Thermodynamics of a spontaneous isothermal chemical reaction. Which of the following is/are correct for spontaneous isothermal
  chemical reaction?
  
  (A) $\Delta H = 0$,  because $\Delta T = 0$
  
  (B) $\Delta S = 0$
  
  (C) $\Delta U = 0$,  because $\Delta T = 0$
  
  (D) $\Delta G &lt; 0$


My answer: $\mathrm{A, C, D}$

Reasoning: 

$U$ is a state function and only depends on the initial and final states and their temperature. Since $\Delta T = 0$, $\Delta U = 0$

$\Delta H = \Delta U + \Delta(PV)$, but $PV = \text{constant}$, so $\Delta H = 0$

$\Delta G &lt;0$ - the requirement for any reaction to be spontaneous. 

But answer given is only option D. 

I think the flaw in my $\Delta H = 0$  argument is that reactants can be solids and liquids as well and we cannot apply $PV = \text{constant}$ to them. Am I right? 

But I  still can't figure out why $\mathrm C$ is wrong.
The internal energy and enthalpy of a pure ideal gas depends only on temperature.  But, the internal energy and enthalpy of an ideal gas mixture depends both on temperature and on the amount of each chemical species that is present.  To be more precise, it is equal to the internal energy and enthalpy per mole of each component as if it were a pure species times the number of moles of that species present, summed over all the species in the mixture:  $$U=\sum{n_iu_i}$$and$$H=\sum{n_ih_i}$$So, even at constant temperature, $$\Delta U=\sum{(\Delta n_i)u_i}$$and$$\Delta H=\sum{(\Delta n_i)h_i}$$You should have been taught this when they were covering the thermodynamics of mixtures and chemical reactions.