Rüchardt experiment equation derivation

[CodyZim]

Homework Statement

The piston is initially at the top of the tube and the air pressure is equal to P0 (ambient pressure).
After it is released and drops to location x, the pressure can be computed as follows. Assuming
the air is an ideal gas with constant specific heats and assuming the process is adiabatic and
reversible, we have

P₀(V₀ + L*A_p)^k = P(V₀ + x*A_p)^k

where x is measured from the bottom of the tube.
For L*Ap /V0 << 1, show that

P - P₀ = k * P₀ * A_p * (L - x) / V₀

Homework Equations

(1-1):
P₀(V₀ + L*A_p)^k = P(V₀ + x*A_p)^k

(1-2):
P - P₀ = k * P₀ * A_p * (L - x) / V₀

PV^k = constant

k = Cp/Cv

The Attempt at a Solution

I tried binomial theorem, and rearranging to solve for P - P₀. I've taken up to second level Differential equations and I cannot think or find a way to get the equation to equal that when I set L*Ap / V0 = 0.

L*Ap / V0 = 0.
P* (x * A_p / V₀)^k - P₀ = 0

Binomial theorem:

(x + y)ⁿ = (n/k) xⁿ * y⁰

I realized right away this method wouldn't work, because the n in my case isn't determined yet, but I know its a constant.

thats as far as I get to determining the equation.

Any help is much appreciated! Thanks!

 

[mfb]

k is not even an integer.

You can use a taylor series.