Grabeh ang galing ko pala sa algebra!
Naderive ko kasi yung equation for the adiabatic reaction sa thermochem! (nosebleed!)
Nyahahaha! Pinasulat saken ni Ma'am Quinto yung solution ko:

Equations:
Cvln(T2/T1) = - nRln (V2/V1) ---(1)
Cp - Cv = nR ---(2)
PV = nRT ---(3)
§ = Cp/Cv ----(4)

Derive:
P1V1§ = P2V2§

Steps:
1. Substitute (2) in terms of R to (1)

Cvln(T2/T1) = -(Cp - Cv)ln (V2/V1)

Cvln(T2/T1) = -Cpln (V2/V1) + Cvln (V2/V1) ----(5)

2. Transpose the Cv to the left side, then factor Cv.

Cv[ln(T2/T1) - ln (V2/V1)]= -Cpln (V2/V1)

3. Divide both sides by Cv then substitute (4)

ln(T2/T1) - ln (V2/V1)= §ln (V2/V1) ----(6)

4. Substitute (6) with (3) in terms of T

ln(P2V2/P2V2) - ln (V2/V1) = §ln (V2/V1) ----(7)

5. Simplify (7) using rules on logarithms

ln(P2/P1) + ln(V2/V1) - ln(V2/V1) = ln (V2/V1)-§

ln(P2/P1) = ln (V1§/V2§) ----(8)

6. Simplify the natural logarithms in (8).

eln(P2/P1) = eln (V1§/V2§)

P2/P1 = V1§/V2§

7. Simplifying further.

P1V1§ = P2V2§