"The total kinetic energy of the molecules in an ideal gas with a volume V at pressure P and temperature T is equal to the total kinetic energy of the molecules present in the same volume of another ideal gas at the same pressure and at temperature 2T"-Justify the statement.

Explanation

Suppose, the number of molecules present in volume V of the first gas=`n_(1)`. If the root mean square velocity of the molecules in the first gas is `c_(1)`, and the mass of each molecule is `m_1`, then from kinetic gas equation we have,
`PV=(1)/(3)m_(1)n_(1)c_(1)^(2) or, PV=(2)/(3)n_(1)xx(1)/(2)m_(1)c_(1)^(2)`
or, `PV=(2)/(3)E_(1)` [where `E_(1)=n_(1)xx(1)/(2)m_(1)c_(1)^(2)`=total kinetic energy of the molecules.]
Similarly, if the number of molecules in volume V of the second gas=`n_(2)`, mass of each molecule`=m_(2)` and root mean square velocity of molecules=`c_(2)`, then-
`PV=(1)/(3)m_(2)n_(2)c_(2)^(2)=(2)/(3)n_(2)xx(1)/(2)m_(2)c_(2)^(2) or, PV=(2)/(3)E_(2)` ltBrgt [where `E_(2)`=total kinetic energy of the molecules of secondd gas]
As, P and V of both the gases are equal, so, `E_(1)=E_(2)`.