about 0.5 cubic meters. For our purposes, we can treat the ball like it is full of nitrogen (N2).
If some joker replaces the gas in the ball with hydrogen (H2) at the same pressure and temperature, then how does the internal energy of the ball change? How do the velocities of the molecules within in the ball change?
The internal energy depends upon the temperature of the gas and the amount of gas, or the pressure of the gas and the volume of the gas (it depends on how you write out the equation). Either way, when you have a fixed volume for the beach ball, and you replace the nitrogen gas with a hydrogen gas at the same temperature and pressure, you have the same internal energy (pressure, volume, number and temperature are all the same).
Now, the rms-speed of the gas depends upon the temperature of the gas and the mass of the individual molecules. The speed DECREASES with an INCREASE in gas mass. Since hydrogen has LESS mass than nitrogen, the rms-speed will be GREATER.
the problem is a little confusing with respect to the internal energy. You can ask your professor about it, but it seems to imply that the volume remains the same. That's sort of the most important aspect of this question. The internal energy depends only upon the temperature of the gas and the number of gas molecules. You can use the ideal gas law to show that if a new gas is introduced at the same pressure and temperature, and occupies the same volume, it HAS to have the same number of molecules.