Thermodynamics- total kinetic energy and rms velocity

xxx23

Find total kinetic energy and root mean square velocity of the molecules of 10 liters of helium gas at an equilibrium pressure of 105 N m−2. Density of helium is 0.1786 gram/liter.


I am having trouble with the question, I know:

K=(Nm<v^2>)/2

PV=(Nm <v^2>)/3

but I am not sure how to apply this.

Any help would be appreciated.

Andrew Mason

Can you find the temperature? What is the relationship between temperature and kinetic energy?

AM

xxx23

I have partial solutions from my tutorial but they are not making too much sense...

From the above equations, it goes:

K= 1/2 Nm <v>^2 so if divide each side by V
K= 1/2 ρ V <v>^2

from here the solution goes:

PV=1/3 Nm <v>^2

K=3/2 PV I can see that 2K=3PV

but i do not understand where the following equations come from:


K=1/3 ρ V <v>^2 because this is what i need to rearrange to find Vrms.

Andrew Mason

You did not answer my questions.

AM

xxx23

I know that as the temp increases so does the kinetic energy.

1/2 m <v>^2 = 3/2 KT

Andrew Mason

You haven't answered my first question: what is the temperature of the Helium? (hint: assume it is an ideal gas).

AM

arkyy

I'm not sure if you still need help with this question, but I'll show how I'd work it out:

Firstly, to find the RMS (root mean squared) velocity of the molecules, we'll start with your equation,

PV=(Nm <v^2>)/3 Now we don't know N, and while we could calculate it, its easier to rearrange this equation to form,
P = (Nm<v^2>)/3V (multiplying both sides by V)
P = (1/3)p<v^2> (using Nm/V = p (density) )

Now we know p the density, and P the pressure so rearrange
<v^2> = 3P/p
<v^2> = 3 x 105 / (0.1786 x 10^-3 / 10^-3 ) [ensuring p is in kg/m^3]
<v^2> = 1763.7178...
RMS = 42.0 ms^-1 (3sf) [square root the <v^2>]

Secondly to find the total Kinetic energy

Start with what we have, so P = Nm<v^2> / 3V
and K= 1/2 Nm <v^2>

So 3PV = Nm<v^2>
3PV/2 = 1/2 Nm<v^2>
and since K = 1/2 Nm<v^2>
We can state that K = 3/2 PV and can now substitute your values in to find the answer.

There are many ways of solving these as they are all rearrangements of the same equation.