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)`.