The initial average kinetic energy of the molecules was E , when a gas sample is at $27^{\circ} \mathrm{C}$. When the gas is heated to $327^{\circ} \mathrm{C}$, then the final average kinetic energy will be

The average kinetic energy of an ideal gas is directly proportional to its absolute temperature, expressed as $E_{avg} = \frac{3}{2}kT$, where $k$ is Boltzmann's constant and $T$ is in Kelvin.

Converting the given temperatures to Kelvin:
$T_1 = 27 + 273 = 300$ K
$T_2 = 327 + 273 = 600$ K

Since $E_{avg} \propto T$, the ratio of final to initial kinetic energy equals the temperature ratio:
$\frac{E_2}{E} = \frac{T_2}{T_1} = \frac{600}{300} = 2$

Final average kinetic energy: $E_2 = 2E$