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QUESTION

# What's the difference between isothermal process, adiabatic process, and isovolumetric process?

They essentially mean:

• isothermal ~ no change in temperature
• adiabatic ~ no heat flow involved
• isovolumetric ~ no change in volume

These are generally used as assumptions to simplify an experiment, or simply to put a situation into a more manageable problem in a chemistry course. Specifically, you should see something like this in a Physics course or a Physical Chemistry course (among others, if applicable).

I've written out some examples of what you can figure out by knowing that the situation is assumed to be such that these terms hold true.

First, some definitions:

• DeltaU is the internal energy of a system.
• DeltaH is the of a system.
• q_"rev" is the reversible/most efficient heat flow that can occur.
• w_"rev" is the reversible/most efficient work that can be performed or that something else can perform upon you.

And the following are some equations we'll end up using (Physical Chemistry: A Molecular Approach, McQuarrie).

Enthalpy vs. Internal Energy:

\mathbf(DeltaH = DeltaU + Delta(PV))

First Law of Thermodynamics:

\mathbf(DeltaU = q + w)

ISOTHERMAL PROCESS

Here, DeltaT = 0.

For an ideal gas, that automatically means the change in internal energy color(blue)(DeltaU = 0) and the change in enthalpy color(blue)(DeltaH = 0), because for an ideal gas, the internal energy and enthalpy are only dependent on the temperature.

By using this equation, you can determine the relationship between heat flow and work now, and it greatly simplifies the problem:

cancel(DeltaH) = cancel(DeltaU) + Delta(PV)

= Delta(PV)

= PDeltaV + VDeltaP

color(blue)(w_"rev" = -PDeltaV = VDeltaP)

Furthermore, using DeltaU = q + w, the first law of thermodynamics:

0 = q_"rev" + w_"rev"

color(blue)(q_"rev" = -w_"rev" = -VDeltaP)

You couldn't say these were true unless it was an ideal gas at isothermal conditions! In my opinion, this is one situation that I find fairly straightforward.

Here, color(blue)(q = 0), so using the first law of thermodynamics again:

color(blue)(DeltaU) = cancel(q_"rev")^(0) + color(blue)(w_"rev" = -PDeltaV)

Additionally, using the enthalpy equation from earlier:

DeltaH = DeltaU + Delta(PV)

= w_"rev" + PDeltaV + VDeltaP

= -cancel(PDeltaV) + cancel(PDeltaV) + VDeltaP

Thus, color(blue)(DeltaH = VDeltaP) when a process upon an ideal gas is adiabatic.

ISOVOLUMETRIC PROCESS

A similar situation arises when DeltaV = 0, because it means expansion/compression work color(blue)(w_"rev" = 0) (see the above usages of w = -PDeltaV?):

color(blue)(DeltaU = q_V)

where q_V is (presumably reversible) heat flow at a constant volume.

However, it does not matter for enthalpy because if you recall from the adiabatic process, let us work backwards from the relation DeltaH = VDeltaP. If it was not adiabatic, q ne 0, thus:

color(blue)(DeltaH = q_V + VDeltaP)

which does not depend on DeltaV.

This can be a fairly challenging situation, because you don't have any easy relationship where you can just do a simple integration of a dT term, or a dV term, or similar (giving e.g. PDeltaV, VDeltaP, etc).

Unless you know the following relationships, it might be difficult to figure this out in full.

color(blue)(DeltaH) = int_(T_1)^(T_2) C_pdT

= color(blue)(C_p(T_2 - T_1))

where C_p is the constant-pressure heat capacity (doesn't have to be use in a constant-pressure situation though). For an ideal gas, it is assumed to be a constant across small temperature ranges.

From determining DeltaH, you can fairly easily determine q, and thus DeltaU. Conveniently, you also have this relationship:

color(blue)(DeltaU) = int_(T_1)^(T_2) C_VdT

= color(blue)(C_V(T_2 - T_1))

where C_V, the constant-volume heat capacity (doesn't have to be use in a constant-volume situation though), is based on the degrees of freedom for an ideal gas. Without going too much into this, C_V = 3/2 R for a monatomic ideal gas, where R is the universal gas constant, and C_p = C_V + nR.

CHALLENGE: What do you imagine will be the relationships for DeltaU, DeltaH, q_"rev", and w_"rev" for an isobaric process? Hint: It means constant pressure, and it is similar to the isovolumetric process.