Why (does/we assume) gas exert same pressure everywhere in a closed container? I was reading about the gaseous state when this question struck my mind: What made us assume that, at every point inside the container, a gas exerts equal pressure? When one brings a barometer, is it true it measures the same pressure at every point inside? Is this applicable to both ideal and real gases?
 A: 
I was reading gaseous state when this question struck my mind what
made us to assume that at every point inside the container gas exert
equal pressure ?

The equilibrium pressure of a gas, just like the equilibrium temperature of a gas, is a macroscopic property applicable to the collection of gas molecules inside the container, not a microscopic property applicable to individual gas molecules at every point inside the container.
Considering the walls of the container, while the impact forces of individual molecules on the walls of the container will vary, it is the average of the impact forces of a collection of molecules that determines the macroscopic property of pressure. Similarly it is the average kinetic energy of the molecules that determines the macroscopic property of the temperature of a gas, not the kinetic energies of the individual molecules which will vary above and below the average.
Hope this helps.
A: Consider a region of the fluid, and let $S$ be the surface of that region. If you take $\int_Sp\vec n$, where $\vec n$ is the normal vector, this is the net force that the pressure of the outside fluid is exerting on the mass of the fluid inside the region. If we take this force to be the only force acting on the fluid, then for the fluid inside the region to not be accelerating, this force must be zero. For this to be zero for every region, the pressure must be equal everywhere.
For fluids in a gravitational field, the force on the fluid is the pressure force plus its weight, so the pressure force must be equal in magnitude, and opposite in direction, to its weight. This pressure force is called "buoyancy", and there must be a gradient in the pressure to cause buoyancy to be equal to weight. However, for small differences in height, this change in pressure is small, and can be disregarded for many purposes.
A: An imbalance of pressure would itself cause an internal flow in the gas. So if the gas has reached equilibrium the pressure must be the same everywhere.
The above is for a gas in ordinary circumstances, without any applied field such as a gravitational field. If there is such a field then the gas flows until the pressure gradient provides a force which just balances the effects of the field.
To calculate these effects more fully one can use the concept of chemical potential and the second law of thermodynamics.
There remains the fact that thermodynamic quantities such as pressure also undergo fluctuations. The above comments about uniformity apply to the time-averaged pressure at any point.
Generalization to fluids
The arguments above apply to fluids more generally, not just to gases (and therefore it is not restricted to ideal gas). As long as the fluid can flow then any pressure gradient will cause a flow so when a fluid reaches equilibrium in a closed container the pressure must be uniform.
A: *

*The average kinetic energy of a gas molecule (temperature) is the same everywhere in the container, because when molecules of different energy collide, they are statistically likely to distribute the energy more evenly after the collision.  Temperature evens out. The average speed of the molecules is also therefore the same everywhere.


*The average velocity of gas molecules everywhere is zero, assuming there are no currents.  (in reality, although currents can indeed affect the relative pressure at different points, you need really strong currents to make a difference.  Currents caused by convection are too small).


*The average density of molecules is the same everywhere.  If you imagine any dividing plane, then given (1) and (2), if there were more molecules on one side, then there would be net flow to the other side.


*The average pressure at any point on the container's surface is proportional to the speed and density of gas molecules there, because it's the total rate of momentum transfer to the container wall, which is proportional to the molecule speed times the collision rate, and the collision rate is proportional to the speed and density, which we've already determined are the same everywhere.
A: As with so many things, it's a matter of how much detail you need in your model to answer questions about the phenomenon you're studying.
Local changes in pressure propagate at the speed of sound in the medium. Assuming the pressure is equal everywhere is a simplified model, but good enough to answer questions about anything that occurs on time scales orders of magnitude longer than it takes a pressure wave to propagate through the vessel. The difference might matter for modeling an explosion, but not for pumping up a bicycle tire.
A: It depends on the resolution of your measuring device.
A gas contains on the order of $10^{22}$ molecules zipping about. The pressure on a wall of the container is due to the tiny force applied by these molecules when they collide with the wall. If you could take a snapshot of each of the walls at an instant in time there would be a certain number of molecules colliding with each wall. However, this number would be different from wall to wall. If you could measure small enough pressures, there would be a difference. I don't know if we have devices that can measure pressures so sensitively.
On a macro scale, the differences from wall to wall are imperceptible. It suffices to say the pressure is constant at every point on the wall of the container.
A: 'Gas' is a continuum (i.e. modeled as 'there are no discrete gas particles' there are gas 'parcels'). That continuum is going to be in hydrostatic equilibrium. The density, and therefore pressure, does not change significantly for gas. The result is a constant pressure in any axis.
If you measure 'every point inside the container', its seems like you are interested in a scale below the scale of the continuum itself, at which point the conceptual model will breakdown.
A: Addition to other answers:
The assumption is NOT true for a container that is large enough along the direction of the local gravity vector for differences due to gravitational force on the molecules to matter.
eg if you make a container that is about 5000 metres tall (!!!) and with rigid walls then the pressure at the top will be about half an atmosphere (about 50 kPa) less than that at the bottom. Most containers are "rather less tall" than this and the differences are usually able to be neglected.
