Final answer:
Without a change in temperature or additional information relating internal energy to pressure for helium, we cannot accurately calculate the change in internal energy of a helium-filled toy balloon with a gauge pressure of 0.200 atm. Thus, we're unable to provide the greater internal energy value with the information given.
Step-by-step explanation:
To determine how much greater the internal energy of the helium in a helium-filled toy balloon is at 0.200 atm gauge pressure compared to zero gauge pressure, we can use the ideal gas law and the fact that internal energy for an ideal gas is related to its temperature. Since gauge pressure is the pressure above atmospheric pressure, at 0.200 atm, the absolute pressure is 1.200 atm (assuming atmospheric pressure is 1 atm). The change in internal energy ΔU associated with a change in volume at constant temperature can be derived from ΔU = nC₀ΔT, where n is the number of moles, C₀ is the molar heat capacity at constant volume, and ΔT is the change in temperature.
We're given volume (10.0 L) and gauge pressure (0.200 atm), but since we're assuming constant temperature and ideal gas behavior, there is no change in internal energy provided by a change in temperature.
However, the question presupposes a method to compare the internal energy, which is not directly provided by the given information since we need either a change in temperature or specific data on how the internal energy is related to pressure for helium. Given only pressure and volume without a temperature change or additional properties, we cannot calculate a change in internal energy.
Thus, we do not have sufficient information to provide a numerical answer to the question as stated. It's important to remember that for an ideal gas and fixed number of particles, any change in internal energy would be associated with a change in temperature, which we do not have here.