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Gas density is a measure of how much mass of a gas is present in a given volume. It is mathematically expressed as the mass of the gas divided by its volume: \( d = \frac{m}{V} \). In simpler terms, density tells you how 'heavy' or 'light' a gas is in a particular space.
Understanding gas density is crucial in this exercise because we are dealing with equal densities of two different gases—helium (\( \text{He} \)) and nitrogen (\( \text{N}_2 \)).
This is important to grasp because even though these gases have drastically different molar masses, the problem sets them equal by density.
- Helium is much lighter than nitrogen due to its smaller molar mass.- But in our containers, they have the same density, meaning the mass per unit volume is equal.
This equal density implies a specific relationship between quantities of each gas present, which will later help us understand their pressure differences.

Molar mass is the mass of one mole of a substance and is typically expressed in grams per mole (g/mol). It plays a key role when using the ideal gas law to determine other properties of the gas.
In this exercise, helium has a molar mass of 4 g/mol while nitrogen's molar mass is 28 g/mol, a significant difference. This difference in molar mass is crucial because:

• It allows us to calculate the number of moles from a given mass, using the formula \( n = \frac{m}{M} \).
• It determines how much gas is present for a fixed amount of mass and thus can greatly impact the pressure of that gas.
• With equal densities given, a lighter molar mass gas (helium) will result in more moles than a heavier molar mass gas (nitrogen).

This disparity highlights how, even with the same amount of mass per volume (density) between two containers, their chemical identity (molar mass) will influence pressures differently.

Pressure in the context of gases is the force exerted by the gas particles when they collide with the walls of their container. The ideal gas law establishes the relationship between pressure and other related variables: \( PV = nRT \).
In this particular exercise, we compared the pressure in two containers of gas that have the same density but different chemical natures. Under the conditions set by the problem, the pressures are affected mainly by the number of moles and not the physical space or temperature, since these are constant.
The step-by-step solution led us to compare the pressures by observing their molar mass differences, resulting in a pressure relationship of \( P_{N_2} = \frac{1}{7} P_{He} \).
This relationship tells us that despite having the same gas density and volume, the gas particles' different identities cause nitrogen's pressure to be significantly less than that of helium:

• Because helium is a lighter gas, its molecules are more numerous in the same volume, explaining why \( P_{He} \) is greater.
• The difference encapsulated in the 1/7 ratio reflects pure chemical nature decisions about how gas atoms and molecules behave under uniform physical conditions.

Thus, understanding gas pressure relationships requires balancing density, molar mass, and how these factors influence the molecular kinetic energy observed as pressure.