Question

What if no expansion work is done; how can we calculate dA?

a) dA = dU - PdV
b) dA = dU + PdV
c) dA = dH - TdS
d) dA = -TdS

Answer 1

The correct equation to calculate the change in Helmholtz free energy
`(dA)` when no expansion work is done is
`dA = dU` , which is derived from option b)
`dA = dU + PdV` when PdV equals zero.

Step-by-step explanation:

When determining the change in Helmholtz free energy
`(dA)` under the condition that no expansion work is done the correct relationship to use is given by option b)
`dA = dU + PdV` .
This equation takes into account the change in internal energy
`(dU)` and the pressure-volume work, which in the case of no expansion is simply zero since
`PdV` represents the work involved with expansion or contraction. Thus if no work is done
`(PdV = 0), dA` simplifies to
`dA = dU` indicating that the change in Helmholtz free energy is equal to the change in internal energy under these specific conditions.

In this case, since the volume of the system does not change, the term
`PdV` is zero, so the equation reduces to
`dA = dU` . This indicates that the change in the Helmholtz free energy is identical to the change in internal energy when no work is being performed by the system (such as in an isochoric process).