Question

Consider the following unbalanced chemical equation in which element X is unknown: $$ \mathrm{X}(s)+\mathrm{F}_{2}(g) \rightarrow \mathrm{XF}_{3}(s) $$ If \(9.15 \mathrm{~g}\) of element \(\mathrm{X}(s)\) is completely reacted with \(4.00 \mathrm{~L}\) of fluorine gas at \(250^{\circ} \mathrm{C}\) and \(2.50 \mathrm{~atm},\) what is the identity of \(\mathrm{X} ?\)

Short answer

The identity of the unknown element \(X\) can be found by following these steps: 1. Balance the chemical equation: \(\mathrm{X}(s)+2\,\mathrm{F}_{2}(g) \rightarrow \frac{3}{2}\,\mathrm{XF}_{3}(s)\) 2. Calculate the moles of \(\mathrm{F}_{2}(g)\) using the ideal gas law: \(PV = n_\mathrm{F_2}RT\) 3. Determine the mole ratio of \(X\) to \(\mathrm{F}_{2}\) from the balanced equation: \(\frac{n_\mathrm{X}}{n_\mathrm{F_2}}=\frac{1}{2}\) 4. Calculate the moles of \(X\): \(n_\mathrm{X} = \frac{1}{2}\times n_\mathrm{F_2}\) 5. Calculate the molar mass of \(X\): \( M_\mathrm{X} = \frac{9.15\,\mathrm{g}}{n_\mathrm{X}}\) 6. Identify the unknown element \(X\) by comparing the calculated molar mass with those in the periodic table.

Step-by-step solution

1: Balance the chemical equation

To balance the chemical equation, place coefficients in front of the chemical formulas so that the number of atoms of each element is equal on both sides of the equation. It is clear that the given equation lacks balance regarding the number of fluorine atoms. To fix this, add a coefficient of 2 in front of \(\mathrm{F}_{2}(g)\) and a coefficient of 3/2 in front of \(\mathrm{XF}_{3}(s)\), so we can equalize the number of fluorine atoms on both sides of the equation. The balanced chemical equation is: $$ \mathrm{X}(s)+2\,\mathrm{F}_{2}(g) \rightarrow \frac{3}{2}\,\mathrm{XF}_{3}(s) $$

2: Calculate the moles of fluorine gas

We can use the ideal gas law, PV=nRT, to find the moles (n) of fluorine gas involved in the reaction. Given: volume (V) = 4.00 L, pressure (P) = 2.50 atm, and temperature (T) = \(250^{\circ}\)C or 523 K (remember to convert Celsius to Kelvin by adding 273). The ideal gas constant (R) is 0.0821 L*atm/mol*K. Then, we can calculate moles of F2: $$ n_\mathrm{F_2}=\frac{PV}{RT} = \dfrac{(2.50\,\mathrm{atm})(4.00\,\mathrm{L})}{(0.0821\,\mathrm{L\cdot atm\, mol^{-1}\cdot K^{-1}})(523\,\mathrm{K})} $$

3: Calculate the mole ratio of X to F2

Using the balanced chemical equation from step 1, determine the mole ratio of X to F2: $$ \frac{n_\mathrm{X}}{n_\mathrm{F_2}}=\frac{1}{2} $$

4: Calculate the moles of X

Use the mole ratio and the moles of F2 calculated in step 2 to solve for the moles of X: $$ n_\mathrm{X} = \frac{1}{2}\times n_\mathrm{F_2} $$

5: Calculate the molar mass of X

Given the mass of X (9.15 g) and the moles of X in step 4, we can find the molar mass (M) of X by dividing the mass by the moles: $$ M_\mathrm{X} = \frac{9.15\,\mathrm{g}}{n_\mathrm{X}} $$

6: Identify the unknown element X

Finally, compare the molar mass calculated in step 5 with the molar masses of elements in the periodic table to identify the element X.

Key concepts

Stoichiometry

Stoichiometry is the field of chemistry that is concerned with the relative quantities of reactants and products in chemical reactions. It is a key concept when it comes to understanding reactions, as it involves using balanced chemical equations to calculate the amounts of reactants needed or products formed.

In the given exercise, stoichiometry comes into play when we look at the balanced chemical equation and determine the required mole ratios. This ratio allows us to predict how much of any one substance in a reaction is necessary to react completely with a given amount of another substance. It is essential to first correctly balance the chemical equation, as this ensures the law of conservation of mass is obeyed and provides us with the correct stoichiometric coefficients necessary for further calculations.

Ideal Gas Law

The ideal gas law is a fundamental equation in chemistry which relates the pressure, volume, temperature, and quantity (in moles) of an ideal gas. It is usually expressed as \( PV=nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is temperature in Kelvin.

To use the ideal gas law in the provided problem, we convert the given conditions for the fluorine gas, such as temperature and pressure, into the amount of substance in moles. This crucial step combines the concepts of ideal gas behavior with stoichiometry and lays the groundwork for later determining the molar mass of the unknown element \( X \. This interplay is a great example of how different areas of chemistry work together to solve practical problems.

Mole Concept

The mole concept is a method of quantifying the amount of substance. One mole of any substance contains Avogadro's number (\( 6.022 \times 10^{23} \) particles) of that substance. It is a central concept in chemistry because it provides a link between the microscopic particles and the macroscopic amount of material we work with in the laboratory.

Understanding the mole concept is crucial when calculating the molarity of a solution, determining reaction yields, and translating between mass and number of atoms or molecules in a given sample. In our exercise, after we have used the ideal gas law to calculate the moles of fluorine, we apply the mole concept to find the stoichiometrically equivalent moles of element \( X \) based on the balanced reaction equation.

Molar Mass Calculation

The molar mass of a substance is the mass in grams of one mole of that substance. It’s essentially the molecular weight of a compound expressed in grams. The unit for molar mass is g/mol.

Calculating molar mass is fundamental when you need to convert between mass and moles of a substance. In our problem, once the moles of the unknown element \( X \) are figured out from the previously calculated moles of fluorine, we use this quantity and the given mass of \( X \) to determine its molar mass. This is done by dividing the mass of \( X \) by its moles, allowing us to compare the result with known molar masses of elements to identify the unknown element. This step highlights the direct application of the mole concept and the importance of molar mass calculation in chemical identification.