Why Does RMS Velocity Increase During Adiabatic Compression?

[UCstudent]

rms velocity and compression. Please help!

Homework Statement

Explain why the rms velocity of a gas increases when it is adiabatically compressed.

Homework Equations

None

The Attempt at a Solution

I know that compressing a gas decreases the volume, but I don't know details on how it would effect rms velocity

 

[netgypsy]

Do you understand what adiabatically compressed means? If you don't, google it :-)

 

[gnulinger]

If you compress a gas adiabatically, no heat is transferred from the gas to its surroundings. You know that compressing a gas increases the temperature (and you can calculate by how much using your adiabatic relations).

$T*V^{γ-1} = constant$
$T_{0}*V_0^{γ-1} = T_1*(V_0 + ΔV)^{γ-1}$
Thus
$T_1 = T_{0}*(V_0/(V_0 + ΔV))^{γ-1}$
And
$ΔT = T_1 - T_0$

Then
$ΔE = (3/2)N k_b ΔT= (1/2)N m (Δv)^2$
Or
$Δv = \sqrt{(3/2) (k_b ΔT)/m}$

 

[netgypsy]

And I was going to make you figure it out yourself. :-(

 

[asteriatic]

. Please help!

The rms (root mean square) velocity of a gas is a measure of the average speed of the gas molecules. When a gas is adiabatically compressed, it means that the compression is done without any heat exchange with the surroundings. This type of compression is typically fast and does not allow enough time for the molecules to exchange energy with each other.

As a result, when the gas is compressed, the molecules are forced closer together, increasing the gas pressure and temperature. This increase in pressure and temperature causes the molecules to move faster, resulting in an increase in the rms velocity. This is because the molecules have less space to move around and collide with each other, leading to a higher average speed.

Additionally, the increase in temperature also leads to an increase in the kinetic energy of the molecules, which is directly related to their velocity. This increase in kinetic energy translates to an increase in the rms velocity of the gas.

In summary, adiabatic compression leads to an increase in the rms velocity of a gas due to the increase in pressure, temperature, and kinetic energy of the gas molecules.