ideal-gaspressurethermodynamics
In a container the pressure on the walls due to a gas can be calculated using the ideal gas law $\bigl(pV=NRT\bigr)$. However, for a column of water, or the pressure at earth's surface due to a column of air, the pressure can be calculated as $P=F/A=\rho gh$.
When does one use which model or is it two completely different principles?
For a liquid bridge between two spheres, would the water pressure be a function of the kinetic behaviour (ideal gas law) or the weight ($\rho gh$)?
Simplify the equation to make it more convenient to you by using the definition of a mole. A mole , generally denoted by 'n' is the mass of the substance taken divided my its molecular weight. On solving the only unknown in this equation , you get the mass of air contained in the volume you obtained. Now , you can just plug in this value into the definition of density,since density is the mass per unit volume. It would be wise to make sure the units of M and m are in the units you want the answer in or usually , we take the SI units. As pentane already mentioned, evaluating n and then solving for m is also a valid option. Note that as Ilja said the density of a substance doesn't depend on the mass taken or the volume independently, it is the mass of the gas contained per volume of that gas (in this case , air). The reason you are calculating mass here IS for the very same reason.So should you take a different mass or volume of air , the density doesn't change as the mass changes with volume to keep their ratio constant. I hope you get the general idea.
Remember that ideal gases do not interact. They don't interact with themselves or with other ideal gases, so in a mixture of ideal gases each component of the mixture behaves as if the other gases were not present.
So for every type of gas present the ideal gas equation holds:
$$ P_i = n_i\frac{RT}{V} \tag{1} $$
And the total pressure is just the sum of all these partial pressures:
$$ P_{tot} = \sum_i P_i = \sum_i n_i\frac{RT}{V} \tag{2} $$
This can be rewritten in various ways using:
$$ \rho_i = \frac{M_i n_i}{V} $$
where $M_i$ is the molecular weight of the gas $i$, so $M_i n_i$ is the total mass present of species $i$. For example we can rewrite this as:
$$ \frac{V\rho_i}{M_i} = n_i $$
and susbtitute this into equation (1) to get the equation you give:
$$ P_i = n_i\frac{RT}{V} = \frac{V\rho_i}{M_i}\frac{RT}{V} = \frac{\rho_i}{M_i}RT $$
However it isn't useful to talk about a partial molar volume as all the gas molecules are uniformly distributed. It would only make sense to talk about $V_i$ if the two gases were separated by a movable partition so they were at the same temperature and pressure but had different volumes:
Best Answer
Are you interested in the relative pressure difference between two points or the absolute pressure?
The first example is for absolute pressure, and the second example is for relative pressure. If I have a glass of water in front of me, the pressure difference from bottom to top is about 1/100th of the total absolute pressure.
Liquids also have a state function, similar to air. You can calculate the absolute pressure of a liquid from its temperature and density, it's just not a very accurate calculation since it's a sharply sloping function.
Likewise, you can use an analog to ρgh for gases. For constant gravity, I would write:
$$ \Delta P = \mu g \\ \mu = \int \rho dh $$
This would be (mostly) valid for calculating the relative difference in air pressure between two cities at different altitudes. If density is constant, this is the same as your ρgh approximation.