Question

A mixture of \(1.00 \mathrm{~g} \mathrm{H}_{2}\) and \(1.00 \mathrm{~g}\) He is placed in a \(1.00-\mathrm{L}\) container at \(27^{\circ} \mathrm{C}\). Calculate the partial pressure of each gas and the total pressure.

Short answer

The partial pressure of hydrogen gas (H₂) is \(12.16 \mathrm{~atm}\), partial pressure of helium gas (He) is \(6.144 \mathrm{~atm}\), and the total pressure of the mixture is \(18.30 \mathrm{~atm}\).

Step-by-step solution

Identify relevant information

Given: Mass of hydrogen (H₂) = 1.00 g Mass of helium (He) = 1.00 g Volume of the container (V) = 1.00 L Temperature (T) = 27°C Note: Temperature must be converted to Kelvin (K) for calculations. So, T = 27 + 273.15 = 300.15 K.

Calculate the moles of each gas in the mixture

To calculate the moles of each gas, we'll use the molar mass of each element. Molar mass of hydrogen (H₂) = 2.016 g/mol Molar mass of helium (He) = 4.0026 g/mol Moles of H₂ (n_H₂) = mass_H₂ / molar_mass_H₂ = (1.00 g) / (2.016 g/mol) = 0.496 moles Moles of He (n_He) = mass_He / molar_mass_He = (1.00 g) / (4.0026 g/mol) = 0.250 moles

Apply the Ideal Gas Law for each gas in the mixture

The Ideal Gas Law formula is: PV = nRT, where P is pressure, V is volume, n is moles, R is the gas constant, and T is temperature. We will use the formula to find the partial pressure of each gas in the mixture. For hydrogen (H₂): P_H₂ = (n_H₂ * R * T) / V For helium (He): P_He = (n_He * R * T) / V Note: The gas constant R = 0.0821 L atm / K mol

Calculate partial pressures of each gas

Using the Ideal Gas Law, we can determine the partial pressures of each gas. For hydrogen (H₂): P_H₂ = (0.496 moles * 0.0821 L atm / K mol * 300.15 K) / 1.00 L = 12.16 atm For helium (He): P_He = (0.250 moles * 0.0821 L atm / K mol * 300.15 K) / 1.00 L = 6.144 atm

Calculate the total pressure of the mixture

The total pressure is the sum of the partial pressures of each individual gas in the mixture. Total Pressure (P_total) = P_H₂ + P_He = 12.16 atm + 6.144 atm = 18.30 atm The partial pressure of hydrogen gas is 12.16 atm, partial pressure of helium gas is 6.144 atm, and the total pressure of the mixture is 18.30 atm.

Key concepts

Partial Pressure

In a gas mixture, each gas exerts its own pressure, known as its partial pressure. This is a fundamental concept in chemistry because it allows us to understand how different gases behave when mixed. When gases like hydrogen and helium are placed in a container, each one contributes to the total pressure inside according to its amount or moles.
The partial pressure of a gas can be calculated using the Ideal Gas Law, expressed as \( P = \frac{nRT}{V} \). Here, \(P\) is the partial pressure, \(n\) is the number of moles of the gas, \(R\) is the gas constant, \(T\) is the temperature in Kelvin, and \(V\) is the volume of the container. It is important to remember that the total pressure is simply the sum of the partial pressures of all gases present.

Mole Calculation

The amount of gas present is measured in moles, which is a basic unit in chemistry to describe quantities of substance. To convert grams to moles, you need the molar mass of the substance, which is a measure of the mass of one mole of that substance.

• For hydrogen (\(H_2\)), the molar mass is \(2.016 \text{ g/mol}\).
• For helium (He), the molar mass is \(4.0026 \text{ g/mol}\).

Therefore, the formula to find the moles from mass is: \( ext{Moles} = \frac{ ext{Mass}}{ ext{Molar Mass}}\).
For example, with 1.00 g of hydrogen, you have \( \frac{1.00}{2.016} \approx 0.496\) moles of hydrogen. This calculation is crucial for determining other properties of gases, such as their pressures.

Gas Constant

The gas constant \(R\) is a key factor in the Ideal Gas Law, which helps in calculating properties of gases like pressure, volume, and temperature. Its value is typically \(0.0821 \text{ L atm } / \text{K mol}\) when dealing with common laboratory conditions.
This constant helps establish a relationship between the variables in the Ideal Gas Law, \(PV = nRT\). This allows us to compute the behavior of gases under different conditions by providing a bridge between physical quantities and the properties of gases.

Temperature Conversion

Temperature needs to be in Kelvin for gas law calculations because Kelvin is an absolute scale and works directly in thermodynamic equations. The conversion from Celsius to Kelvin is straightforward: simply add 273.15 to the Celsius temperature.
For instance, a given temperature of \(27^{\circ} \text{C}\) becomes \(27 + 273.15 = 300.15 \text{ K}\).

• This conversion ensures accuracy in calculations, as Kelvin directly relates to particle energy.
• Higher temperatures indicate more energetic particles, affecting pressure calculations.

Always ensuring temperature is in Kelvin is critical for using gas laws accurately.