Gas pressure is created when gas molecules collide with the wall of the container creating a force. Gas temperature is a measure of how fast the molecules are moving / vibrating.
However, they both seem to be concerned by "kinetic energy" of the molecules, or in other words, the "collision" they impose on the target. How do we visualize the difference between pressure and temperature of gas? Is there any obvious difference between the two?
The same question in another form:
A gas is hot when the molecules collided with your measuring device.
A gas have high pressure when the molecules collided with your measuring device.
So, what is the difference between the two "collisions" in the physical sense and how do we visualize the difference?
For Simplicity,
How can a Hot gas be Low Pressured? ( They are supposed to have High Kinetic Energy since it is Hot. Therefore should be High Pressured at all times! But no. )
How can a High Pressured gas be Cold? ( They are supposed to collide extremely frequently with the walls of the container. Therefore should be Hot at all times! But no. )
Answer
Let us assume we have a function, fs(x,v,t), which defines the number of particles of species s in the following way: dN=fs(x,v,t) d3x d3v which tells us that fs(x,v,t) is the particle distribution function of species s that defines a probability density in phase space. We can define moments of the distribution function as expectation values of any dynamical function, g(x,v), as: ⟨g(x,v)⟩=1N∫d3x d3v g(x,v) f(x,v,t) where ⟨Q⟩ is the ensemble average of quantity Q.
If we define a set of fluid moments with similar format to that of central moments, then we have: number density [# (unit volume)−1]: ns=∫d3v fs(x,v,t)average or bulk velocity [length (unit time)−1]: Us=1ns∫d3v v fs(x,v,t)kinetic energy density [energy (unit volume)−1]: Ws=ms2∫d3v v2 fs(x,v,t)pressure tensor [energy (unit volume)−1]: Ps=ms∫d3v (v−Us)(v−Us) fs(x,v,t)heat flux tensor [energy flux (unit volume)−1]: (Qs)i,j,k=ms∫d3v (v−Us)i(v−Us)j(v−Us)k fs(x,v,t)etc. where ms is the particle mass of species s, the product of AB is a dyadic product, not to be confused with the dot product, and a flux is simply a quantity multiplied by a velocity (from just dimensional analysis and practical use in continuity equations).
In an ideal gas we can relate the pressure to the temperature through: ⟨Ts⟩=13Tr[PsnskB] where Tr[] is the trace operator and kB is the Boltzmann constant. In a more general sense, the temperature can be (loosely) thought of as a sort of pseudotensor related to the pressure when normalized properly (i.e., by the density).
How can a Hot gas be Low Pressured?
If you look at the relationship between pressure and temperature I described above, then you can see that for low scalar values of Ps, even smaller values of ns can lead to large Ts. Thus, you can have a very hot, very tenuous gas that exerts effectively no pressure on a container. Remember, it's not just the speed of one collision, but the collective collisions of the particles that matters. If you gave a single particle the enough energy to impose the same effective momentum transfer on a wall as 1023 particles at much lower energies, it would not bounce off the wall but rather tear through it!
How can a High Pressured gas be Cold?
Similar to the previous answer, if we have large scalar values of Ps and even larger values of ns, then one can have small Ts. Again, from the previous answer I stated it is the collective effect of all the particles on the wall, not just the individual particles. So even though each particle may have a small kinetic energy, if you have 1023 hitting a wall all at once, the net effect can be large.
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