Biogeochemical cycles and chemical reservoirs

A cycle connects chemical reservoirs through transfers

A reservoir contains an inventory of a constituent within a chosen boundary. A flux describes its transfer per time across a boundary or between reservoirs. State whether an inventory counts molecules, an element such as carbon, or a mass-equivalent convention.

Chemical reactions can change species while conserving the chosen element. Physical transport can move a species without changing its identity. A cycle diagram should distinguish those operations and identify what enters or leaves the represented system.

Inventory and concentration have different balances

Take an original hypothetical reservoir containing 1,260 mol of carbon atoms, written mol C. For five days assign constant input 84.0 mol C day⁻¹ and output 63.0 mol C day⁻¹, including every counted carbon form within the boundary. Net accumulation is 21.0 mol C day⁻¹, so final inventory is 1,365 mol C. The separate integrated input and output are 420 and 315 mol C.

The initial stock/output quotient is 1,260/63.0=20.0 days. This changing stock is not at steady state, and the quotient is neither the age of every carbon atom nor a trajectory for a well-mixed exponential system. If output instead equals input at 84.0 mol C day⁻¹, the stock remains 1,260 mol C and its quotient is 15.0 days, despite continuing throughput.

For a separate invented mixture, combine 0.650 m³ at 3.20 mmol C L⁻¹ with 1.85 m³ at 0.800 mmol C L⁻¹. Stipulate additive volumes, conservation of dissolved total carbon and no reaction or gas exchange. Convert cubic metres to litres: the amounts are 2,080 and 1,480 mmol C.

Their total 3,560 mmol C in 2,500 L gives 1.424 mmol C L⁻¹. An unweighted concentration mean would be 2.00 mmol C L⁻¹ and would ignore the unequal volumes. Adding 0.500 m³ carbon-free water changes volume to 3,000 L and concentration to 1.18666667 mmol C L⁻¹ while retaining the same carbon amount. This dilution does not destroy carbon.

A steady reservoir can have continuing throughput

The change of an inventory equals inputs minus outputs plus any formation or loss included in its definition. If all relevant terms balance, the inventory can remain steady while material continuously enters and leaves. A constant concentration additionally depends on the reservoir volume.

A residence-time estimate from inventory divided by an output flux belongs to its stated turnover model. It is not automatically an individual molecule’s age, a response time after perturbation or a universal lifetime independent of mixing and transport.

Chemical forms and time averages need their own definitions

Specify another original inventory: 12.0 mol CO₂, 5.00 mol CH₄ and 0.800 mol calcite CaCO₃. Each formula unit contains one carbon atom, so total inventory is 17.8 mol C. This is a declared phase inventory, with no claim that the components equilibrate. Knowing the elemental total alone does not identify its chemical forms.

The formal net CH₄+2O₂→CO₂+2H₂O balances C₁H₄O₄ and zero charge on each side. Assign extent 2.50 mol solely for bookkeeping: CO₂ becomes 14.5 mol, CH₄ becomes 2.50 mol, calcite stays 0.800 mol; the carbon total remains 17.8 mol C. The net consumes 5.00 mol O₂ and forms 5.00 mol H₂O. Neither the equation nor this extent supplies a reaction rate or a handling procedure.

For a separate stationary model, feed 15.0 mol C day⁻¹ into two independent parallel well-mixed first-order pools. Fractions 0.800 and 0.200 enter pools with assigned exponential times τ₁=4.00 days and τ₂=36.0 days. There is no transfer between pools; age means time since entry. At steady state, each stock is its input flux times its τ: 48.0 and 108 mol C, total 156 mol C.

The total stock/output quotient is 156/15.0=10.4 days. Mean transit time of exiting material is input-fraction weighted: 0.800×4.00+0.200×36.0=10.4 days. Mean age of stored carbon is stock weighted: (48.0×4.00+108×36.0)/156=26.1538462 days. Those averages describe different populations.

Changing the slow-pool time to 18.0 days and waiting for the new stipulated steady state gives stocks 48.0 and 54.0 mol C. Their total 102 mol C has turnover quotient and mean exit transit 6.80 days; mean stock age is 11.4117647 days. This is a comparison of steady states, not the time course of the transition. The table keeps the weights visible.

An element can move through several chemical forms

Carbon, nitrogen, sulfur and other elements occur in species with different oxidation states, interactions and mobility. Biological transformations, mineral reactions and physical exchanges can connect their reservoirs. Chemical form matters when interpreting the transport and effects of an element.

An elemental balance alone does not identify the transformation pathway. More than one chemical network can give the same net transfer. Isotopic, species-resolved and time-dependent observations can provide further evidence within their own models.

A cycle model needs a spatial and time boundary

A local sample, catchment, water body and global reservoir describe different scales. Seasonal changes, mixing and transport delays can affect which simplified balance is useful. Report the interval over which a flux or average is defined.

The original reservoirs below illustrate inventory and flux reasoning with invented values. They are not current estimates of planetary inventories, environmental targets or proof that one local measurement represents a global cycle.

Worked examples

Try it yourself

1. Change only output to 84.0 mol C day⁻¹. Find final stock after five days and stock/output. Does steady stock imply no transfers?

   Show the answer and reasoning

   1,260 mol C; 15.0 days; transfers still continue.

   Input and output each total 420 mol C, giving zero net change with nonzero throughput.

2. Add 0.500 m³ carbon-free water after the stated mixture. Find carbon amount and concentration.

   Show the answer and reasoning

   3,560 mmol C; 1.18666667 mmol C L⁻¹.

   The amount stays fixed and total volume becomes 3,000 L.

3. Apply formal CH₄+2O₂→CO₂+2H₂O at extent 2.50 mol to the stated inventory. Find the new species and O₂/H₂O account.

   Show the answer and reasoning

   CO₂ 14.5 mol; CH₄ 2.50 mol; CaCO₃ 0.800 mol; O₂ consumed and H₂O formed 5.00 mol each.

   The carbon total remains 17.8 mol C. The formal extent supplies no kinetic evidence.

4. Change the slow-pool time to 18.0 days and compare the new stationary stocks and mean times.

   Show the answer and reasoning

   Stocks 48.0/54.0 mol C; stock/output and exit transit 6.80 days; mean stored age 11.4117647 days.

   New stocks follow input times assigned pool time; the age average remains stock weighted.

What to carry forward

• An inventory changes by its compatible net input over time.
• Concentration mixing needs an amount and volume balance.
• A conserved element can move among different chemical species.
• Stock turnover, stored age and exit transit require stated weights and assumptions.