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ab and bc curves in the figure represent...

ab and bc curves in the figure represent an isothermal and an adiabatic process respectively. Show that at the point b the ratio of the slopes of isothermal and adiabatic curves is equal to `C_P/C_V`.

Text Solution

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For an adiabatic process `PV^gamma` =constant
`therefore P_gamma V^(gamma-1) dV+V^gamma dP=0`
`therefore (P_gamma V^gamma)/V dV+V^gamma dP=0`
`therefore dP=-P_gamma/V dV`
`therefore ("dP"/"dV")_"adiabatic"=-gamma P/V`….(1)
For an isothermal process PV = constant,
`therefore VdP+PdV=0`
`therefore ((dP)/(dV))_"isothermal"=-P/V` ...(2)
`("dP"/"dV")_"adiabatic"/("dP"/"dV")_"isothermal"=(-gamma P/V)(-V/P)=gamma=C_P/C_V`
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