Problem 44 - Chemistry: Principles and Reactions

Question

The contents of a tank of natural gas at \(1.20 \mathrm{~atm}\) is analyzed. The analysis showed the following mole percents: \(88.6 \% \mathrm{CH}_{4}, 8.9 \% \mathrm{C}_{2} \mathrm{H}_{6}\), and \(2.5 \%\) \(\mathrm{C}_{3} \mathrm{H}_{3}\). What is the partial pressure of each gas in the tank?

Step-by-Step Solution

Answer

Answer: The partial pressures of each gas in the tank are: - Methane (CH₄): 1.0632 atm - Ethane (C₂H₆): 0.1068 atm - Propane (C₃H₈): 0.0300 atm

Step 1: Calculate mole fractions of each gas

To find the mole fractions, we'll divide the mole percent by 100. This will give us the mole fraction for each gas. Mole fraction of \(\mathrm{CH}_{4}\): \(\frac{88.6}{100} = 0.886\) Mole fraction of \(\mathrm{C}_{2} \mathrm{H}_{6}\): \(\frac{8.9}{100} = 0.089\) Mole fraction of \(\mathrm{C}_{3} \mathrm{H}_{8}\): \(\frac{2.5}{100} = 0.025\)

Step 2: Use Dalton's law of partial pressures

According to Dalton's law of partial pressures, the partial pressure of each gas can be calculated by multiplying the mole fraction of each gas by the total pressure. Partial pressure of \(\mathrm{CH}_{4} = 0.886 \times 1.20\mathrm{~atm} = 1.0632\mathrm{~atm}\) Partial pressure of \(\mathrm{C}_{2} \mathrm{H}_{6} = 0.089 \times 1.20\mathrm{~atm} = 0.1068\mathrm{~atm}\) Partial pressure of \(\mathrm{C}_{3} \mathrm{H}_{8} = 0.025 \times 1.20\mathrm{~atm} = 0.0300\mathrm{~atm}\)

Step 3: Final Answer

The partial pressures of each gas in the tank are as follows: Partial pressure of \(\mathrm{CH}_{4} = 1.0632\mathrm{~atm}\) Partial pressure of \(\mathrm{C}_{2} \mathrm{H}_{6} = 0.1068\mathrm{~atm}\) Partial pressure of \(\mathrm{C}_{3} \mathrm{H}_{8} = 0.0300\mathrm{~atm}\)

Key Concepts

Dalton's Law of Partial Pressures

Understanding the behavior of gas mixtures often requires insight into how the individual gases contribute to the overall pressure. This is where Dalton's Law of Partial Pressures comes into play. Put simply, this law states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of individual gases.

For instance, in a natural gas tank with methane (CH₄), ethane (C₂H₆), and propane (C₃H₈), we calculate each gas's partial pressure independently. This law is particularly useful because it lets us treat complex gas mixtures in a straightforward, analytical manner by breaking them down into simpler components.

Mole Fraction

A mole fraction is a way of expressing the concentration of a constituent in a mixture. To calculate it, you divide the number of moles of the constituent by the total number of moles in the mixture. The beauty of mole fractions lies in their simplicity and direct relation to the composition, as they do not change with the pressure or volume of the gas.

In our exercise, the mole fraction was given in percent and then converted into a decimal form for each gas component to use in further calculations. It's this mole fraction that gets multiplied by the total pressure in Dalton's Law to yield the partial pressures.

Gas Mixture

A gas mixture contains more than one type of gas particle. Each distinct gas in a mixture contributes to the total pressure and has specific properties like volume, temperature, and quantity—expressed in moles. When solving problems involving gas mixtures, considering how each gas behaves under given conditions is crucial.

Moreover, in a gas mixture, the individual gases often act as if they are alone occupying the whole volume, this assumption is based on the ideal gas behavior and is used to simplify calculations like those involving Dalton's Law and mole fractions.

Natural Gas Composition

Natural gas is predominantly composed of methane (CH₄), but it also contains other hydrocarbons like ethane (C₂H₆) and propane (C₃H₈), as well as varying amounts of other substances. The percentage of each component in natural gas can impact its energy content and how it burns, which is why understanding the composition is essential for various applications, such as energy production.

Knowing the natural gas composition is also essential for calculating partial pressures in a mixture. As demonstrated in the textbook exercise, you can use the mole percents to find the mole fractions and then apply Dalton's Law of Partial Pressures to determine the individual pressures of each component gas.