Question

A \(2.747-\mathrm{g}\) sample of manganese metal is reacted with excess \(HCl\) gas to produce \(3.22 \mathrm{L} \mathrm{H}_{2}(g)\) at \(373 \mathrm{K}\) and 0.951 atm and a manganese chloride compound (MnCl_). What is the formula of the manganese chloride compound produced in the reaction?

Short answer

The formula of the manganese chloride compound produced in the reaction is MnCl.

Step-by-step solution

Find the number of moles of hydrogen gas

Using the ideal gas law (PV=nRT), we can find the number of moles of hydrogen gas (H₂) produced by rearranging the equation: n = PV/RT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant (0.0821 L atm/mol K), and T is the temperature. Given: P = 0.951 atm V = 3.22 L T = 373 K R = 0.0821 L atm/mol K Plug the values into the equation: n = (0.951 atm × 3.22 L) / (0.0821 L atm/mol K × 373 K) Calculate the number of moles of hydrogen gas (H₂): n = 0.124 moles of H₂

Find the moles of manganese reacted

Since manganese metal reacts with hydrochloric acid to produce hydrogen gas, the stoichiometry of the reaction can be represented as: Mn + x HCl → MnCl_x + x/2 H₂ Given that the moles of H₂ produced are 0.124, we can solve for the moles of manganese reacted: Moles of Mn = moles of H₂ * 2 / x 0.124 moles of H₂ * 2 = 0.248 moles Mn / x

Calculate the grams of manganese per mole

To find the grams of manganese per mole (molar mass), we can use the mass of manganese given in the problem statement (2.747 grams): molar mass of Mn = mass of Mn / moles of Mn molar mass of Mn = 2.747 g / (0.248 moles Mn / x)

Find the value of x

We know the molar mass of Mn is 54.94 g/mol. Plug this value into the equation from step 3 to solve for x: 54.94 g/mol = 2.747 g / (0.248 moles Mn / x) Rearrange the equation to find x: x = 2.747 g * (0.248 moles Mn / 54.94 g/mol) Calculate the value of x: x ≈ 1.25 As x must be a whole number, we can round it to the nearest integer: x ≈ 1

Write the formula of the manganese chloride compound

Now that we have the value of x, we can write the formula of the manganese chloride compound: MnCl_x MnCl_1 or simply MnCl The formula of the manganese chloride compound produced in the reaction is MnCl.

Key concepts

Stoichiometry

Understanding stoichiometry is crucial for solving problems in chemistry. It deals with the quantitative relationships of the substances as they participate in chemical reactions. Essentially, stoichiometry allows us to predict the amounts of products formed in a reaction based on the amounts of reactants used, or vice versa.

To apply stoichiometry, we start with a balanced chemical equation, which provides the mole ratios of reactants to products. These ratios serve as a conversion factor between different substances in a reaction. In our exercise with manganese chloride, the stoichiometry dictates that one mole of manganese reacts with a certain number of moles of hydrochloric acid to produce manganese chloride and hydrogen gas. By analyzing the mole ratio, we can determine the formula of the manganese chloride compound.

Ideal Gas Law

The ideal gas law is a fundamental equation in chemistry that relates the pressure, volume, temperature, and number of moles of a gas. The formula is expressed as
\( PV = nRT \)
where
\( P \) is pressure,
\( V \) is volume,
\( n \) is the number of moles,
\( R \) is the ideal gas constant, and
\( T \) is temperature in Kelvin. In the context of our exercise, we use the ideal gas law to determine the number of moles of hydrogen gas produced. This allows us to infer the number of moles of manganese that reacted, providing a crucial link in identifying the stoichiometry of the manganese chloride compound.

Mole Concept

The mole concept is a vital part of chemistry that provides a bridge between the atomic world and the macroscopic world we observe. One mole of any substance contains Avogadro's number of entities, whether they're atoms, molecules, or ions.

In the problem at hand, the mole concept is applied to relate the mass of manganese used to the number of moles of manganese. Knowing the moles allows us to understand the stoichiometric relationship in the chemical reaction. This not only tells us how much manganese reacted but also helps in calculating the molar mass of manganese and, subsequently, determining the stoichiometry of the final manganese chloride compound.

Molar Mass

Molar mass is the mass of one mole of a particular substance and is expressed in grams per mole (g/mol). It's a physical property that is often used in stoichiometry calculations to convert between the mass of a substance and the amount in moles. With molar mass, we can relate a substance's macroscopic mass to its microscopic atomic count.

For instance, in our problem, the molar mass of manganese is used to determine the moles of manganese that reacted with hydrochloric acid. Once we know the number of moles, we can figure out the mole ratio of manganese to chlorine in the compound, thus revealing the chemical formula for manganese chloride.