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

[fluidistic]

Homework Statement

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.

Homework Equations

Some.

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.

 

[anathema]

Thank you for your post. It seems like you have made some progress in solving the problem. To find the period of the motion, you will need to solve the differential equation you have set up. This can be done by using standard methods for solving second-order differential equations, such as separation of variables or using the characteristic equation. Once you have obtained a solution for x(t), you can find the period by finding the time it takes for the piston to complete one full oscillation, which is the time it takes for x(t) to return to its initial value.

I would also like to point out that in order to find the frequency, you will need to divide the period by 2π. This is because frequency is defined as the number of oscillations per unit time, and the period is the time it takes for one full oscillation. So, the frequency will be 1/2π times the inverse of the period.

I hope this helps. Good luck with your calculations!

 

[nlightner]

I would first like to commend you on your attempt at solving this problem and utilizing the relevant equations. Your approach is correct in setting up the differential equation for the motion of the piston. However, finding the frequency of oscillation would require solving this equation, which may be difficult without additional information such as the initial conditions and specific values for the variables.

In general, the frequency of oscillation for a system with a small mass attached to a spring (in this case, the piston) is given by the equation f = 1/(2π) * √(k/m), where k is the spring constant and m is the mass. In your case, the spring constant would be related to the pressure and volume of the gas in each compartment, so it may be necessary to further manipulate the equation to solve for k.

Additionally, the conditions of the experiment being adiabatic or isothermal may affect the frequency as well. In an adiabatic process, there is no heat exchange with the surroundings, so the temperature may change as the gases expand and compress. In an isothermal process, the temperature remains constant. This may affect the pressure and volume of the gases, and therefore the frequency of oscillation.

Overall, without more information and specific values, it may be difficult to determine the frequency of oscillation in this system. I would recommend consulting with a professor or reference material for further guidance on solving the differential equation and finding the frequency.