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Show that the Slope of Adiabatic Curve is more steeper than that of Isothermal Curve?

Today we are going to share another important topic, i.e. The slope of the Adiabatic Curve is more steeper than that of Isothermal Curve.

Show that the Slope of Adiabatic Curve is more steeper than that of Isothermal Curve.

The slope of the Adiabatic Curve is more steeper than that of Isothermal Curve: We know that for an Isothermal process:
           PV = K     ……..(i)
Differentiating eq.(i) we get ,P△V + V△P = 0

     or     P△V =  -V△P
     or    △P/△V = -P/V …….(ii)
Also for adiabatic process,
       PVγ= K    ……..(iii)
Differentiating eq.(iii) we get ,γ.PVγ-1△V + Vγ△P = 0
    or     γ.PVγ-1△V = – Vγ△P
    or     △P/△V = – γ.PVγ-1/ Vγ ……….(iv)
From Eq. (ii) and (iv) , where γ > 1
It is clear that{ Slope of adiabatic curve > Slope of Isothermal Curve }
∴ The adiabatic curve at any point is more steeper than Isothermal Curve.
The adiabatic & Isothermal curve in case of Compression & expansion is as follows:
Slope of Adiabatic Curve is more steeper than that of Isothermal Curve
Fig. (i) For expansion
Slope of Adiabatic Curve is more steeper than that of Isothermal Curve
Fig (ii). For compression
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