We know that

Qp = ΔH and qv = ΔE

At constant pressure

ΔH = ΔE + PΔV ----------------------------i

Where ΔV is the change in volume, thus above equation can be written as

ΔH = ΔE + P (V2 – V1)

=ΔE + (PV2 – PV1) ---------------------------ii

Where V1 is the initial volume and V2 is the final volume of the system

For ideal gases:

PV = nRT

So we have

PV1 = n1RT

PV2 = n2RT

Here n1 is the number of moles of the gaseous reactants and n2 is the number of moles of the gaseous products.

Substituting these in equation ii, we get

ΔH = ΔE + (n2RT – n1RT)

=ΔE + (n2 – n1) RT

ΔH = ΔE + Δng RT

Where Δng = n2 – n1 is the difference between the number of moles of the gaseous products and those of the gaseous reactants.

Substituting the value of ΔE and ΔH the above equation becomes

qp = qv + Δng RT

The above equation gives us the relationship between heat of reaction at constant pressure and that at constant volume.