# Visualising gas temperature and gas pressure

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:

1. A gas is hot when the molecules collided with your measuring device.

2. 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. )

# Background

Let us assume we have a function, $f_{s}(\mathbf{x},\mathbf{v},t)$, which defines the number of particles of species $s$ in the following way: $$dN = f_{s}\left( \mathbf{x}, \mathbf{v}, t \right) \ d^{3}x \ d^{3}v$$ which tells us that $f_{s}(\mathbf{x},\mathbf{v},t)$ 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, $g(\mathbf{x},\mathbf{v})$, as: $$\langle g\left( \mathbf{x}, \mathbf{v} \right) \rangle = \frac{ 1 }{ N } \int d^{3}x \ d^{3}v \ g\left( \mathbf{x}, \mathbf{v} \right) \ f\left( \mathbf{x}, \mathbf{v}, t \right)$$ where $\langle Q \rangle$ is the ensemble average of quantity $Q$.

# Application

If we define a set of fluid moments with similar format to that of central moments, then we have: $$\text{number density [\# \ (unit \ volume)^{-1}]: } n_{s} = \int d^{3}v \ f_{s}\left( \mathbf{x}, \mathbf{v}, t \right) \\ \text{average or bulk velocity [length \ (unit \ time)^{-1}]: } \mathbf{U}_{s} = \frac{ 1 }{ n_{s} } \int d^{3}v \ \mathbf{v}\ f_{s}\left( \mathbf{x}, \mathbf{v}, t \right) \\ \text{kinetic energy density [energy \ (unit \ volume)^{-1}]: } W_{s} = \frac{ m_{s} }{ 2 } \int d^{3}v \ v^{2} \ f_{s}\left( \mathbf{x}, \mathbf{v}, t \right) \\ \text{pressure tensor [energy \ (unit \ volume)^{-1}]: } \mathbb{P}_{s} = m_{s} \int d^{3}v \ \left( \mathbf{v} - \mathbf{U}_{s} \right) \left( \mathbf{v} - \mathbf{U}_{s} \right) \ f_{s}\left( \mathbf{x}, \mathbf{v}, t \right) \\ \text{heat flux tensor [energy \ flux \ (unit \ volume)^{-1}]: } \left(\mathbb{Q}_{s}\right)_{i,j,k} = m_{s} \int d^{3}v \ \left( \mathbf{v} - \mathbf{U}_{s} \right)_{i} \left( \mathbf{v} - \mathbf{U}_{s} \right)_{j} \left( \mathbf{v} - \mathbf{U}_{s} \right)_{k} \ f_{s}\left( \mathbf{x}, \mathbf{v}, t \right) \\ \text{etc.}$$ where $m_{s}$ is the particle mass of species $s$, the product of $\mathbf{A} \mathbf{B}$ 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: $$\langle T_{s} \rangle = \frac{ 1 }{ 3 } Tr\left[ \frac{ \mathbb{P}_{s} }{ n_{s} k_{B} } \right]$$ where $Tr\left[ \right]$ is the trace operator and $k_{B}$ 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 $P_{s}$, even smaller values of $n_{s}$ can lead to large $T_{s}$. 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 $10^{23}$ 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 $P_{s}$ and even larger values of $n_{s}$, then one can have small $T_{s}$. 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 $10^{23}$ hitting a wall all at once, the net effect can be large.

By the Ideal gas law, $PV=nRT$, or "pressure times volume equals the number of molecules times a constant times temperature". So, all else being the same, as the temperature goes up, the pressure goes up in an exact ratio.

However, all else does not have to be the same. So, for instance, if you reduce the number of molecules in a container ($n$), the pressure ($P$) will go down even though the temperature ($T$) may stay the same.

Edit: A thermometer or pressure gauge measures the molecules that collide with it. A thermometer measures the average energy of the collisions. A pressure gauge measures the average collision energy times the number of collisions per second.

As an example, the pressure at the top of the Sun's photosphere is 0.86 millibar, or less than a thousandth of our air pressure at sea level. But, the temperature is far higher: 4400 Kelvin, or about fifteen times our air temperatures. (The sun's temperature is far higher as you go further out, but that's another story.)

• I understand the ideal gas law. But the question was what is the significant difference between Temperature and Pressure in the physical way. Commented Nov 14, 2015 at 4:20

Of course, they are relate to each other but that doesn't mean they are the same things.

Temperature is the average kinetic energy of the molecules while pressure is the force they exert perpendicularly on any surface. Of course, more the temperature, more would be the pressure.

