# System of a gas separated with a piston, frequency of oscillation

1. Apr 19, 2009

### fluidistic

1. The problem statement, all variables and given/known data
In a chamber whose section is $$A$$ and length is $$L$$ has two compartments separated by a mobile piston whose mass is $$m$$.
When the system is in equilibrium, the left compartment has $$N_1$$ particles of an ideal gas in a volume of $$AL_1$$ while the right compartment has $$N_2$$ particles of the same gas in a volume of $$AL_2$$.
If the gases are at the temperature $$T_0$$ and the piston is moved by a distance $$\Delta x$$ from its equilibrium position, what will be the frequency of oscillation if :
1)The conditions of the experiment are adiabatic
2)The conditions of the experiment are isothermal.

2. Relevant equations

Some.

3. The attempt at a solution

Let's do part 2).
I've calculated the force exerted by the gas on the piston in each compartment and as they have opposite direction I can write their sum as $$A \left ( \frac{P_0V_0}{V_0+A\Delta x} \right) - \left( \frac{N_2 k_B T_0}{L_2 - \Delta x} \right )$$.
Thanks to Newton it's the same as writing $$m \frac{d^2x}{dt}=A \left ( \frac{P_0V_0}{V_0+A\Delta x} \right) - \left( \frac{N_2 k_B T_0}{L_2 - \Delta x} \right ) \Leftrightarrow \frac{d^2x}{dt}=[A \left ( \frac{P_0V_0}{V_0+A\Delta x} \right) - \left( \frac{N_2 k_B T_0}{L_2 - \Delta x} \right ) ] \frac{1}{m}$$. Which is the differential equation of motion of the piston. I don't have a clue about how to find the period of the motion, in order to reach the frequency.
Sorry about not posting an image but it seems that PF is experiencing problems and it doesn't work. (Nor reading latex. I see latex images as ones I already posted a month ago).
Thank you for helping.