CBSE-XI-Physics
27: Specific Heat Capacities of Gases
- #7Show that the slope of the p-V diagram is greater for an adiabatic process compared to an isothermal process.Ans : In an isothermal process,
PV = k ...(i)
On differentiating it w.r.t V, we get
`` V\frac{\,\mathrm{\,d\,}P}{\,\mathrm{\,d\,}V}+P=0``
`` \frac{\,\mathrm{\,d\,}P}{\,\mathrm{\,d\,}V}=-\frac{P}{V}``
`` \frac{\,\mathrm{\,d\,}P}{\,\mathrm{\,d\,}V}=-\frac{k}{{V}^{2}}[\text{U}\,\mathrm{\,sing\,}(\,\mathrm{\,i\,}\left)\right],\text{k = constant}``
k = constant
In an adiabatic process,
`` P{V}^{\gamma }=K....\left(\,\mathrm{\,ii\,}\right)``
On differentiating it w.r.t V, we get
`` {V}^{\gamma }\frac{\,\mathrm{\,d\,}P}{\,\mathrm{\,d\,}V}+\gamma P{V}^{\gamma -1}=0``
`` \frac{\,\mathrm{\,d\,}P}{\,\mathrm{\,d\,}V}=-\frac{\gamma P{V}^{\gamma -1}}{{V}^{\gamma }}``
`` \frac{\,\mathrm{\,d\,}P}{\,\mathrm{\,d\,}V}=-\frac{\gamma K}{{V}^{\gamma +1}}[\text{U}\,\mathrm{\,sing\,}(\,\mathrm{\,ii\,}\left)\right],\gamma >1\text{and}``
K is constant
`` \gamma \text{and}\frac{\,\mathrm{\,d\,}P}{\,\mathrm{\,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 `` \gamma >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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