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Question
Show that the slope of the p−V diagram is greater for an adiabatic process compared to an isothermal process.
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Solution
In an isothermal process,
PV = k ...(i)
On differentiating it w.r.t V, we get
`"V" (dP)/(dV) + "P" = 0`
`(dP)/(dV) = -"P"/"V"`
`(dP)/(dV) = - "k"/"V"^2` [ Using (i)] , k = constant
k = constant
In an adiabatic process,
PVγ = K ...(ii)
On differentiating it w.r.t V, we get
`"V"^gamma(d"P")/(d"V")+ gamma"PV"^(gamma-1) = 0`
`(d"P")/(d"V") = -( gamma "P""V"^(gamma-1))/"V" ^ (gamma+1) ["Using" (ii) , γ > 1 ]` and
K is constant
`gamma and (d"P")/(d"V")`
are the slope of the curve and the ratio of heat capacities at constant pressure and volume, respectively; P is pressure and V is volume of the system.
By comparing the two slopes and keeping in mind that γ >1 , we can see that the slope of the P-V diagram is greater for an adiabatic process than an isothermal process.
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