A slightly different angle for this one to most questions - I demonstrate undergraduate level physics labs as a postgraduate student, but the problem is probably undergraduate level.

In a lab experiment, a piston with a magnet is placed within a sealed tube, and forced to oscillate by an electromagnet driven by an oscillating signal. At a certain driving frequency, the piston achieves resonance, its natural frequency being related to the forces of the magnets, gravity, and the force provided by the compression/rarefaction of the gas in the sealed tube as the piston moves. The position of the piston in the tube can be changed, varying the amount of air above and below the piston, which changes the natural frequency. By measuring the variation of natural frequency with the volume of air below the piston, the students are asked to derive a value for γ=Cp/Cv, the ratio of the heat capacities for air, based on the assumptions that air is an ideal gas and that the process is adiabatic.

The problem is that the experiment simply doesn't give the correct answer. Rather than giving the expected γ ~= 1.4 for air, the answer is nearer 0.4. This is an accepted problem with the experiment - i.e. it's not a case of the experiment being performed differently to the lab script! My suspicion is that there is a problem with the equations/assumptions presented in the lab book (getting 0.4 instead of 1.4 looks suspiciously like it could be solved by changing a γ to γ-1), but the re-derivation of those equations is eluding me.

It is stated that:

• P V^γ is constant for an adiabatic change in an ideal gas, P being pressure and V being the volume
• Kg = γPA2/V, the force per unit displacement of the gas, P being the pressure, A the area of the piston (i.e. cross section of the tube), and V the volume.
• f0 = 1/(2 π) √((Km 2Kg) / (M)), the natural frequency of the piston, Km being the force per unit displacement due to the magnet and gravity, Kg being the force due to the compression of the gas, and M the mass of the piston.
• f0² = (γPA²) / (2 π² M) 1/V (Km) / (4 π² M), which follows from the third equation by substituting in the second and squaring. By plotting resonance frequency squared against 1/V, this equation then describes a straight line of the form y=mx c, and γ can be derived from the gradient (the values of P, A, and M being known or measurable).

Getting the stated result in the second equation has me stumped. I can get the steps that P V^γ = P' V'^γ, where P' and V' are the pressure and volume after the piston has moved some distance x from the initial position with pressure P and volume V. Therefore, P = P' (V'/V)^γ, and Fg = P A = P' A (V'/V)^γ. V can be expressed as x A, where x is the position of the piston in the tube multiplied by the cross section of the tube. I suspect that the next step towards the second equation is to somehow pull down the power of γ, e.g. via a Taylor expansion, but being a non-integer value the γ never "disappears" in a Taylor expansion (you end up with powers of ~0.4, -0.6, and so on). Once you arrive at F'g = γ P A (V'/V) (assuming it's correct!), I think it's relatively straightforward to continue.