Question

Derive the relationship between ∆H and ∆U for an ideal gas. Explain each term involved in the equation.

Answer 1

We know that the change in internal energy equals the heat absorbed at constant volume, i.e., U = q_V.m. However, the majority of chemical reactions are carried out in flasks or test tubes under constant atmospheric pressure U = qp – PV under constant pressure, where qp is the heat absorbed by the system and – PV is the expansion work done by the system.

Let's call the beginning state with subscript 1 and the end state with subscript 2.

The above equation can be rewritten as

U₂ – U₁= P(V₂ – V₁)

On rearranging, we get

q_p = (U₂ + pV₂) – (U₁ + pV₁)   .......(1)

Now, we can define another thermodynamic function, the enthalpy H [Greek word enthalpies, to warm or heat content] as:

H = U + pV  ......(2)

So, equation (1) becomes

q_p = H₂ – H₁ = ΔH

Despite the fact that q is a path-dependent function, H is a state function because it is reliant on state functions U, p, and V. As a result, H is path-independent. As a result, qp is path-independent.

Equation (2) can be written as for finite changes at constant pressure.

ΔH = ΔU + ΔpV

Since, P is constant, we can write:

ΔH = ΔU + PΔV  ......(3)

It's worth noting that we're actually measuring changes in the enthalpy when heat is absorbed by the system at constant pressure. Remember ΔH= qp , heat absorbed by the system at constant pressure. H is negative for exothermic reactions, which produce heat, and positive for endothermic reactions, which absorb heat from the environment.

As a result, equation (3) becomes U = qv at constant volume (V = 0).

ΔH = ΔU = q_v

Let's have a look at a gaseous reaction. If V_A is the total volume of gaseous reactants, V_B is the total volume of gaseous products, n_A is the number of moles of gaseous reactants, and n_B is the number of moles of gaseous products, then we write using the ideal gas law.,

PV_A = n_ART

PV_B = n_BRT

Thus, PV_B – PV_A = n_BRT – n_ART = (n_B – n_A)RT

Or, P(V_B – V_A) = (n_B – n_A)RT

Or, pΔV = Δn_gRT  ......(4)

The number of moles of gaseous products minus the number of moles of gaseous reactants is denoted by ng.

Equation (3) is obtained by substituting the value of PV from equation (4) in equation (3).

ΔH = ΔU + Δn_gRT  ......(5)