/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 104 A \(10.0-\mathrm{L}\) vessel con... [FREE SOLUTION] | 91影视

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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 N鈧� molecule (approx. 0.3 nm).

Step by step solution

01

Calculate the number of moles of N鈧� gas

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

Calculate the number of molecules

To calculate the number of molecules in the vessel, use Avogadro's number (6.022 脳 10虏鲁 particles per mole): Number of molecules = number of moles 脳 Avogadro's number Number of molecules = 0.428 moles 脳 6.022 脳 10虏鲁 molecules/mole Number of molecules 鈮� 2.57 脳 10虏鲁 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 = \(\frac{10.0 L}{2.57 脳 10^{23} molecules}\) Volume occupied by one molecule 鈮� 3.89 脳 10鈦宦猜� 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) = \((3.89 脳 10鈦宦猜� L/molecule)^{1/3}\) Convert the volume to \(\mathrm{m^{3}}\): 1 L = 1 脳 10鈦宦� \(m^{3}\) Nearest-neighbor distance (cube side length) = \((3.89 脳 10鈦宦猜� 脳 10鈦宦� \mathrm{m^{3}}/molecule)^{1/3}\) Nearest-neighbor distance 鈮� 1.54 脳 10鈦烩伕 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 N鈧� 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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