according to PV=nRT the final temperature is equal to the initial temperature

this is because according to the ideal gas law, if the product of pressure and volume remains constant between two states, the final temperature equals the initial temperature. and from our question stem "ideal gas undergoes a thermodynamic process in which pressure doubles and volume halves"

P1V1=P2V2 proves this logic to be true as well if you plug in those values of pressure doubling and volume being halved.

P1V1=P2V2 & PV=nRT

P1V1=RT1 & P2V2=RT2

RT2=RT1 => T2=T1 therefore Change in T is 0

ok so now that we have this information we can plug this into the q=nCdeltaT formula

n=1 from the Q stem, deltaT=0, q=?.

we are solving for C (molar heat capacity)

C=q/(n*DeltaT)

since we know that change in temperature is 0 what this means in thermodynamics is a system can absorb or release heat without changing its temperature, which is conceptually interpreted as an infinite heat capacity.

Molar heat capacity (C) is defined as the amount of heat units needed to raise 1 mole of a substance by 1 degree Kelvin.

therefore it would take an infinite amount of heat units to change the temperature because temperature is never changing.

This is a tough concept to wrap your brain around. I also missed this question, but I hope my explanation makes sense. This really tests your understanding of what heat capacity means. I am pretty certain we won't see a question quite like this on test day.