Yes.
I tried to convince myself it was the closest answer (which would be a 100 times different than ##13.82~k_B##) before you suggested the one state population scenario.
And you argued that that it is most probable for the bosons to occupy the same state. Which gives only one most probable arrangement, when there is no constraint of temperature involved.
So, I think you're right and zero is the correct choice, I think.
There are six options given
##13.82~k_B##
##693.1~k_B##
##zero##
##1000~k_B##
##5909.693~k_B##
##6909~k_B##
And no the question does not ask for entropy as a function of temperature.
Thanks for your suggestion @PeterDonis :bow:
The question clearly says "the temperature of ##n=19~\rm {mol}## of argon gas ..." why did you use ##n=1##?
How did you find the change in temperature to add with?
So, ##\Delta T## is change in temperature, right?
Now, say, initially the temperature is ##T_i## and finally ##T_f##.
What is the change in temperature?
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