Relation between energy annd pressure

[Haseeb Ali]

Me and and my friend were having discussion about the motion of molecules of gas.We talked about their velocity ,kinetic energy and much more. He asked me to derive a relation between pressure and energy. I was unable to explain him that... Can anyone explain the relation?

 

[Bystander]

He asked me to derive a relation between pressure and energy.

How would you begin? What information do you have?

 

[Haseeb Ali]

How would you begin? What information do you have?

P=density*Vx*2

T= 2/3k 1/2mv*2

 

[BvU]

You could read up on the gas law, e.g. here
Your equations feature in there, but one appears molecular to me and the other macroscopic:

${1\over 2} mv^2 = {3\over 2} kT$ average kinetic energy for a molecule.

$P = \rho v_x^2 = {mass\over V} v_x^2$ may be ok if in the right context.

I can wrangle a bit with formulas: $$ {1\over 2} mv^2 = {3\over 2} kT\ \Rightarrow\ {1\over 2} mv_x^2 = {1\over 2} kT$$ (the energy is equally distributed over the three degrees of freedom),

$\displaystyle \ \Rightarrow\ v_x^2 = {kT\over m}$

mass = number of molecules * mass of a molecule = number of moles * $N_A$ * mass of a molecule (Avogadro number); write mass = $n\; N_A\; m$

Leaves

$pV = n\;N_A\;m \ {kT\over m} = n\; N_A \;kT$;

then use

$R_G = N_A \; k$ (gas constant) to get the ideal gas law

$pV = nR_GT$.

pressure times volume has the dimension of energy.

It's not the whole story, but quite a big part of it.

 

[HalfLostAndSearching]

The relation between energy and pressure in a gas is described by the ideal gas law, which states that the pressure of a gas is directly proportional to its temperature and the number of molecules present, and inversely proportional to its volume. This can be represented by the equation: P = (nRT)/V, where P is pressure, n is the number of moles of gas, R is the gas constant, T is the temperature, and V is the volume.

In terms of energy, the kinetic energy of the gas molecules is directly related to their velocity, which is represented by the root mean square velocity (urms) in the equation: urms = √(3RT/M), where M is the molar mass of the gas. This shows that as the temperature (T) increases, the kinetic energy and velocity of the gas molecules also increase.

As the kinetic energy of the gas molecules increases, they collide more frequently and with greater force against the walls of the container, resulting in an increase in pressure. This is because pressure is defined as the force per unit area, and the increased collisions of the gas molecules against the container walls result in a higher force per unit area.

Therefore, there is a direct relationship between the energy of gas molecules and the pressure of the gas. As the energy of the gas molecules increases, so does the pressure. This is why increasing the temperature of a gas also increases its pressure, as seen in the ideal gas law equation.

In summary, the relation between energy and pressure in a gas is explained by the ideal gas law, which shows that as the kinetic energy of the gas molecules increases, so does the pressure of the gas. This relationship is important in understanding the behavior of gases and is a fundamental concept in thermodynamics and fluid mechanics.