How do I get the enthalpy of combustion from the change in temperature of the surrounding water in a bomb calorimetry experiment?

Well, the Maxwell Relations are generally a good place to start. I get for a usual bomb calorimeter experiment:

#DeltaH_C ~~ DeltaU_C + Deltan_"gas"RT_"room"#

where:

• #DeltaU_C# and #DeltaH_C# are KNOWN to be in #"kJ/mol"#.
• #n# is the mols of gas on each side of the BALANCED combustion reaction on the scale of your experiment's reaction.
• #T_"room"# is the temperature of the surroundings in #"K"#.
• #R = "8.314472 J/mol"cdot"K"# is the universal gas constant.

for the reaction:

#"CH"_4(g) + 2"O"_2(g) -> 2"H"_2"O"(l) + "CO"_2(g)#

And remember that if your units don't work, it will not be right! Be sure to critically evaluate your units!

CHALLENGE: What are the units of #Deltan_(gas)RT_(ro om)# as-written?

For systems that conserve mass and energy, the Maxwell Relation for the internal energy is

#dU = TdS - PdV#

and the Maxwell Relation for the enthalpy is

#dH = TdS + VdP#,

where #T# is temperature, #S# is entropy, #P# is pressure, and #V# is volume.

Therefore, for conservative systems, we must have:

#TdS = dU + PdV = dH - VdP#

As a result, we obtain the relationship between enthalpy and internal energy:

#dH = dU + PdV + VdP#

Using the reverse product rule,

#dH = dU + d(PV)#

And thus, for non-infinitesimal changes:

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

Here, the #Delta(PV)# term represents

• a change to the initial system volume at a constant initial pressure (#PDeltaV#), followed by
• a change to the initial system pressure at a constant final volume (#VDeltaP#), followed by
• a change to both the initial system pressure and volume at the same time (#DeltaPDeltaV#),

i.e. #Delta(PV) = PDeltaV + VDeltaP + DeltaPDeltaV#.

In ordinary lab bench conditions, the atmospheric pressure is constant, so...

#DeltaH ~~ DeltaU + PDeltaV#

The change in system volume is most significantly due to gases by a long shot, so assuming #"CO"_2# is the ideal gas formed in the combustion reaction

#"CH"_4(g) + 2"O"_2(g) -> 2"H"_2"O"(l) + "CO"_2(g)#,

at constant system volume and at a constant temperature of the surroundings,

#color(blue)barul(|stackrel(" ")(" "DeltaH_C ~~ DeltaU_C + Deltan_"gas"RT_"room"" ")|)#

where:

• #n# is the mols of gas on each side of the BALANCED combustion reaction on the scale of your experiment's reaction.
• #T_"room"# is the temperature of the surroundings in #"K"#.
• #R = "8.314472 J/mol"cdot"K"# is the universal gas constant.

Note that you must base it off of the scale of your reaction... you cannot simply say that #Deltan_"gas" = -"2 mols"#.