Thermodynamics question. It's a homework question but I'm not looking for help on the entire thing, just help with determining if I'm overthinking one of my steps. Pic related is the question. I'm start by trying to find either the pressure or volume at the second state, or the cross-sectional area of the piston (A). Any one will suffice.
I've already established the ratio of specific heats is 1.4 and for the purposes of this problem the compression was isentropic.
I start off with force balance. The internal gas and spring are pushing/pulling right, the ambient pressure is pushing left.
[math]p_2A + kx_1 = p_aA[/math]
Using isentropic relations I can rewrite [math]p_2 = p_1(\frac{v_1}{v_2})^\gamma[/math] and then I can also rewrite [math]v_2[/math] in terms of area:
[math]p_2 = \frac{p_1v_1^\gamma}{(v_1 - x_1A)^\gamma} [/math].

So I ultimately end up with:
[math]\frac{p_1v_1^\gamma}{(v_1 - x_1A)^\gamma}*A +kx_1 = p_aA[/math]

I can't solve this by hand and I didn't even consider trying. I solved it numerically for A. But I feel like I'm severely overthinking things. I don't feel like it's my professor's intent to have us solve for this numerically, and I feel as if I should've done something much simpler which leads me to believe I'm missing something very basic. Am I overthinking this or can anyone spot something I've missed? I've stopped myself from moving forward on this problem for 2 days because I'm certain I missed something.