A mass of diatomic gas $(\gamma = 1 .4)$ at a pressure of $2$ atmospheres is compressed adiabatically so that its temperature rises from $27^o C$ to $927^o C.$ The pressure of the gas in the final state is ...... $atm$
AIPMT 2011, Medium
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For an adiabatic process
$\frac{{{T^\gamma }}}{{{P^{\gamma - 1}}}} = constant$
$\therefore \,{\left( {\frac{{{T_i}}}{{{T_f}}}} \right)^\gamma } = {\left( {\frac{{{p_i}}}{{{p_f}}}} \right)^{\gamma - 1}}\,\,;\,\,{p_f} = {p_i}{\left( {\frac{{{T_f}}}{{{T_i}}}} \right)^{\frac{\gamma }{{\gamma - 1}}}}$ $...(i)$
$Here,\,{T_i} = {27^ \circ }C = 300\,K,\,{T_f} = {927^ \circ }C = 1200\,k$
${p_i} = 2\,atm,\,\gamma = 1.4$
Substituting these values in eqn $(i)$, we get
${P_f} = \left( 2 \right){\left( {\frac{{1200}}{{300}}} \right)^{\frac{{1.4}}{{1.4 - 1}}}}$
$ = \left( 2 \right){\left( 4 \right)^{1.4/0.4}} = 2{\left( {{2^2}} \right)^{7/2}} = \left( 2 \right){\left( 2 \right)^7} = {2^8} = 256\,atm$
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