Final answer:
To double the average speed of an ideal gas initially at 300 K, the temperature must be quadrupled to 1200 K due to the quadratic relationship between speed and kinetic energy.
Step-by-step explanation:
To find the temperature needed to double the average speed of the molecules of an ideal gas that is initially at 300 K, we can use the relationship between temperature and average kinetic energy of the gas molecules. The average kinetic energy of a molecule in an ideal gas is proportional to its absolute temperature. Specifically, the average kinetic energy KE is given by KE = (3/2)kT, where k is the Boltzmann constant and T is the temperature in kelvins.
Since the kinetic energy is also proportional to the square of the average speed (u) of the molecules (KE ∝ u^2), we can say that when the temperature is doubled, the kinetic energy is doubled, but to double the average speed, you need to quadruple the kinetic energy because of the squaring relationship. Therefore, to double the speed (which implies quadrupling the kinetic energy), we need to quadruple the temperature.
Thus, if the initial temperature is 300 K, and we want to double the average speed, we need to calculate: T2 = 4 * T1, which gives T2 = 4 * 300 K = 1200 K.
The correct answer to the question is (b) 1200 K.