Question

A compound exists in the gaseous phase both as monomer (A) and dimer (B). The molecular mass of \(\mathrm{A}\) is \(48 .\) In an experiment, \(96 \mathrm{~g}\) of the compound was confined in a vessel of volume \(33.61\) and heated to \(546 \mathrm{~K} .\) What is the pressure developed if the compound exists as dimer to the extent of \(50 \%\) by weight under these conditions? (a) \(2 \mathrm{~atm}\) (b) \(4 \mathrm{~atm}\) (c) \(3 \mathrm{~atm}\) (d) \(0.5 \mathrm{~atm}\)

Short answer

The pressure developed is approximately 2.25 atm, which is not an option among the provided choices.

Step-by-step solution

Calculate the number of moles of monomer and dimer

Given that the molecular mass of A is 48 and 50% exists as dimer by weight, the weight of the monomer is 96 g * 50% = 48 g and the weight of the dimer is also 48 g. Thus, the number of moles of monomer, n(A), is calculated by dividing the weight by the molecular mass: n(A) = 48 g / 48 g/mol = 1 mol. The molecular mass of dimer B is twice that of A, so M(B) = 2 * 48 g/mol = 96 g/mol. The number of moles of dimer, n(B), is n(B) = 48 g / 96 g/mol = 0.5 mol.

Calculate the total number of moles

The total number of moles is the sum of the moles of A and B. So, n(total) = n(A) + n(B) = 1 mol + 0.5 mol = 1.5 moles.

Apply the ideal gas equation

To find the pressure, use the ideal gas equation PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the universal gas constant (0.0821 atm L/mol K), and T is the temperature in Kelvin. Substitute the corresponding values into the equation to get P = (nRT) / V. Therefore, P = [(1.5 moles) * (0.0821 atm L/mol K) * (546 K)] / 33.61 L.

Calculate the pressure developed

Now, calculate the pressure using the values from the previous step. P = [(1.5 moles) * (0.0821 atm L/mol K) * (546 K)] / 33.61 L = 2.25 atm.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that describes the relationship among pressure (P), volume (V), temperature (T), and amount of gas in moles (n) for an ideal gas. It is expressed as PV = nRT, where R stands for the ideal gas constant. The value of R can vary based on the units used; in the context of the given problem, it is given as 0.0821 atm L/mol K. The Ideal Gas Law assumes that gas particles are point particles moving randomly and do not exert forces on one another except during elastic collisions.

Understanding how to apply the Ideal Gas Law is crucial when tackling chemical equilibrium problems involving gases. In the exercise, by applying the Ideal Gas Law, we can solve for the pressure developed inside a vessel when a gas or mixture of gases is confined at a certain temperature and volume. This law is particularly useful in conditions where the gas behaves ideally, which is usually at high temperature and low pressure.

Molecular Mass Calculation

Calculating molecular mass is a key step in chemistry that involves summing the atomic masses of all the atoms in a molecule. This calculation is essential for converting between grams and moles, which is a necessary conversion for using the Ideal Gas Law. In the context of the given problem, the molecular mass of the monomer A and dimer B is directly used to find the number of moles of each species.

To calculate the moles, we divide the given mass of the substance by its molecular mass. Given the molecular mass of monomer A as 48 g/mol, and knowing that the dimer B is simply two A units bonded together, the molecular mass of B becomes 96 g/mol. This understanding allows for precise calculations in gaseous phase reactions and is vital for quantifiably describing the composition of a gas mixture.

Gaseous Phase Reactions

Gaseous phase reactions involve reactants and products in the gas phase and can often reach a state of chemical equilibrium. Chemical equilibrium occurs when the rate of the forward reaction equals the rate of the reverse reaction, leading to a constant composition of the reaction mixture over time. In a situation where a compound exists as both a monomer and a dimer, understanding the equilibrium between the two states is crucial for predicting and calculating the behavior of the system under various conditions.

In the provided exercise, we are dealing with a scenario in which 50% of the compound exists as a dimer by weight. This constrained equilibrium impacts the pressure calculation as the presence of monomers and dimers affects the total number of gas particles. Problems in this domain require not only the application of the Ideal Gas Law but also a considered approach to the molecular forms present and their respective proportions within the mixture, an insight that is vital in predicting and controlling the outcomes of gaseous reactions.