# Difference between pressure and temperature

If I am given the average kinetic energy of the molecules of a gas or a liquid, how can I tell if the fluid will burn me/crush me/both if I immerse my hand in it?

Equivalently, what is the difference between heat transfer and momentum transfer at a molecular level?

Both pressure and temperature can be thought of as forms of kinetic energy density, but they are divided over different quantities. Pressure is proportional to kinetic energy per unit volume, while temperature is proportional to kinetic energy per particle. The conversion factor between the two measures (per-volume vs. per-particle) is the number density (particles per unit volume). A gas can be at high temperature and low pressure if it has low number density; likewise, a gas can be at low temperature and high pressure if it has high number density. The relationship between these three quantities (pressure, temperature, density) is contained in the equation of state for the material. For ideal gases, this equation of state is the ideal gas law, $P=\frac{N}{V}kT$.

• Thank you for your answer. I can see the difference mathematically for a gas, but I don't see the physics. As in why does few high energy particles cause something to heat up and many lower energy particles cause something to be crushed? How would all this play out for a liquid? – jumpmonkey Sep 7 '18 at 15:22

If I am given the average kinetic energy of the molecules of a gas or a liquid, how can I tell if the fluid will burn me/crush me/both if I immerse my hand in it?

Kinetic energy of the molecules reasonably defines the temperature of fluids: more directly for gases, less directly for liquids.

Given the temperature of a fluid, the thermal effect of immersing your hand in it will mostly depend, on the thermal conductivity of the fluid, which, along with its temperature, will determine how quickly the heat will be transferred from the fluid to your skin or vice versa, depending on the temperature difference.

This could be expressed as $q=-k\nabla T$, where $q$ is heat flux density, $k$ is thermal conductivity of the fluid and $\nabla T$ is temperature gradient.

As an example, under normal conditions, thermal conductivity of water is about $0.6\space Wm^{-1} K^{-1}$, while thermal conductivity of air it is only about $0.03\space Wm^{-1} K^{-1}$.

This is why $80^{\circ}C$ water would be unbearable, while $80^{\circ}C$ suana will is just fine.

Although pressure and temperature in fluids are intimately related, high pressure does not necessarily imply high temperature (bottom of the ocean) and low pressure does not necessarily imply low temperature (boiling water in a kettle), so one can be crushed without being burned or burned without being crushed.

I would add that, while the pressure in the ideal gas is caused by collisions and depends on the frequency of collisions and kinetic energy of gas molecules, pressure in real fluids, and, particularly, in liquids, is, substantially, due to the repulsion between molecules and could increase under compression forces even when the temperature and, therefore, the kinetic energy of molecules is relatively low.

Equivalently, what is the difference between heat transfer and momentum transfer at a molecular level?

For a momentum transfer at a molecular level to translate into significant heat transfer at a macro level, there have to be a lot of molecular level transfers per unit contact area, per unit of time, which, among other things, could be affected by fluid density.

For instance, thermal conductivity of air decreases as its density decreases.