Question

A \(15.0\) -L rigid container was charged with \(0.500\) atm of krypton gas and \(1.50\) atm of chlorine gas at \(350 .{ }^{\circ} \mathrm{C}\). The krypton and chlorine react to form krypton tetrachloride. What mass of krypton tetrachloride can be produced assuming \(100 \%\) yield?

Short answer

The mass of krypton tetrachloride produced can be calculated by first finding the moles of krypton and chlorine using the ideal gas law, determining the limiting reactant, calculating the moles of krypton tetrachloride produced, and finally converting the moles to mass using the molar mass of KrCl4. The balanced chemical equation for this reaction is \( Kr + 4\ Cl_2 \rightarrow KrCl_4 \). The limiting reactant is determined by dividing the moles of each reactant by their respective coefficients in the balanced equation, and finding the smallest result. Assuming a 100% yield, the mass of krypton tetrachloride produced can be calculated as: \( Mass \, of \, KrCl_4 = moles \, of \, KrCl_4 \cdot molar \, mass \, of \, KrCl_4 \)

Step-by-step solution

Write the balanced chemical equation for the reaction

Krypton and chlorine react to form krypton tetrachloride, and the balanced chemical equation for the reaction is: \( Kr + 4\ Cl_2 \rightarrow KrCl_4 \)

Calculate the initial moles of reactants using the ideal gas law (PV=nRT)

The ideal gas law relates pressure, volume, temperature, and number of moles for a gas. Rearrange the ideal gas law to solve for n (moles): \( n = \frac{PV}{RT} \) Use the given pressure, volume, and temperature for each gas to calculate the initial moles. (Note: Temperature should be in Kelvin, so add 273.15 to the given temperature in Celsius) For krypton: \( n_{Kr} = \frac{(0.500 \ atm)(15.0 \ L)}{(0.0821 \ L \cdot atm \cdot mol^{-1} \cdot K^{-1})(350^{\circ}C + 273.15)} \) For chlorine: \( n_{Cl_2} = \frac{(1.50 \ atm)(15.0 \ L)}{(0.0821 \ L \cdot atm \cdot mol^{-1} \cdot K^{-1})(350^{\circ}C + 273.15)} \)

Determine the limiting reactant

To determine the limiting reactant, divide the moles of each reactant by their respective coefficients in the balanced chemical equation, and find the smallest result. For krypton: \( \frac{n_{Kr}}{1} \) For chlorine: \( \frac{n_{Cl_2}}{4} \) Whichever is smaller, that reactant is the limiting reactant.

Calculate the moles of krypton tetrachloride produced

Since we are assuming a 100% yield, the moles of krypton tetrachloride produced depends on the limiting reactant. Use the mole ratio from the balanced equation to calculate moles of krypton tetrachloride: \( moles \, KrCl_4 = moles \, limiting \, reactant \cdot \frac{1 \, mol \, KrCl_4}{1 \, mol \, limiting \, reactant} \)

Convert moles of krypton tetrachloride to mass

Now that we have the moles of krypton tetrachloride produced, we can find its mass using the molar mass of KrCl4: Molar mass of KrCl4 = M(Kr) + 4 * M(Cl) = 83.80 g/mol + 4 * 35.45 g/mol Mass of KrCl4 = moles of KrCl4 * molar mass of KrCl4

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental formula used in chemistry that relates the pressure, volume, number of moles, and temperature of a gas. It is expressed as \( PV = nRT \), where \( P \) is pressure in atmospheres (atm), \( V \) is volume in liters (L), \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is temperature in Kelvin (K).
This equation helps us predict the behavior of gases by converting one or more known parameters into unknown quantities. In the context of the exercise, we rearrange the formula to solve for the number of moles \( n = \frac{PV}{RT} \).

• **Pressure (P)**: The exertion of force by the gas molecules against the walls of its container; measured in atm.
• **Volume (V)**: The space occupied by the gas; provided in liters.
• **Temperature (T)**: Must always be in Kelvin for calculations; convert from Celsius by adding 273.15.
• **Ideal Gas Constant (R)**: Typically \(0.0821 \, L\cdot atm\cdot mol^{-1}\cdot K^{-1}\).

Using the values provided in the problem, we can accurately determine the moles of both krypton and chlorine, setting the stage for further calculations related to the reaction.

Limiting Reactant

In a chemical reaction, the limiting reactant is the substance that is completely consumed first, limiting the amount of product that can be formed. This is crucial because it determines the maximum yield of the product. The limiting reactant can be identified by comparing the mole ratios of the reactants to the coefficients in the balanced chemical equation.
In our exercise:

• **Krypton (Kr)**: Divide the moles of krypton by its coefficient in the balanced reaction, which is 1.
• **Chlorine (Cl_2)**: Divide the moles of chlorine by its coefficient, which is 4.
  

The reactant with the smaller result is the limiting reactant because it predicts how much product can be formed. This step ensures that we use the correct quantities to calculate the maximum mass of krypton tetrachloride produced. It’s like having all ingredients but running out of sugar in a recipe - without enough sugar, you can't make as much cake.

Chemical Equation

A chemical equation is a symbolic representation of a chemical reaction. It shows the reactants converting into products and must be balanced to obey the law of conservation of mass. In the balanced chemical equation:
\[ Kr + 4Cl_2 \rightarrow KrCl_4 \]
This means that one mole of krypton reacts with four moles of chlorine to produce one mole of krypton tetrachloride.
Balancing ensures:

• The same number of each type of atom on both sides of the equation.
• Mass is conserved throughout the reaction.
• The correct stoichiometric coefficients, which determine the mole ratios.

In our problem, the balanced equation tells us that for every mole of \( KrCl_4 \) produced, one mole of krypton and four moles of chlorine are needed. Understanding and writing balanced equations is key to solving stoichiometry problems as it provides the necessary ratios to compute reactors and products. This is a bit like baking; you need to use specific amounts of ingredients to achieve the intended outcome.