The building blocks of ordinary matter are atoms and molecules.  Molecules are two or more atoms held together by a chemical bond.  Atoms themselves are made up of more fundamental particles, i.e. protons, neutrons, and electrons.  In a solid, the atoms and molecules are densely packed and held in place by intermolecular forces.  The atoms in a solid can be modeled as being held together by tiny springs that permit them to vibrate back and forth about their equilibrium position, but not to exchange positions with other atoms.  Solids are nearly incompressible.  In a liquid the atoms and molecules are also densely packed.  They cannot easily escape from one another, but they are free to move with respect to each other.  Liquids are nearly incompressible.  In a gas, intermolecular forces are weak and short ranged, and the atoms and molecules can move about nearly independently.  Gases are compressible.  Gases and liquids are fluids, i.e. collections of atoms or molecules that are free to move with respect to each other.

Assume we have a collection of gas molecules in a container in gravity-free space.  Then each molecule moves along with constant velocity in a straight line, until it hits another molecule, or a container wall.  A collision between two molecules is similar to a collision between two balls.  The molecules exchange momentum, but the total momentum of the two molecules is conserved.  When a molecule hits a wall, it bounces back.  Its momentum changes.  To change the molecule's momentum, the wall must exert a force on the molecule.  Newton's third law tells us that the molecule exerts a force on the wall.  The greater the number of molecules hitting a wall, the greater is the force on the wall.  In a container with different size walls, the bigger walls will receive more hits that the smaller walls and therefore experience a greater force.  We define the pressure P in the container as the magnitude of the normal force F on a wall divided by the surface area A of the wall.  In gravity free space the pressure is the same everywhere inside the container.

P = F/A

In SI units, the units of pressure are N/m² = Pa (pascal).

The faster the molecules move in the container, the greater is the change in momentum when they bounce off a wall, and the more often do they hit the walls.  

Assume a molecule moves horizontally with speed v back and forth between two infinitely-massive walls, which are a distance L apart.  When it hits the right wall its momentum changes from p₁ = +mv to p₂ = -mv.  The change in the molecule's momentum is Dp_mol = p₂ - p₁ = -2mv.  The time interval between successive hits on the right wall is Dt = 2L/v.  So the average force the wall exerts on this molecule is F_mol = Dp_mol/Dt = -2mv/(2L/v) = -mv²/L.  By Newton's third law, the average force that the molecule exerts on the wall is F_wall = mv²/L, it is proportional to the square of the speed of the molecule.

The force exerted by a gas on the walls of a container is proportional to the square of the speed of the molecules, the pressure is therefore proportional to the kinetic energy of the molecules.  Not all the molecules have the same kinetic energy.  The pressure in a container is proportional to the average kinetic energy of the molecules.  If all the molecules in a container would be at rest, their kinetic energy would be zero and the pressure would be zero.  The pressure is also proportional to the particle density r_particle of the molecules in the container.  The particle density is the number of particles (molecules) in the container, divided by the volume of the container.  If we double the number of molecules in the container, the pressure will double.

The density r of a substance is defined as its mass per unit volume, r = M/V.

Our general definition of the average pressure exerted by a normal force F on an area A is P = F/A.  If the pressure varies from point to point, then the pressure at a specific point exerted on a surface element dA is given by P = dF/dA, where dF is the normal force acting on dA.

Problems:

Liquids in a constant gravitational field

Consider a large pool of water on the surface of the earth and a box-shaped volume of water at some depth in the pool.  Imagine it enclosed by some weightless container.

The volume of water is in equilibrium and stays in place.  It does not rise and it does not fall.  The net force on it must be zero.  The vertical component of the net force is

F_net = P_bottomA - P_topA - Mg = 0.
F_net = P_bottomA - P_topA - rhAg = 0.
P_bottom - P_top = rhg.

The pressure in the pool increases with depth.  If we let h denote the vertical distance of a point below the surface of the water, then we can write the pressure at this point as

P = P₀ +  rgh.

P is the pressure at depth h and P₀ is the pressure at the surface.  Very often this pressure is atmospheric pressure.  The atmospheric pressure at sea level at room temperature is approximately

1 atmosphere = 101 kPa = 14.7 pounds per square inch (psi).

The pressure at a point below the surface of a liquid in a constant gravitational field depends only on the depth of that point and the pressure at the surface.  Any change in the pressure at the surface is therefore transmitted to every point in the liquid.  This is called Pascal's law.

Problems:

Buoyancy

Now consider again the box-shaped volume of water in equilibrium at some depth in the pool.  The upward force provided by the surrounding water must exactly balance the force of gravity acting on the water in the box.  The upward force provided by the surrounding water must be equal to the weight of water in the box.

If we replace the volume of water with a box of the same shape containing some other material, then the net upward force provided by the surrounding water does not change.  It depends only on the difference in the pressure at the top and at the bottom of the box.  But the weight of the box changes, and therefore the net force on the box changes.  If the weight is greater than that of the corresponding volume of water, the net force is downward and the box will accelerate downward and fall.  If the weight is less than that that of the corresponding volume of water, the net force is upward and the box will accelerate upward and rise.  This is Archimedes' principle.  It holds for all fluids, i.e. it holds for all liquids and gases.

An object partially or wholly immersed in a gas or liquid is acted upon by an upward buoyant force B equal to the weight w of the gas or liquid it displaces.

B = w

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