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Chapter 13: Problem 104

A \(10.0-\mathrm{L}\) vessel contains \(12 \mathrm{g}\) of \(\mathrm{N}_{2}\) gas at \(20^{\circ} \mathrm{C}\) (a) Estimate the nearest-neighbor distance. (b) Is the gas dilute? [Hint: Compare the nearest-neighbor distance to the diameter of an \(\left.\mathrm{N}_{2} \text { molecule, about } 0.3 \mathrm{nm} .\right]\)

Short Answer

Expert verified
Answer: Yes, the nitrogen gas is considered dilute as the nearest-neighbor distance of 1.54 nm is significantly larger than the diameter of an N₂ molecule (approx. 0.3 nm).

Step by step solution

01

Calculate the number of moles of N₂ gas

To determine the number of moles of N₂ gas, we will use the formula: Number of moles = mass / molar mass The molar mass of N₂ is 28.02 g/mol. Number of moles = \(\frac{12 g}{28.02 g/mol}\) Number of moles ≈ 0.428 moles of N₂
02

Calculate the number of molecules

To calculate the number of molecules in the vessel, use Avogadro's number (6.022 × 10²³ particles per mole): Number of molecules = number of moles × Avogadro's number Number of molecules = 0.428 moles × 6.022 × 10²³ molecules/mole Number of molecules ≈ 2.57 × 10²³ molecules
03

Calculate the volume occupied by one molecule

To determine the volume occupied by one molecule, we will divide the volume of the vessel by the total number of molecules: Volume occupied by one molecule = \(\frac{10.0 L}{2.57 × 10^{23} molecules}\) Volume occupied by one molecule ≈ 3.89 × 10⁻²³ L/molecule
04

Calculate the nearest-neighbor distance

Since the gas molecules are evenly distributed, we can model the volume occupied by one molecule as a cube. To find the nearest-neighbor distance, we will calculate the length of one side of the cube: Nearest-neighbor distance (cube side length) = \((3.89 × 10⁻²³ L/molecule)^{1/3}\) Convert the volume to \(\mathrm{m^{3}}\): 1 L = 1 × 10⁻³ \(m^{3}\) Nearest-neighbor distance (cube side length) = \((3.89 × 10⁻²³ × 10⁻³ \mathrm{m^{3}}/molecule)^{1/3}\) Nearest-neighbor distance ≈ 1.54 × 10⁻⁸ m or 1.54 nm
05

Determine if the gas is dilute

To determine if the gas is dilute, we will compare the nearest-neighbor distance (1.54 nm) to the diameter of an N₂ molecule (approx. 0.3 nm). If the nearest-neighbor distance is much larger than the diameter of the molecule, the gas is dilute. Since 1.54 nm is significantly larger than 0.3 nm, the gas is dilute.

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Most popular questions from this chapter

A hydrogen balloon at Earth's surface has a volume of \(5.0 \mathrm{m}^{3}\) on a day when the temperature is \(27^{\circ} \mathrm{C}\) and the pressure is \(1.00 \times 10^{5} \mathrm{N} / \mathrm{m}^{2} .\) The balloon rises and expands as the pressure drops. What would the volume of the same number of moles of hydrogen be at an altitude of \(40 \mathrm{km}\) where the pressure is \(0.33 \times 10^{3} \mathrm{N} / \mathrm{m}^{2}\) and the temperature is \(-13^{\circ} \mathrm{C} ?\)
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Estimate the mean free path of a \(\mathrm{N}_{2}\) molecule in air at (a) sea level \((P \approx 100 \mathrm{kPa} \text { and } T \approx 290 \mathrm{K}),\) (b) the top of \(\quad\) Mt. Everest (altitude $=8.8 \mathrm{km}, P \approx 50 \mathrm{kPa},\( and \)T \approx 230 \mathrm{K}),\( and \)(\mathrm{c})$ an altitude of \(30 \mathrm{km}(P \approx 1 \mathrm{kPa}\) and \(T=230 \mathrm{K}) .\) For simplicity, assume that air is pure nitrogen gas. The diameter of a \(\mathrm{N}_{2}\) molecule is approximately \(0.3 \mathrm{nm}\)
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What is the kinetic energy per unit volume in an ideal gas at (a) \(P=1.00\) atm and (b) \(P=300.0\) atm?
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