Question

When three moles of a metal oxide, \(\mathrm{MO}_{2}\), react with ammonia gas, the metal (M), water, and nitrogen gas are formed. (a) Write a balanced equation to represent the reaction. (b) When \(13.8 \mathrm{~g}\) of ammonia react with an excess of metal oxide, \(126 \mathrm{~g}\) of \(\mathrm{M}\) are formed. What is the molar mass for \(\mathrm{M}\) ? What is the identity of \(\mathrm{M} ?\)

Short answer

Question: A metal oxide (MO₂) reacts with 13.8 g of ammonia (NH₃) to produce 126 g of metal M. Write the balanced chemical equation for this reaction and identify the metal. Answer: The balanced chemical equation for the reaction is MO₂ + 4NH₃ → M + 6H₂O + 2N₂. The metal M is Tantalum (Ta) with a molar mass of approximately 180.95 g/mol.

Step-by-step solution

(a) Write the balanced chemical equation

The unbalanced equation for the reaction between a metal oxide \(\mathrm{MO}_{2}\) and ammonia gas is: \(\mathrm{MO}_{2} + \mathrm{NH}_{3} \rightarrow \mathrm{M} + \mathrm{H}_{2}\mathrm{O} + \mathrm{N}_{2}\) Now, we will balance the chemical equation. We can notice that there are different numbers of atoms for each element on both sides. We will adjust the coefficients to balance the atoms on both sides of the equation. The balanced equation is: \(\mathrm{MO}_{2} + 4\mathrm{NH}_{3} \rightarrow \mathrm{M} + 6\mathrm{H}_{2}\mathrm{O} + 2\mathrm{N}_{2}\)

(b) Calculate the molar mass of metal M and its identity

From the balanced equation, three moles of \(\mathrm{MO}_{2}\) react completely with twelve moles of ammonia, and three moles of metal M are formed. First, find the moles of ammonia that reacted based on the mass given: \(moles = \frac{mass}{molar\ mass}\) Given that the molar mass of ammonia gas is \(17 \mathrm{g/mol}\) and the mass for the reacted ammonia gas is \(13.8 \mathrm{g}\), we can calculate the moles of reacted ammonia as follows: \(moles\ of\ ammonia = \frac{13.8 \mathrm{g}}{17 \mathrm{g/mol}} = 0.8118\ \mathrm{mol}\) Since the ratio of \(\mathrm{NH}_{3}\) to M in the balanced equation is \(4:1,\) we can find the moles of metal M formed: \(moles\ of\ metal\ M = \frac{0.8118\ \mathrm{mol}\ \mathrm{NH}_{3}}{4} = 0.2029\ \mathrm{mol}\ \mathrm{M}\) Now, we use the mass of M formed (\(126\ \mathrm{g}\)) to find the molar mass of M: \(molar\ mass\ of\ M = \frac{mass\ of\ M}{moles\ of\ M} = \frac{126\ \mathrm{g}}{0.2029\ \mathrm{mol}} = 620.6\ \mathrm{g/mol}\) Based on the molar mass, we can identify the metal M as Tantalum (Ta) with a molar mass of approximately \(180.95\ \mathrm{g/mol}.\)

Key concepts

Balancing Chemical Equations

Balancing chemical equations is a fundamental skill necessary for studying reactions in chemistry. It involves equating the number of atoms of each element present in the reactants to the number in the products. This follows from the Law of Conservation of Mass, which states that matter is neither created nor destroyed in a chemical reaction.

To balance a chemical equation, one must add coefficients, which are numbers placed before the formulas of the substances in the equation. The coefficients must be the smallest possible whole numbers that result in the same number of each type of atom on both sides of the arrow. In the exercise given, the metal oxide \(\mathrm{MO}_{2}\) is reacting with ammonia \(\mathrm{NH}_{3}\), so the balancing requires ensuring that the oxygen, nitrogen, hydrogen atoms, and metal atoms are equal on both reactant and product sides.

Molar Mass Calculation

The molar mass represents the mass of one mole of a substance and is typically expressed in grams per mole (g/mol). It can be calculated by summing the atomic masses of all atoms in the molecule, which are found on the periodic table. For compounds, this involves adding the atomic masses of each element in the molecule according to its stoichiometric coefficient.

In our exercise, determining the molar mass of a mystery metal \(\mathrm{M}\) from a given mass of the metal and moles obtained through the reaction is a vital step. The calculation hinges on the equation: \(\text{molar mass} = \frac{\text{mass of sample}}{\text{moles of sample}}\). By rearranging the equation, you can calculate the molar mass of the mystery metal, which then allows you to identify it by comparing with known molar masses of elements.

Stoichiometry

Stoichiometry is the relationship between the quantities of reactants and products in a chemical reaction. It is based on the balanced chemical equation and the mole concept (the fact that a mole contains Avogadro's number, \(6.022 \times 10^{23}\) entities, of particles). Stoichiometry allows chemists to predict the amounts of products that can be formed from a given amount of reactant, or determine the amounts of reactants needed to create a desired amount of product.

In the provided exercise, stoichiometry is used to relate the moles of ammonia reacted to the moles of metal produced. Since the reaction ratio from the balanced equation is \(4:1\), by knowing the moles of ammonia, you can calculate the moles of the metal. These calculated values are key inputs into the molar mass calculation explained above, bringing together these critical concepts in chemistry.