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An ideal gas in initially at temperature \mathrm{T}  and volume \mathrm{V}. Its volume is increased by \mathrm{\Delta \mathrm{V}} due to an increase in temperature \mathrm{\Delta \mathrm{T}}, pressure remaining constant. The quantity \mathrm{\delta=\frac{\Delta \mathrm{V}}{\mathrm{V} \Delta \mathrm{T}}} varies with temperature as

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For an ideal gas, \mathrm{\mathrm{PV}=\mathrm{nRT}}

\mathrm{\therefore P d V=n R d T(\text { since } P=\text { constant }) }

\mathrm{ \text { Now, } \frac{d V}{d T}=\frac{n R}{P} }

\mathrm{\text { and } \frac{d V}{V d T}=\frac{n R}{P V}=\frac{n R}{n R T}=\frac{1}{T} }

\mathrm{ \Rightarrow \frac{\Delta V}{V \Delta T}=\frac{1}{T} \Rightarrow \delta T=1}

This is the equation of rectangular hyperbola

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vinayak

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