If r denotes the ratio of adiabatic of two specific heats of...

If $r$ denotes the ratio of adiabatic of two specific heats of a gas. Then what is the ratio of slope of an adiabatic and isothermal $\mathrm{P} \rightarrow \mathrm{V}$ curves at their point of intersection ?
(A) $(1 / \gamma)$
(B) $\gamma-1$
(C) $\gamma$
(D) $\gamma+1$

Key Concepts

P-V Curve Slopes

The slope of a pressure-volume (P-V) curve represents the rate at which pressure changes with volume. For an adiabatic process, this slope is steeper than that of an isothermal process because the pressure decrease is more rapid with increasing volume. The slopes, derived from the respective process relations, reveal that at the point of intersection, the ratio of the slope of the adiabatic curve to that of the isothermal curve is equal to the ratio of specific heats, ?.

Adiabatic Process

An adiabatic process is one in which there is no heat transfer into or out of the system. In thermodynamics, during an adiabatic transformation, the changes in pressure and volume are governed by the relation PV^? = constant, where ? is the ratio of specific heats. This relation implies that pressure and volume change in such a way that any work done is at the expense of the internal energy of the gas, affecting the slope of the P-V curve.

Isothermal Process

An isothermal process is a thermodynamic process conducted at a constant temperature. For an ideal gas undergoing an isothermal change, the pressure and volume obey the relation PV = constant. Since the temperature remains constant, the interdependence between pressure and volume is different from that in an adiabatic process, resulting in a distinct slope on the P-V curve.

Ratio of Specific Heats (?)

The ratio of specific heats, denoted by ?, is a fundamental property of a gas defined as the ratio of its specific heat at constant pressure to that at constant volume. This ratio plays a key role in adiabatic processes, determining how pressure and volume change relative to each other. It directly influences the steepness of the adiabatic curve compared to the isothermal curve on a P-V diagram.

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