While the former is related to the energy, the later is related to the momentum; they are different things.

Pressure is a measure of force per unit area exerted on the 'measuring device', while the temperature is a measure of kinetic energy of the individual molecules of the gas. Thus, high pressure can arise when there are either many slow moving molecules with low kinetic energy colliding with the container, or a few fast moving molecules colliding with the container. Going through the derivation of pressure of a gas using the kinetic theory of gases should help. Wikipedia link: https://en.wikipedia.org/wiki/Kinetic_theory#Pressure_and_kinetic_energy

An example of a difference where the pressure of a reasonably dilute gas depends on something else other than the kinetic energy of the particles is actually just the air on Earth. A classic exercise in statistical mechanics is to consider an ideal gas subject to gravity and find how the pressure varies with altitude.

Of course, in reality the temperature of the air on Earth varies with altitude, but doing this problem by assuming that the gas has a constant temperature provides a pretty reasonable result, that the pressure goes as $P(z) \sim \exp\{-mgz/kT\}$ (don't quote me on this) where $m$ is mean molecular mass. In this case, to a decent approximation, the pressure of the gas varies with height, but the temperature does not, because one now takes into account the gravitational potential and not just the kinetic energy.

1. A gas is hot when the molecules collided with your measuring device.

Not quite. Gas heats your measuring device when the collisions are mostly such that the colliding gas molecule has more kinetic energy than the colliding measuring device molecule.

It's instructive to think colliding molecules as sumo wrestlers: The molecule which has more momentum wins the bout, the winner does work on the loser by throwing him. Winner loses energy, loser gains energy.

The above rule works for straight head-on collisions. For other kind of collisions there are different rules. For example a molecule that experiences a collision on its rear gains energy. And a molecule with lot of kinetic energy rarely experiences rear collisions.

To measure somethings means to compare it with an etalon or a measurement instrument, made by the help of an etalon (or the combination of etalons).

To measure the pressure of a gas inside a volume one take for example a barometer and measures the pressure difference to the outer room. The measured pressure inside the volume is the result of the hitting of gas molecules with some average velocity and an average number of gas molecules on some area of the barometer.

To get the right correlation of how a volume contraction rise the pressure one has to do this contraction very slow. That allows to awoid the rise of the gas temperature (if the volume is not a thermal isolated system of course) and one get the right solution.

To study the relations between heating the gas and the temperatures rise one has to connect with the volume a second compensation volume and now it is possible to measure the temperature in The right Männer. If you take for this a mercury thermometer you could see, that this device is very similar to a mercury barometer. The scales or different.

Mercury thermometer and Mercury barometer (from Wikipedia)

So you are right that there are some similarities and pressure and temperature in closed volumes are somehow connected. Having different scales on it and changing the boundary conditions (or hold or pressure constant or temperature) one can measure temperature and pressure of the gas in a closed volume with the same measurement instrument.

Credits given to all answers posted. They helped me figured this out. Thanks a lot.

Temperature is heavily linked with Kinetic Energy.
Pressure is heavily linked with number of Collisions per Time AND Kinetic Energy.

Example:
A gas is hot when the molecules posses high Kinetic Energy and collides with the measuring device with great force.
A gas is hot not because there are a lot of molecules with low Kinetic Energy colliding with the measuring device. A lot of slow moving molecules does not add up to become Hot.

A gas is high pressured when there are a lot of molecules colliding with the wall either with High or Low Kinetic Energy. Higher Kinetic Energy creates more Pressure since change in momentum after each collision is high.

For short:
Hot gas need not to be pressurized. In other words, Low Pressured gas can still be Hot. This is because they just need to collide with enough force to transfer their Kinetic Energy while still remaining Low in pressure.

Pressurized gas need not to be hot. In other words, High Pressured gas can still be Cold. This is because they just need to collide frequently enough, Slow ( cold ) or Fast ( hot ).

• I dont think you have distilled the answers properly. If $n$ is constant, increasing in pressure results in an increase in temperature. Similarly, increasing temperature results in an increase in pressure. Two intensive properties fully define the state- they are not independent, as your answer suggests (to me). Commented Nov 15, 2015 at 4:59
• @theNamesCross Actually, I am trying to say Hot gas can still stay Low in pressure. The n ( moles ) are not kept constant. I'll try rephrase the answer. Commented Nov 15, 2015 at 8:43
• Correct, but you might explicitly state that in these cases $n$ is NOT constant. Also, remember phase diagrams- depending on the ($P, T$) values, it may not be in a gas phase. Commented Nov 15, 2015 at 15:25