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Chapter 10: Problem 73

A quantity of \(\mathrm{N}_{2}\) gas originally held at \(531.96 \mathrm{kPa}\) pressure in a 1.00 - \(\mathrm{L}\) container at \(26^{\circ} \mathrm{C}\) is transferred to a \(12.5-\mathrm{L}\) container at \(20^{\circ} \mathrm{C}\). A quantity of \(\mathrm{O}_{2}\) gas originally at \(531.96 \mathrm{kPa}\) and \(26^{\circ} \mathrm{C}\) in a \(5.00-\mathrm{L}\) container is transferred to this same container. What is the total pressure in the new container?

Short Answer

Expert verified
The total pressure in the new container is given by Dalton's Law of Partial Pressures, which is the sum of the final pressures for N₂ and O₂ gases. Calculate the moles of each gas using the Ideal Gas Law, then compute their respective final pressures based on the new volume and temperature. Finally, add these partial pressures to obtain the total pressure: \(P_{total} = P_3 + P_4\).

Step by step solution

01

Ideal Gas Law for initial conditions

Using the Ideal Gas Law, \(PV = nRT\), we can calculate the number of moles (n) for each gas in their respective initial conditions. The given values are \(R = 8.314\ \mathrm{J \cdot K^{-1} \cdot mol^{-1}}\), and temperature must be in Kelvin. For N₂ gas: Initial pressure (P₁): \(531.96\ \mathrm{kPa}\) Initial volume (V₁): \(1.00\ \mathrm{L}\) Initial temperature (T₁): \(26^{\circ} \mathrm{C} = 299\ \mathrm{K}\) For O₂ gas: Initial pressure (P₂): \(531.96\ \mathrm{kPa}\) Initial volume (V₂): \(5.00\ \mathrm{L}\) Initial temperature (T₂): \(26^{\circ} \mathrm{C} = 299\ \mathrm{K}\)
02

Calculate moles (n) for N₂ and O₂

Solve the Ideal Gas Law for n: n = \(\frac{PV}{RT}\) For N₂ gas: n₁ = \(\frac{(531.96\ \mathrm{kPa}) * (1.00\ \mathrm{L})}{(8.314\ \mathrm{J \cdot K^{-1} \cdot mol^{-1}}) * (299\ \mathrm{K})}\) For O₂ gas: n₂ = \(\frac{(531.96\ \mathrm{kPa}) * (5.00\ \mathrm{L})}{(8.314\ \mathrm{J \cdot K^{-1} \cdot mol^{-1}}) * (299\ \mathrm{K})}\)
03

Use the Ideal Gas Law for final conditions

Using the calculated moles of both gases and the new final conditions, we can calculate the final pressures for both gases. The final conditions are: Final volume (V₃): \(12.5\ \mathrm{L}\) Final temperature (T₃): \(20^{\circ} \mathrm{C} = 293\ \mathrm{K}\)
04

Calculate final pressure (P) for N₂ and O₂

Using the Ideal Gas Law and the final conditions, we can obtain the final pressures for N₂ and O₂: For N₂ gas: P₃ = \(\frac{n₁ * R * T₃}{V₃}\) For O₂ gas: P₄ = \(\frac{n₂ * R * T₃}{V₃}\)
05

Use Dalton's Law of Partial Pressures

Dalton's Law of Partial Pressures states that the total pressure is the sum of the partial pressures of the individual gases: Total pressure (P_total) = P₃ + P₄ Calculate the total pressure using the values obtained in Step 4.

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