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]

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.

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]

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

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

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 into find the answer.

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