Summary 5: Absolute Enthalpy, Enthalpy Changes and Reference States

The General Energy Balance Equation

The general energy balance equation can describe any process:

\[\frac{d(m_{sys}H_{sys})}{dt} = \sum_i \dot{m}_{in,i} H_{in,i} - \sum_j \dot{m}_{out,j} H_{out,j} + \sum_k \dot{Q}_k + \sum_l \dot{W}_l\]

To solve the energy balance, the enthalpy of the system, and of the flow in & out, must be known.

What affects Enthalpy?

Enthalpy is affected by:

  • Temperature changes
  • Pressure changes
  • Phase changes
  • Formation and separation of solutions
  • Reactions

The problem: absolute enthalpy

The balance asks for \(H_{in}\), \(H_{out}\) values, not changes.

  • No experiment measures the total energy stored in matter.
  • Every measurement is a change: heat added, work done, temperature risen.
  • So no absolute value of \(H\) exists to look up.

What we can calculate

Enthalpy changes between two states are always well defined:

\[\Delta H_{state\:1\rightarrow\:state\:2} = H_{state\:2} - H_{state\:1} \quad \checkmark\]

For a sensible heat effect:

\[\Delta H_{1\rightarrow 2} = \int_{T_1}^{T_2} C_P\,dT = \langle C_P \rangle\,(T_2 - T_1)\]

The reference state: choose a zero

Pick a reference state and set its enthalpy as zero

\[H_{ref} \equiv 0\]

Then any state has an “absolute” enthalpy measured from that reference:

\[H_{state\:1} = \underbrace{H_{ref}}_{=\,0} + \Delta H_{ref\:\rightarrow\:state\:1} = \Delta H_{ref\:\rightarrow\:state\:1}\]

An “absolute” enthalpy is really just a change — measured from a reference point.

Enthalpy relative to a reference state

Figure 1: Enthalpy of two states measured relative to a reference state at \(T_{ref}\)

Reference state calculations

\[H_1 = \displaystyle\int_{T_{ref}}^{T_1} C_P dT + H_{ref}\]

\[H_2 = \displaystyle\int_{T_{ref}}^{T_2} C_P dT + H_{ref}\]

\[\Delta H = H_2 - H_1 = \displaystyle\int_{T_1}^{T_2} C_P dT\]

The individual values \(H_1\) and \(H_2\) depend on the reference state. The difference \(H_2 - H_1\) does not.

Why path independence matters

Enthalpy is a state function — the path between two states does not change \(\Delta H\).

So every calculation can pass through the reference state:

\[\Delta H_{1\rightarrow 2} = \underbrace{\Delta H_{1\rightarrow ref}}_{-H_1} + \underbrace{\Delta H_{ref\rightarrow 2}}_{H_2}\]

The reference state should always cancel

Steady-state heater, one stream in, one out, no work:

\[\dot{Q} = \dot{m}\left(H_{out} - H_{in}\right)\]

With \(H = C_P (T - T_{ref})\) for both streams:

\[\dot{Q} = \dot{m}\left[C_P(T_{out} - T_{ref}) - C_P(T_{in} - T_{ref})\right]\]

\[\dot{Q} = \dot{m} C_P (T_{out} - T_{in})\]

\(T_{ref}\) has vanished. Any reference state gives the same answer.

So why bother with reference states?

If it cancels, why not always work with \(\Delta T\) directly?

  • With many streams at different temperatures, tabulating one \(H\) per stream is tidier
  • Tabulated data (steam tables, formation enthalpies) is already published relative to a fixed reference.
  • Once phase change, mixing or reaction appear, reference states keep the math straightforward.

Choosing a reference state

A reference state is freely chosen — but it must be fully specified:

  • Chemical species — which substances, in what form
  • Phase — solid, liquid or gas
  • Temperature, \(T_{ref}\)
  • Pressure, \(P_{ref}\)

Important

Use the same reference state for every enthalpy term in a calculation.

