Question

Equal weights of methane and hydrogen are mixed in an empty container at ${25}^{\circ }\mathrm{C}$. The fraction of the total pressure exerted by hydrogen is (1) $\frac{1}{2}$ (2) $\frac{8}{9}$ (3) $\frac{16}{17}$ (4) $\frac{1}{9}$

Short answer

The fraction of the total pressure exerted by hydrogen is \frac{8}{9}.

Step-by-step solution

- Determine the molar masses

Calculate the molar masses of methane (CH4) and hydrogen (H2). For methane, the molar mass is 12 (carbon) + 4*1 (hydrogen) = 16 g/mol. For hydrogen, the molar mass is 2*1 (hydrogen) = 2 g/mol.

- Equate given weights to moles

Since equal weights of methane and hydrogen are given, let the weight be W grams for each gas. The number of moles of methane is $\frac{W}{16}$ and for hydrogen is $\frac{W}{2}$.

- Calculate the total moles of gas

The total moles of gas in the container is the sum of the moles of methane and hydrogen: $$\text{Total moles}=\frac{W}{16}+\frac{W}{2}=\frac{W}{16}+\frac{8W}{16}=\frac{9W}{16}$$

- Determine fraction of total pressure by hydrogen

The fraction of the total pressure exerted by hydrogen is the ratio of the moles of hydrogen to the total moles of gas. This is calculated as follows: $$\text{Fraction}=\frac{\frac{W}{2}}{\frac{9W}{16}}=\frac{8W}{18W}=\frac{8}{9}$$

Key concepts

molar mass calculation

Understanding molar mass is crucial in solving many problems in chemistry. Molar mass is the mass of one mole of a substance, usually expressed in grams per mole (g/mol). To calculate the molar mass, sum the atomic masses of all the atoms in a molecule. For example, in the exercise, methane (CH₄) consists of one carbon (C) atom and four hydrogen (H) atoms. The atomic mass of carbon is 12 g/mol, and hydrogen is 1 g/mol. Therefore, the molar mass of CH₄ is:
12 (C) + 4*1 (H) = 16 g/mol.
Similarly, for hydrogen gas (H₂), composed of two hydrogen atoms, the molar mass is: 2*1 (H) = 2 g/mol.

partial pressure

Partial pressure is the pressure that one component of a gas mixture would exert if it were alone in the container. It is essential in understanding how gas mixtures behave. To find the partial pressure, we use Dalton's Law of Partial Pressures, which states that the total pressure exerted by a mixture of gases is the sum of the partial pressures of each gas. Mathematically, this can be expressed as:
P_total = P1 + P2 + ... + Pn
where P_total is the total pressure, and P1, P2,..., Pn are the partial pressures of gases 1, 2,..., n, respectively. In the exercise, we determined the partial pressure fraction exerted by hydrogen. Since the mole fraction of hydrogen is the ratio of the moles of hydrogen to the total moles of gas, we used this ratio to find the fraction of the total pressure.

ideal gas law

The Ideal Gas Law is a cornerstone of understanding gas behavior. This law combines several gas laws (Boyle's, Charles's, and Avogadro's) into one equation: PV = nRT Where:
P = Pressure of the gas
V = Volume of the gas
n = Number of moles of the gas
R = Universal gas constant (8.314 J/(mol·K))
T = Temperature in Kelvin (K)
In the given exercise, although we did not directly use the Ideal Gas Law equation, concepts from it were essential. By calculating the total number of moles, we implicitly relied on the relationships described by this law.

stoichiometry

Stoichiometry involves the calculation of reactants and products in chemical reactions. It is vital for predicting the amounts of substances consumed and produced in a reaction. In the exercise, the stoichiometry of the gases allowed us to determine the number of moles from a given mass. For equal masses of methane and hydrogen, we calculated their moles using their molar masses. These calculations led to understanding the total moles in the container and thereby helped in finding the fraction of pressure exerted by each gas. Always remember these steps:

• Convert given masses to moles using molar masses.
• Use the mole ratio to understand the relationship between different components.
• Apply the results to find properties like partial pressures or total volumes.