The symbols in the following list that represent state functions or are equivalent to state functions are? 1. #S# 2. #H# 3. #q_V# 4. #w#

#D)# is always true, but #B)# is sometimes true, requiring that either non-PV work is zero or the process has constant temperature and volume AND is reversible.

• If non-PV work is zero, then #q_V# is a state function.
• If non-PV work is NOT zero, then #q_V# is a state function only if the process is isothermal (#DeltaT = 0#) and reversible (100% efficient).

Furthermore, only for an ideal gas would this isothermal, constant-volume heat flow be nonzero, because #DeltaU# is a function of only temperature only for ideal gases.

In reality, nothing is reversible, so only the first scenario guarantees that #q_V# becomes a state function.

The first law of thermodynamics is:

#dU = deltaq + deltaw#

where #U# is the internal energy, #q# is heat flow, and #w# is work. #d# indicates a state function and #delta# indicates a path function.

Heat #q# and work #w# are path functions (which depend on how a process is performed), while #U# is a state function (which only depends on initial and final states, not the steps).

However, there is such a thing as non-PV work, #w_"non-PV"#, such that

#deltaw = deltaw_"PV" + deltaw_"non-PV"#

where #deltaw_"PV" = -PdV#.

At constant volume, #q = q_V#, so that #deltaw_"PV" = 0#. Therefore:

#color(blue)(overbrace(dU)^"State function" = overbrace(deltaq_V)^"Path function" + overbrace(deltaw_"non-PV")^"Path function")#

Non-PV work could be, e.g. electrical work, work against gravity, work against friction. This is most applicable to a process that changes the Helmholtz free energy at constant temperature and volume:

#color(green)(dA >= deltaw_("non-PV"))#, #" "##color(green)"constant T and V"#

where "#>#" refers to irreversible and "#=#" refers to reversible non-PV work.

[This reference provides a proof for the Gibbs' free energy being:

#dG >= deltaw_("non-PV")#, #" "##"constant T and P"#]

Helmholtz free energy is a state function, but could POTENTIALLY be equal to the non-PV work if the process ALSO is at constant temperature. It could be equal IF the process is also reversible.

Our conclusion is:

• If non-PV work is zero, then #q_V# is a state function.
• If non-PV work is NOT zero, then #q_V# is a state function only if the process is isothermal (#DeltaT = 0#) and reversible (100% efficient).

Furthermore, only for an ideal gas would this isothermal, constant-volume heat flow be nonzero, because #DeltaU# is a function of only temperature only for ideal gases.

In reality, nothing is reversible, so only the first scenario guarantees that #q_V# becomes a state function.