Final answer:
Adiabatic and isothermal processes cannot intersect at two distinct points on a p-V diagram because the isothermal curve reflects a constant temperature and pressure-volume product, whereas the adiabatic curve shows a change in temperature and thus a more rapid drop in pressure after the initial intersection, making a second intersection impossible.
Step-by-step explanation:
To prove that adiabatic and isothermal processes cannot intersect at two points on a p-V diagram, we must first understand the characteristics of these processes. An isothermal process is characterized by a constant temperature; therefore, for an ideal gas, the product of pressure (p) and volume (V) remains constant, as described by the equation PV = nRT. On the other hand, in an adiabatic process, there is no heat transfer (Q = 0), leading to a change in temperature as the system does work.
On a p-V diagram, an isothermal curve for an ideal gas is hyperbolic, depicting the constant PV product. The adiabatic curve is steeper because it reflects a temperature change: during an adiabatic expansion, the gas cools as it performs work, leading to a decrease in temperature and therefore a lower pressure at a given volume compared to the isothermal case. Since the pressure on the adiabatic curve falls more rapidly compared to the isothermal curve, from any starting point A, the isothermal process will always have a higher pressure than the adiabatic process for a given volume after the initial intersection, preventing a second intersection point.