Question

A mixture contains \(5.0 \mathrm{g}\) of \(\mathrm{He}, 1.0 \mathrm{g}\) of \(\mathrm{Ar},\) and 3.5 g of Ne. Calculate the volume of this mixture at STP. Calculate the partial pressure of each gas in the mixture at STP.

Short answer

The moles of He, Ar, and Ne are approximately 1.25, 0.025, and 0.17, respectively. The total moles in the mixture are approximately 1.45. Using the Ideal Gas Law at STP, the volume of the mixture is approximately \(35.53\:L\). The mole fractions of He, Ar, and Ne are approximately 0.862, 0.017, and 0.121, respectively. At STP, the partial pressures of He, Ar, and Ne are approximately 0.862 atm, 0.017 atm, and 0.121 atm, respectively.

Step-by-step solution

Calculate the number of moles for each gas

In order to find the number of moles, we will use the formula: Number of moles = mass of gas (in g) / molar mass of gas (in g/mol) Molar Mass of He = 4.0026 g/mol Molar Mass of Ar = 39.948 g/mol Molar Mass of Ne = 20.180 g/mol Hence, He moles: \(n_{He}\) = \(\frac{5.0\:g}{4.0026\:g/mol}\) Ar moles: \(n_{Ar}\) = \(\frac{1.0\:g}{39.948\:g/mol}\) Ne moles: \( n_{Ne}\) = \(\frac{3.5\:g}{20.180\:g/mol}\) Now calculate the moles for each gas.

Calculate the total number of moles in the mixture

We can find the total number of moles in the mixture by summing the moles of each gas: \(n_{total}\) = \(n_{He}\) + \(n_{Ar}\) + \(n_{Ne}\) Now calculate the total moles.

Calculate the volume of the mixture at STP

To find the volume, we will use the Ideal Gas Law. At STP (T = 273.15 K and P = 1 atm), the Ideal Gas Law becomes: V = \(nRT / P\) Substitute the calculated values of n_total, R, T, and P into the equation to find the volume at STP.

Calculate the mole fraction of each gas

In order to find the partial pressures of each of the gases, we first need to calculate their respective mole fractions. Mole fraction (\(X_i\)) of a gas i is given by: \(X_i\) = \(n_i\) / \(n_{total}\) Calculate the mole fraction for He, Ar, and Ne.

Calculate the partial pressure of each gas

Now that we have the mole fraction of each gas, we can determine their partial pressures at STP (P = 1 atm). The partial pressure of a gas i is given by: \(P_{i}\) = \(X_i * P_{total}\) Calculate the partial pressures for He, Ar, and Ne at STP.

Key concepts

Stoichiometry

Stoichiometry is like a recipe for chemistry. It tells you how much of each substance you need to react together and what you can expect to get out at the end. In the context of gas mixtures, stoichiometry involves the calculation of moles, masses, and volume relationships of gases, based on balanced chemical equations and the use of conversion factors.

For instance, when you know the mass of each gas in a mixture, stoichiometry helps you convert that mass into moles using the molar mass of each gas. This is essential because all other calculations we need, like volume at Standard Temperature and Pressure (STP) or partial pressures, rely on knowing the number of moles. Finding the moles is the crucial first step that sets the groundwork for the calculations that follow.

Ideal Gas Law

The ideal gas law is the superstar equation that relates the volume, temperature, pressure, and number of moles of an ideal gas. It's expressed as PV = nRT, where P stands for pressure, V for volume, n for the number of moles, R for the ideal gas constant, and T for temperature.

For gases at STP, where T is 273.15 K and P is 1 atm, the ideal gas law simplifies the calculation of volume. Knowing the total moles of gas from stoichiometry, you can directly calculate the volume the gas mixture would occupy at STP. It's a neat way to predict how much space a gas will take up under standard conditions.

Molar Mass

Molar mass is the weight of one mole of a substance, usually expressed in grams per mole (g/mol). It's like knowing how heavy a dozen eggs would be if you knew the weight of one.

When dealing with gases, knowing the molar mass lets you convert between the mass of a sample and the number of moles. This conversion is crucial in gas mixture calculations because you typically know the mass of each gas (like in our textbook problem), and you need to find the number of moles to use the ideal gas law effectively.

Partial Pressure

Imagine a group of people in a room, each talking at their own volume. The partial pressure is like how loud each person is contributing to the noise in the overall room. It's the pressure that each gas in a mixture would exert if it were alone in the container at the same temperature.

In gas mixtures, you can find the partial pressure by multiplying the mole fraction of each gas by the total pressure. This concept is based on Dalton's Law of Partial Pressures, which states that the total pressure in a mixture of gases is the sum of the individual pressures each gas would exert if it were alone.

Mole Fraction

The mole fraction is a way to express concentration, telling you what portion of the total moles a particular gas contributes in a mixture. It's like looking at a fruit salad and figuring out the percentage of strawberries, based on the total number of fruit pieces.

You calculate the mole fraction by dividing the number of moles of one gas by the total moles in the mixture. Once you know the mole fractions, you can use them to determine the partial pressures of each gas in the mixture, shedding light on how each gas participates in the total pressure.

Standard Temperature and Pressure (STP)

Standard Temperature and Pressure (STP) acts like a baseline or a set of 'standard' conditions that chemists use to compare gases. When gases are measured at STP (0°C and 1 atm), one mole of any ideal gas occupies 22.4 liters.

By performing calculations at STP, we can predict or compare how different gases will behave under the same conditions. It provides a common ground for scientists to talk about gas volumes and enables easier calculation of gas behavior using the ideal gas law because the values for temperature and pressure are constants (273.15 K and 1 atm respectively). Knowing the STP conditions is crucial for any gas-related calculations and helps keep results consistent and comparable.