Two samples of a gas initially at same temperature and pressure are compressed from a volume `V` to ` (V)/(2)` . One sample is compressed isothermally and the other adiabatically . In which sample is the pressure greater?

Explanation

Let `" "` Intial volume , `V_(1) = V` and pressure . `P_(1) = P , `
final volume, `V_(2)= (V)/(2)` and final pressure , `P_(2) = ?`
For isothermal compression `P_(2)V_(2) = P_(1)V_(1)` or `P_(2) = (P_(1)V_(1))/(V_(2)) = (PV)/(V/2) = 2P`
For adiabatic compression `P_(2)' = P_(1)[(V_(1))/(V_(2))]^(gamma) implies P_(2)' = P[(V)/(V//2)]^(gamma) = 2^(gamma) P implies P_(2)' = 2^(gamma) P gammagt1`
`therefore " " 2^(gamma) gt2` and `P_(2)'gtP_(2)`
Pressure during adiabatic compression is greater than the pressure during isothermal compression.