An ideal gas undergoes an adiabatic process. When the natural logarithm of its pressure ( P ) is plotted against the natural logarithm of its absolute temperature ( T ), the resulting graph is a straight line with a slope of 7 2 . The ratio of specific heats ( C_p C_v ) for this gas is :

Step-by-step solution

For an adiabatic process, the relationship between pressure and volume is given by: PV^ = constant Using the ideal gas equation PV = nRT , we can substitute V = nRT P to get the relation between P and T : P ( T P )^ = constant P^ 1- T^ = constant Taking the natural logarithm on both sides, we get: (1- ) P + T = constant Rearranging this into the equation of a straight line ( y = mx + c ) for P versus T : P = ( - 1 ) T + C The slope of this line is given as 7 2 . Equating the theoretical slope to the given value: -