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### Definition of enthalpy

The first law of thermodynamics postulates the conservation of energy of a closed system. The law states that the heat Q added to a closed contributes to the increase ΔU of the inner energy and the work W done by the system on its surroundings. This balance is written as follows:

Q = ΔU + W
(1)

In typical thermodynamical systems, the most common type of work is pressure-volume (P ⋅ V) work. Let us define the enthalpy H of a system as

H ≡ U + P ⋅ V
(2)

The change of enthalpy H during a process is:

ΔH = ΔU + Δ(P ⋅ V) = ΔU + ΔP ⋅ V + P ⋅ ΔV
(3)

For processes with constant pressure we have ΔP = 0, so:

ΔH = ΔU + P ⋅ ΔV
(4)

Comparing equations (1) and (4) we see that ΔH = Q for processes at constant pressure, so the change in enthalpy is simply equal to the heat released/absorbed by the process.

### Specific heat

Specific heats are defined as:

Cv =
∂U

∂T
at constant volume v
(5a)
Cp =
∂H

∂T
at constant pressure p
(5b)

We will remember:

ΔU = Cv ΔT
(6a)
ΔH = Cp ΔT
(6b)
ΔW = P ΔV
(6c)

For ideal gas (1 mol) we have P V = R T, so:

H = U + P V = U + R T
(7)

and

Cp =
∂H

∂T
=
∂U

∂T
+
∂ (R T)

∂T
= Cv + R
(8)

### Enthalpy in chemistry

Chemical reactions are thermodynamical processes that involve transfer of energy. They usually happen at constant atmospheric pressure of 1 atm. So we can apply the above equations.

• Endothermic reactions require an input of energy to proceed and are signified by a positive change in enthalpy ΔH > 0.
• Exothermic reactions release energy upon completion and are signified by a negative change in enthalpy ΔH < 0.