During a reversible adiabatic change, how does the temperature of a perfect gas change?

In a reversible adiabatic process involving a perfect gas, the relationship between temperature and volume is governed by the principles of thermodynamics, specifically the adiabatic condition where there is no heat exchange with the surroundings.

For an ideal gas undergoing a reversible adiabatic process, a key relationship involves the initial and final states of the gas defined by its temperature and volume. For a perfect gas, the following equation applies:

T_f * V^(γ-1) = T_i * V_i^(γ-1)

Where T represents temperature, V represents volume, and γ (gamma) is the heat capacity ratio (C_p/C_v). Rearranging this equation allows us to express the final temperature (T_f) in terms of the initial temperature (T_i) and the volumes:

T_f = T_i * (V_i / V_f)^(γ - 1)

In practical terms, this can be rewritten as:

T_f = T_i * (V_i/V_f)^(1/c)

Here, c is typically used somewhere similar to γ. This shows that the temperature decreases as the volume increases during an adiabatic expansion (when moving from an initial state to a final state) and vice versa.

Thus, the