If the absolute temperature of a gas is quadrupled, what happens to the root-mean-square speed of the molecules?

The root-mean-square speed measures the average speed of the gas molecules. The relation of root-mean-square speed and the absolute temperature is based on the kinetic molecular theory of gases.
The root-mean-square speed define as ν_rms 
where:
ν_rms = √(3RT/M)
R = universal gas constant; 0.8206 L-atm/mol-K
T = absolute temperature
M = molecular weight of gas particles

So, if the temperature of the gas is quadrupled the root-mean-square speed will be doubled.
Proof:
Since T is quadrupled, then T=4T
Substitute to the formula of root-mean-square speed,
ν_rms = √[3R(4T)/M]
          =   2√(3RT/M)       since the square root of 4 is 2
Therefore, root-mean-square speed is doubled when the absolute temperature is quadrupled.

Determine the  wavelength of light emitted when an electron  in an li2+ ion  makes a transition from an orbital with n  =  6 to an orbital with n  = 5.

☆ionisation energy increases across a period from na -> ar☆ionisation energy decreases down a group from li -> fr☆ionic radius decreases across a period from na -> cl☆ionic radius increases down a groupfrom f -> ii hope it clarifies your doubts