Final answer:
When the temperature of a diatomic gas is doubled, causing it to dissociate into atoms, the root mean square speed of the resulting atoms increases by a factor of the square root of 2. This occurs because the kinetic energy is conserved, but each atom now has half the mass of the original molecule.
Step-by-step explanation:
The root mean square (rms) speed of a molecule in an ideal gas is a measure of the molecule's typical speed, and it's closely related to the kinetic energy of the molecule. The formula for rms speed is derived from the kinetic energy equation KE = \( \frac{1}{2}mv^2 \), where KE is kinetic energy, m is mass, and v is the speed of the molecule. When the temperature is doubled, the kinetic energy also doubles because temperature is proportional to kinetic energy for gases.
Now, considering the diatomic molecule dissociates into two monatomic gases, each atom will have half the mass of the original diatomic molecule. However, since kinetic energy remains conserved, we can derive that the new rms speed of the atom, using Urms = \( \sqrt{ \frac{3kT}{m} } \), where k is the Boltzmann constant, T is temperature, and m is mass, will increase by a factor of \( \sqrt{2} \) due to the halving of the mass (m) and doubling of the temperature (T). Therefore, the new rms speed will be v' = v\( \sqrt{2} \).