Question

The closed containers of the same capacity and at the same temperature are filled with \(44 \mathrm{~g}\) of \(\mathrm{H}_{2}\) in one and \(44 \mathrm{~g}\) of \(\mathrm{CO}_{2}\) in the other. If the pressure of carbon dioxide in the second container is \(1 \mathrm{~atm}\). That of hydrogen in the first container would be : (a) 1 atm (b) \(10 \mathrm{~atm}\) (c) 22 atm (d) 44 atm

Short answer

The pressure of hydrogen in the first container would be 22 atm.

Step-by-step solution

Understanding the Concept

The problem is based on Avogadro's Law, which states that equal volumes of gases at the same temperature and pressure contain an equal number of molecules. However, the problem is actually an application of the Ideal Gas Law, which is: PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature. Since the containers are of the same capacity and at the same temperature, V and T remain constant, and therefore, the pressure of a gas is directly proportional to the number of moles of the gas. We also know the molar mass of H2 is 2 g/mol, and CO2 is 44 g/mol.

Calculating Moles of Hydrogen and Carbon Dioxide

Calculate the moles of H2 and CO2 using the molar mass: Moles of H2 = mass of H2 / molar mass of H2 = 44 g / 2 g/mol = 22 moles. Moles of CO2 = mass of CO2 / molar mass of CO2 = 44 g / 44 g/mol = 1 mole.

Applying the Ideal Gas Law Principle

Since both gases are at the same temperature and are contained in vessels of the same volume, the ratio of their pressures will be the same as the ratio of their moles. The pressure of CO2 is given to be 1 atm, so the pressure of hydrogen will be: Pressure of H2 = moles of H2 / moles of CO2 * Pressure of CO2 = 22 moles / 1 mole * 1 atm = 22 atm.

Key concepts

Avogadro's Law

Avogadro's Law is a foundational principle in chemistry that informs our understanding of how gases behave under same conditions of temperature and pressure. It states that equal volumes of all gases, at the same temperature and pressure, have the same number of molecules.

In practice, Avogadro's Law allows us to predict how a gas will expand or contract with changes in volume, pressure, and temperature, assuming the number of gas molecules remains constant. In the context of our exercise, Avogadro's Law implies that if we have two containers of equal capacity at the same temperature, and they contain the same number of molecules, they will exhibit the same pressure.

This law is vital when studying gas mixtures and reactions, and it forms the basis for the molar volume of a gas, which is the volume one mole of a gas occupies at standard temperature and pressure (STP). Understanding this principle is crucial for mastering gas law calculations, and it seamlessly integrates with the Ideal Gas Law for more complex scenarios.

Molar Mass

Molar Mass is another essential concept, particularly when dealing with gases and their reactions. It represents the mass of one mole of a substance, usually expressed in grams per mole (g/mol).

The molar mass is a bridge between the macroscopic scale of grams and the microscopic scale of molecules, and it's fundamental for conversions in stoichiometry. For gases, knowing the molar mass allows us to determine the number of moles present in a given sample, as seen in the problem where the molar masses of hydrogen (H2) and carbon dioxide (CO2) are used to calculate the moles of each gas.

Moles of H2 and CO2

From the molar masses provided, we understand that 44 grams of H2 corresponds to significantly more molecules than 44 grams of CO2 because hydrogen has a much lower molar mass. The resultant calculation of moles is pivotal for finding the gas pressures using the Ideal Gas Law.

Pressure and Volume Relationship

The Pressure and Volume Relationship, often expressed as Boyle's Law, is another fundamental principle within the realm of gas laws. It states that for a fixed amount of gas at constant temperature, the pressure of the gas is inversely proportional to its volume.

In the context of the Ideal Gas Law, when temperature (T) and the amount of gas (n) are kept constant, as volume (V) decreases, pressure (P) increases, and vice versa. However, in our exercise, since the temperature and volume are consistent for both gases, we focus on the direct relationship between pressure and the number of moles as indicated by the Ideal Gas Law.

Moles Calculation

Calculating moles is a routine task in chemistry, serving as the cornerstone for reaction stoichiometry, gas laws, and concentration calculations. To determine the number of moles (n), we typically divide the given mass of a substance by its molar mass.

Applying to the Exercise

In our exercise, we use the given masses of hydrogen and carbon dioxide and their respective molar masses to find moles. This calculation is crucial because, under constant conditions of volume and temperature, the pressure of the gas is directly proportional to its number of moles. Thus, by finding the mole ratio, and given that the volume and temperature are the same for both gases, we can directly compare pressures.

Consequently, the moles calculation directly influences our final solution involving pressure determination. It underscores the importance of accurate calculation and comprehension of moles in understanding the behavior and interactions of gases.