Common reference states

  • Triple point of water (0.01 \(^\circ\)C, liquid)
  • Pure species at 298.15 K, 1 bar
  • Dry air at 101.325 kPa and 0 \(^\circ\)C
  • Elements at 298.15 K, 1 bar

Choose one that is convenient for the data or problem.

Choosing well

Two rules of thumb:

  • Pick a state that makes as many terms as possible zero. If a feed enters at 25 \(^\circ\)C, choosing \(T_{ref} = 25\ ^\circ\)C makes that stream’s enthalpy zero and removes it from the balance.
  • Pick a state your data is already referenced to. Using steam tables means your reference is the triple point of water, whether you say so or not.

Always state the reference explicitly in the write-up.

When a reference state isn’t needed

A reference state is a tool, not a requirement.

The energy balance only ever contains enthalpy differences. If those differences can be written down directly, the reference state never has to be named.

Rule of thumb

If every enthalpy term in the balance can be paired up as \(H_{out} - H_{in}\) for the same species in the same phase, skip the reference state.

Case 1: Single stream, sensible heat only

Heating water from 20 \(^\circ\)C to 80 \(^\circ\)C:

\[\dot{Q} = \dot{m}\left(H_{out} - H_{in}\right) = \dot{m} C_P (T_{out} - T_{in})\]

No reference states are needed

Writing \(H_{in} = C_P(20 - T_{ref})\) and \(H_{out} = C_P(80 - T_{ref})\) gives the same answer

Case 2: The data is already a change

Much tabulated data is published as a \(\Delta H\), not an \(H\):

\(\Delta H_{vap}\) — latent heat of vaporisation

\(\Delta H^\circ_{rxn}\) — heat of reaction

\(\Delta H_{soln}\) — heat of solution or mixing

These are used directly. Boiling 2 kg/s of water at 100 \(^\circ\)C:

\[\dot{Q} = \dot{m}\,\Delta H_{vap} = 2 \times 2257 = 4514\ \text{kW}\]

Case 3: Build a path instead

Enthalpy is a state function, so any convenient path works — a reference state is just one particular choice of path.

Adiabatic reactor: react at \(T_{in}\), then heat the products to \(T_{out}\)

\[\Delta H_{sys} = \underbrace{\xi\,\Delta H_{rxn}(T_{in})}_{\text{reaction}} + \underbrace{\sum_k n_k C_{P,k}(T_{out} - T_{in})}_{\text{sensible heat}} = 0\]

Every term is a difference between two real states of the process.

Which method to use?

There is no hard and fast rule ::: {.incremental} - Simple problem — one or two streams, one species, no reaction? Work with differences

  • Complex problem — several streams, phase changes, reactions, recycles? Use a reference state

  • Steam tables or \(\Delta H^\circ_f\) data? These already use a reference state :::

Important

Whichever method is used, state it explicitly. An unstated reference state and an unstated path are equally hard to trobleshoot.

Example 5.1: Building enthalpies from a reference state

Air is heated in a steady-state duct heater from 25 \(^\circ\)C to 150 \(^\circ\)C at a mass flow rate of 0.50 kg s\(^{-1}\). \(C_P = 1.01\) kJ kg\(^{-1}\) K\(^{-1}\)

Using a reference state of air, gas, 0 \(^\circ\)C, 1 atm:

  1. Tabulate the absolute enthalpy of the inlet and outlet streams.
  2. Calculate the required heat, \(\dot{Q}\).

Example 5.2: Avoiding a reference state

Repeat the heat calculation of Example 5.1 without a reference state

Problem 5.1: Adiabatic mixing of two streams

Two liquid streams are combined continuously at steady state:

Stream Fluid \(\dot{m}\) (kg s\(^{-1}\)) \(T\) (\(^\circ\)C) \(C_P\) (kJ kg\(^{-1}\) K\(^{-1}\))
A Ethanol 1.50 20 2.44
B Water 0.80 75 4.18

There is no work or heat loss. Assume heat capacities are constant, and neglect the enthalpy of mixing.

  1. Choosing a reference state of the pure liquids at 20 \(^\circ\)C, find the outlet temperature.
  2. Repeat with a reference state of the pure liquids at 0 \(^\circ\)C.