Question

A \(1.00-\mathrm{g}\) sample of a metal \(\mathrm{X}\) (that is known to form \(\mathrm{X}^{2+}\) ions) was added to a \(0.100 \mathrm{~L}\) of \(0.500 \mathrm{M}\) \(\mathrm{H}_{2} \mathrm{SO}_{4}\). After all the metal had reacted, the remaining acid required \(0.0334 \mathrm{~L}\) of \(0.500 \mathrm{M} \mathrm{NaOH}\) solution for neutralization. Calculate the molar mass of the metal and identify the element.

Short answer

The molar mass of the metal is approximately \(30.031\, g/mol\). And the metal is identified as Magnesium (Mg).

Step-by-step solution

Calculate the amount of NaOH used in the neutralization

The molarity of NaOH is 0.500 M and the volume of NaOH used is 0.0334 L. The moles of NaOH used can be calculated using the equation Moles = Molarity x Volume. Therefore, Moles of NaOH = \(0.500 M \times 0.0334 L = 0.0167\, moles\)

Determine the moles of H2SO4 that reacted with the metal

Since NaOH and H2SO4 chemically react in a 1:1 ratio, the moles of H2SO4 neutralized by NaOH is also 0.0167 moles. The initial moles of H2SO4 can be calculated from its molarity and volume, which is \(0.500 M \times 0.100 L = 0.050\, moles\). Therefore, the moles of H2SO4 that reacted with the metal would be \(0.050 moles - 0.0167 moles = 0.0333\, moles\)

Calculate the moles of metal X

The reaction between X and H2SO4 would follow the pattern: \(X + H_2SO_4 -> XSO_4 + 2H\) . Therefore there is a 1:1 molar ratio between metal X and H2SO4. This means the moles of metal X that reacted will be the same as the moles of H2SO4 that reacted which is 0.0333 moles.

Calculate the molar mass and identify metal X

The molar mass of the metal can be found by using the equation molar mass = mass / moles. The mass of the metal is given as 1.00 g and we have calculated the moles as 0.0333. So, the molar mass = \(1.00 g / 0.0333 \,moles = 30.031\, g/mol\). Based on the periodic table, the metal with a molar mass close to 30.031 g/mol is Magnesium (Mg) which has a molar mass of 24.31 g/mol.

Key concepts

Understanding Neutralization Reactions

A neutralization reaction occurs when an acid and a base react to form water and a salt. This type of chemical reaction is important in our daily lives and in numerous scientific processes. In our example, the strong acid \( \mathrm{H}_2\mathrm{SO}_4 \) reacts with the base \( \mathrm{NaOH} \) in equal proportions, meaning they neutralize each other well.
Neutralization is a dynamic process that depends heavily on the properties and amounts of the acid and base involved. This means if you know the molarity and the volume of one reactant, you can find out how much of the other reactant is needed to completely neutralize it.
When working through a neutralization problem, keep in mind the following points:

• Always pay attention to the ratio in which the reactants neutralize each other. Often it will be a 1:1 relationship, like with \(\mathrm{NaOH}\) and \(\mathrm{H}_2\mathrm{SO}_4\).
• Understand that the remaining acid or base present after the reaction can tell you more about what happened with the initial reactants.
• Think about how the reaction's stoichiometry will impact the rest of the equation's calculations.

Concept of Moles and Molarity

Moles and molarity are fundamental concepts in chemistry that quantify substances and reactions. Understanding these is crucial when calculating how reactions occur, especially in solutions.
In the given problem, molarity refers to the number of moles of a solute per liter of solution. Here, you are told the molarity and volume of the \(\mathrm{H}_2\mathrm{SO}_4\) solution and of the \(\mathrm{NaOH}\) used in the neutralization.
Here are the core ideas you need to grasp:

• Moles: These represent the number of molecules and allow chemists to work with the macroscopic quantities of substances. You calculate moles using the formula: \[\text{Moles} = \text{Molarity} \times \text{Volume (in liters)}\]
• Molarity: This helps in understanding the concentration of a solution and is indicated by the letter \(M\). It is central in determining how substances in chemical reactions are mixed.
• Volume: Often given in liters, the volume of a solution, in consideration with its molarity, helps to find out how many moles of a solution are present, which can be crucial to determining reaction yields.

Basics of Stoichiometry

Stoichiometry is a section of chemistry that involves calculating the relative quantities of reactants and products in chemical reactions. This concept helps predict how much product will form from a given amount of reactants.
In the current exercise, we see stoichiometry come into play when determining the quantity of \(\mathrm{H}_2\mathrm{SO}_4\) that reacted with the metal. By knowing the ratios of the reactants and measuring the leftover reactants, we can deduce how much acid participated in the reaction.
Here’s what to focus on:

• Chemical Ratios: The molar ratio between reactants is key. For example, the metal \(X\), reacts with \(\mathrm{H}_2\mathrm{SO}_4\) in a 1:1 molar relationship, allowing us to match moles of the metal to moles of \(\mathrm{H}_2\mathrm{SO}_4\), which eventually leads to the molar mass calculation.
• Balance of Reaction: Every chemical reaction should be balanced, meaning the number of atoms on the reactant side equals that on the product side.
• Stoichiometric Coefficients: These coefficients in the balanced equation tell us how much of each substance is involved, which is vital for accurately calculating moles and molar masses.

Lean on stoichiometry to understand how substances combine and transform, laying the groundwork for outcomes such as determining molar mass, as in this problem.