Why slope of Adiabatic curve is more than the Isothermal curve in a PV Graph ?
Unlike the adiabatic process, there is no exponent (γ) involved in the isothermal equation, resulting in a less steep curve on the PV graph.
In a PV (pressure-volume) graph, the adiabatic curve typically has a steeper slope compared to the isothermal curve. This difference in slope arises from the different thermodynamic processes represented by each curve and their respective relationships between pressure, volume, and temperature.
1. Adiabatic Process:
— In an adiabatic process, there is no heat exchange with the surroundings (Q = 0). This means that the change in internal energy (ΔU) of the system is solely due to work done on or by the system <Q=ΔU+W, Q=0>
— The adiabatic equation for an ideal gas is given by: PV^γ = constant, where γ is the heat capacity ratio (specific heat at constant pressure divided by specific heat at constant volume).
— For an adiabatic process, γ > 1. This means that the adiabatic curve on a PV graph is steeper than the isothermal curve because the pressure-volume relationship is more sensitive to changes in volume (because it is raised to the power γ) for adiabatic processes.
2. Isothermal Process:
— In an isothermal process, the temperature of the system remains constant (ΔT = 0). This means that any change in internal energy (ΔU) due to work done on or by the system is balanced by an equal amount of heat exchange with the surroundings.<Q=ΔU+W, ΔU=0 because ΔT = 0>
— The equation for an isothermal process is: PV = constant. Unlike the adiabatic process, there is no exponent (γ) involved in the isothermal equation, resulting in a less steep curve on the PV graph.
In summary, the steeper slope of the adiabatic curve compared to the isothermal curve on a PV graph reflects the greater sensitivity of pressure to changes in volume in adiabatic processes, which is influenced by the heat capacity ratio (γ) of the gas.
