. These elements are known for having similar chemical properties, often forming complex compounds due to their partially filled f-orbitals.
Lanthanides are unique in that they generally exhibit +3 oxidation states in compounds, making them highly reactive with hydrogen to form hydrides like $La{H}_{2.90}$.
Lanthanides are crucial in various technologies, from electronics to optics, owing to their magnetic, luminescent, and catalytic properties.

• Lanthanoid contraction refers to the gradual decrease in atomic and ionic radii.
• Lanthanides are often used in alloys to improve strength and workability.

Oxidation States

Understanding oxidation states in chemistry is crucial as they help define how electrons are transferred in reactions.
In lanthanide compounds like $La{H}_{2.90}$, the elements can exist in different oxidation states, crucial for calculating the charge balance.
The common oxidation state for lanthanum is +3, yet in certain nonstoichiometric compounds, it can also exist as +2 due to variations in bonding environments.

• Oxidation states determine how an element interacts within a compound.
• Multiple oxidation states can exist in a compound, affecting its chemical properties.

Charge Balance

Charge balance in chemical compounds is a pivotal concept. It ensures that the sum of positive charges equals the sum of negative charges, maintaining the compound's neutrality.
For compounds like $La{H}_{2.90}$, the positive charges contributed by $L{a}^{2+}$ and $L{a}^{3+}$ ions must balance the negative charges contributed by hydride ($H^{-}$) ions.
This balance is achieved through equations that relate charges to stoichiometry, such as the equation $2x+3y=z$.

• Charge balance is essential for predicting the stability of a compound.
• Ensures that total charge within the compound is neutral.

Stoichiometry

Stoichiometry refers to the quantitative relationships between the elements in chemical reactions and compounds. It's vital for understanding how elements combine to form compounds, such as $La{H}_{2.90}$.
In such nonstoichiometric compounds, the conventional atom ratios aren't whole numbers, requiring careful calculation to understand proportions.
For $La{H}_{2.90}$, the stoichiometry is reflected in the molecular formula, necessitating equations like $x+y=1$ to calculate component ratios.

• Stoichiometry helps in calculating reactant and product masses in reactions.
• Derived from balanced chemical equations to determine reactant-product relationships.