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Chapter 8: Problem 55

A quantity of \(10 \mathrm{~g}\) of a mixture of \(\mathrm{C}_{2} \mathrm{H}_{6}\) and \(\mathrm{C}_{5} \mathrm{H}_{10}\) occupy \(4480 \mathrm{ml}\) at \(1 \mathrm{~atm}\) and \(273 \mathrm{~K}\). The percentage of \(\mathrm{C}_{2} \mathrm{H}_{6}\) by mass, in the mixture is (a) \(30 \%\) (b) \(70 \%\) (c) \(50 \%\) (d) \(60 \%\)

Short Answer

Expert verified
Answer: The percentage by mass of ethane (C2H6) in the mixture is 30%.

Step by step solution

01

Calculate n_total using Ideal Gas Law

Using the Ideal Gas Law (PV = nRT), you can find the total number of moles (n_total) in the mixture. Given that P = 1 atm, V = 4480 ml (or 4.48 L), R = 0.0821 L atm/mol K, and T = 273 K: n_total = PV / RT n_total = (1 * 4.48) / (0.0821 * 273) n_total = 0.2 moles So, there are a total of 0.2 moles in the 10 g mixture of ethane and pentene.
02

Calculate mole fractions of ethane and pentene in the mixture

Let x be the mole fraction of ethane (C2H6) in the mixture. Then, the mole fraction of pentene (C5H10) will be (1-x). Next, find the moles of each gas by multiplying the mole fraction by n_total: Moles of ethane = x * n_total Moles of pentene = (1 - x) * n_total
03

Calculate the mass (m) of each gas

Using the molar masses of ethane (M_C2H6 = 30 g/mol) and pentene (M_C5H10 = 70 g/mol), calculate the mass of each gas: Mass of ethane = moles of ethane * M_C2H6 = x * n_total * 30 Mass of pentene = moles of pentene * M_C5H10 = (1 - x) * n_total * 70
04

Write the equation for the total mass of the mixture

The total mass of the mixture is given as 10 g. So, the sum of the masses of ethane and pentene should be equal to 10 g: Mass of ethane + Mass of pentene = 10 x * n_total * 30 + (1 - x) * n_total * 70 = 10
05

Calculate the value of x

Now, solve the equation for x from step 4. You already know that n_total = 0.2 moles: x * 0.2 * 30 + (1 - x) * 0.2 * 70 = 10 6x + 14 - 14x = 10 Now, combine the terms with x: -8x = -4 Divide both sides by -8: x = 0.5 So, the mole fraction of ethane (C2H6) in the mixture is 0.5.
06

Calculate the percentage of ethane by mass in the mixture

Use the mass of ethane from step 3 to calculate the percentage of ethane by mass in the mixture. Remember that x = 0.5: Mass of ethane = x * n_total * 30 = 0.5 * 0.2 * 30 Mass of ethane = 3 g Finally, calculate the percentage of ethane by mass: Percentage of ethane by mass = (Mass of ethane / Total mass of mixture) * 100 Percentage of ethane by mass = (3 / 10) * 100 Percentage of ethane by mass = 30% The correct answer is (a) 30%.

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