For which of the following is deltaH equal to deltaE(internal energy)?

Well, only #(2)#, but only if we assume constant atmospheric pressure...

1. #2"CO"(g)+"O"_2(g) -> 2"CO"_2(g)#
2. #"H"_2(g)+"Br"_2(g) ->2"HBr"(g)#
3. #"C"(s)+2"H"_2"O"(g) ->2"H"_2+"CO"_2#
4. #"PCl"_5(g) ->"PCl"_3(g)+"Cl"_2(g)#

By definition,

#H = E + PV#

where #H# is enthalpy, #E# is internal energy, and #P# and #V# are known from the ideal gas law.

Taking the change, we get:

#color(green)(DeltaH) = DeltaE + Delta(PV)#

#= DeltaE + P_2V_2 - P_1V_1#

#= DeltaE + (P_2 - P_1 + P_1)(V_2 - V_1 + V_1) - P_1V_1#

#= DeltaE + (DeltaP + P_1)(DeltaV + V_1) - P_1V_1#

#= DeltaE + P_1DeltaV + V_1DeltaP + DeltaPDeltaV + cancel(P_1V_1 - P_1V_1)#

#= color(green)(DeltaE + P_1DeltaV + V_1DeltaP + DeltaPDeltaV)#

Now, we have assumed that pressure is constant, so:

#DeltaH = DeltaE + P_1DeltaV#

Under this assumption, if #DeltaV = 0# too, then #DeltaH = DeltaE#. Liquids are generally incompressible, and so are solids. Thus, if the mols of gas don't change, the volume of the system pretty much doesn't change.

Hence, we can say #color(blue)(DeltaH ~~ DeltaE)# #color(blue)" if "# #color(blue)(Deltan_"gas" = 0)#.