Question

Which is larger: ΔS at constant pressure or ΔS at constant volume? Provide a conceptual rationale.

Short answer

ΔS at constant pressure is higher than ΔS at constant volume because heat transferred at constant pressure is higher than heat transferred at constant volume.

Step-by-step solution

ΔS Formula and comparison of ΔS at constant pressure and ΔS at constant volume

ΔS at constant pressure is higher than ΔS at constant volume.

The formula for ΔS is

$\mathbf{\Delta S}\mathbf{=}\frac{\mathbf{Heat}\mathbf{Transferred}}{\mathbf{Temperature}}$

Reason

Heat Transferred at constant volume(q_v) is equal to Internal energy change as work done at constant volume is zero.

${\mathbf{\Delta S}}_{\mathbf{v}}\mathbf{=}\frac{{\mathbf{q}}_{\mathbf{v}}}{\mathbf{T}}\mathbf{=}\frac{\mathbf{\Delta U}}{\mathbf{T}}$

While heat transferred at constant pressure (q_p) is equal to of the sum Internal energy change and work done.

${\mathbf{\Delta S}}_{\mathbf{p}}\mathbf{=}\frac{{\mathbf{q}}_{\mathbf{p}}}{\mathbf{T}}\mathbf{=}\frac{\mathbf{\Delta U}\mathbf{+}\mathbf{P\Delta U}}{\mathbf{T}}$

On comparing these two equations we can say that: ${\mathbf{\Delta S}}_{\mathbf{p}}\mathbf{>}{\mathbf{\Delta S}}_{\mathbf{v}}$.