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Chapter 8: Problem 89

The partial pressure of \(\mathrm{CH}_{4}(g)\) is 0.175 atm and that of \(\mathrm{O}_{2}(g)\) is 0.250 atm in a mixture of the two gases. a. What is the mole fraction of each gas in the mixture? b. If the mixture occupies a volume of \(10.5 \mathrm{L}\) at \(65^{\circ} \mathrm{C}\), calculate the total number of moles of gas in the mixture. c. Calculate the number of grams of each gas in the mixture.

Short Answer

Expert verified
The mole fractions of CH4 and O2 are approximately 0.412 and 0.588, respectively. The total number of moles in the mixture is approximately 0.137 mol, and the mass of each gas in the mixture is approximately 0.904 g for CH4 and 2.58 g for O2.

Step by step solution

01

Calculate the mole fraction of each gas

To find the mole fraction of each gas, simply divide the individual partial pressure of the gas by the total pressure of the mixture. Let's denote the mole fraction of methane (CH4) as X_CH4 and the mole fraction of oxygen (O2) as X_O2. Find the total pressure of the mixture by adding the partial pressures of CH4 and O2: Total Pressure (P_total) = P_CH4 + P_O2 P_total = 0.175 atm (CH4) + 0.250 atm (O2) = 0.425 atm Now, calculate the mole fraction of each gas: X_CH4 = P_CH4 / P_total X_O2 = P_O2 / P_total
02

Calculate the mole fractions

Calculate the mole fractions using the formula from step 1: X_CH4 = 0.175 atm / 0.425 atm ≈ 0.412 X_O2 = 0.250 atm / 0.425 atm ≈ 0.588
03

Calculate the total number of moles in the mixture

We can use the Ideal Gas Law to find the total number of moles in the mixture. The Ideal Gas Law is given by PV = nRT, where P is the total pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature in Kelvin. First, we need to convert the temperature from Celsius to Kelvin: Temperature (T) = 65℃ + 273.15 = 338.15 K Now plug the values into the Ideal Gas Law formula and solve for n: (0.425 atm) * (10.5 L) = n * (0.0821 L atm/mol K) * (338.15 K) Divide both sides by the product of the gas constant and the temperature: n = (0.425 atm * 10.5 L) / (0.0821 L atm/mol K * 338.15 K)
04

Find the total number of moles

Calculate the total number of moles using the formula from step 3: n ≈ 0.137 mol
05

Calculate the number of moles of individual gases in the mixture

Now that we know the total number of moles in the mixture, we can find the number of moles of each individual gas by multiplying the total number of moles by the mole fraction of each gas: Moles of CH4 = n * X_CH4 Moles of O2 = n * X_O2
06

Find the number of moles of individual gases

Calculate the number of moles of CH4 and O2 using the formula from step 5: Moles of CH4 ≈ 0.137 mol * 0.412 ≈ 0.0564 mol Moles of O2 ≈ 0.137 mol * 0.588 ≈ 0.0806 mol
07

Calculate the mass of individual gases in the mixture

Now we can calculate the mass of each individual gas in the mixture by multiplying the number of moles of each gas by its molar mass: Mass of CH4 = moles of CH4 * molar mass of CH4 Mass of O2 = moles of O2 * molar mass of O2
08

Find the mass of individual gases

Calculate the mass of each gas using the formula from step 7: Mass of CH4 ≈ 0.0564 mol * (16.04 g/mol) ≈ 0.904 g Mass of O2 ≈ 0.0806 mol * (32.00 g/mol) ≈ 2.58 g To sum up, the mole fraction of CH4 is ≈ 0.412 and O2 is ≈ 0.588, the total number of moles in the mixture is ≈ 0.137 mol, and the mass of each gas in the mixture is ≈ 0.904 g for CH4 and ≈ 2.58 g for O2.

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Most popular questions from this chapter

Small quantities of hydrogen gas can be prepared in the laboratory by the addition of aqueous hydrochloric acid to metallic zinc. $$\mathrm{Zn}(s)+2 \mathrm{HCl}(a q) \longrightarrow \mathrm{ZnCl}_{2}(a q)+\mathrm{H}_{2}(g)$$ Typically, the hydrogen gas is bubbled through water for collection and becomes saturated with water vapor. Suppose 240\. mL of hydrogen gas is collected at \(30 .^{\circ} \mathrm{C}\) and has a total pressure of \(1.032\) atm by this process. What is the partial pressure of hydrogen gas in the sample? How many grams of zinc must have reacted to produce this quantity of hydrogen? (The vapor pressure of water is 32 torr at \(30^{\circ} \mathrm{C}\).)

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