Approximately how many times larger is that distance compared to the size of a small molecule like N2? Explain why this justifies the assumption that a gas like N2 is ideal at room temperature and pressure.
The correct answer and explanation is:
Sure! Here’s a detailed explanation about the size comparison and why nitrogen gas (N₂) behaves ideally at room temperature and pressure:
Answer:
The average distance between nitrogen (N₂) molecules in a gas at room temperature and pressure is roughly 10,000 times larger than the size of an individual N₂ molecule.
Explanation:
- Size of an N₂ molecule:
A nitrogen molecule is about 0.3 nanometers (3 × 10⁻¹⁰ meters) in diameter. This is an extremely small scale representing the actual size of the molecule. - Distance between molecules in a gas:
At standard temperature and pressure (STP), gases are very sparse. The average distance between neighboring N₂ molecules can be estimated using the ideal gas law and typical molecular number densities. It turns out that this distance is on the order of 30 nanometers (3 × 10⁻⁸ meters) or even more, which is about 100 times larger than the molecular diameter. More accurate estimates usually put the mean free path (the average distance a molecule travels before colliding) around 60 to 70 nanometers, making the gap about 200 times larger. However, when considering molecular spacing in the bulk gas, the distance between centers of molecules is roughly 10,000 times the molecular size. - Why this justifies the ideal gas assumption:
The ideal gas model assumes:
- Gas molecules have negligible volume compared to the container volume.
- There are no intermolecular forces except during brief collisions.
- Collisions between molecules are perfectly elastic.
Because the molecules are so far apart (thousands of times their own size), the actual volume they occupy is negligible compared to the total volume of the gas. This means the gas behaves as if molecules are just point particles. Moreover, at room temperature, the kinetic energy of molecules is high enough that weak intermolecular forces (like Van der Waals forces) have little effect over the large distances separating molecules.
Thus, N₂ molecules mostly move independently without significant attraction or repulsion, validating the ideal gas assumptions and making nitrogen behave like an ideal gas at room temperature and atmospheric pressure.