Final answer:
The equation in question is derived from thermodynamic principles and involves the ratio of specific heats for an adiabatic process. The specific steps needed for the derivation are not provided, but they involve Poisson's law and the ideal gas law.
Step-by-step explanation:
The question relates to the derivation of the equation t₂/t₁ = (p₂/p₁)^((k-1)/k), which appears to be based on thermodynamics and the relationship between pressure, volume, and temperature in a gas, as described by adiabatic processes. Specifically, this equation seems to be a reformulation of Poisson's law, which describes the adiabatic expansion or compression of an ideal gas and involves the specific heat capacities at constant pressure and constant volume. The derivation requires an understanding of the ideal gas law and the relationship between the state variables of an ideal gas when it undergoes an adiabatic process. The index k (sometimes denoted as γ) represents the ratio of the specific heats (Cp/Cv), and it is characteristic of an adiabatic process where no heat is exchanged with the environment. Let's start by writing the ideal gas law for an initial state (p₁, V₁, T₁) and a final state (p₂, V₂, T₂). Next, we'll apply the adiabatic condition (pV^k = constant) to both the initial and final states and then rearrange to find the relationship involving temperatures and pressures. After that, we'll take logarithms and exponents to reach the final form of the equation. However, without more context, it's difficult to provide a step-by-step derivation here.