Conservation of Mass

Conservation of mass is the principle that the total mass of a system remains constant when no matter enters or leaves it, within the framework of classical mechanics and ordinary chemistry. Materials may change their physical state or chemical composition, but the mass of all products equals the mass of all starting materials to ordinary experimental accuracy. The principle underlies chemical calculations and the accounting of material flows. In modern physics, it requires qualification: mass–energy equivalence means that changes in a system’s energy can change its mass, and the sum of individual particles’ rest masses is not universally conserved. (www1.grc.nasa.gov)

System boundaries and the classical statement

For a fixed collection of matter, the classical statement is

Applying this statement requires a clearly defined system and boundary. A closed system exchanges no matter with its surroundings, although it may exchange energy. An open system allows matter to cross its boundary. An isolated system exchanges neither matter nor energy. In an open system, the mass inside the boundary can increase or decrease without violating conservation: the change must be accounted for by incoming and outgoing material. (ocw.mit.edu)

Conservation of mass does not imply conservation of volume or density. A gas can be compressed into a smaller volume while retaining the same mass; its density then increases. For a uniform sample,

where $\rho$ is density and $V$ is volume. The conserved quantity is the product $\rho V$, not necessarily either factor separately. (www1.grc.nasa.gov)

Chemical reactions and stoichiometry

In an ordinary chemical reaction, atoms are redistributed among substances rather than transformed into atoms of different elements. Consequently, a complete reaction equation must contain the same number of atoms of each element on both sides. This atomic accounting provides the basis for balancing a chemical equation and for stoichiometry, the quantitative calculation of reactant and product amounts. (openstax.org)

For example, the combustion of methane is represented by

Both sides contain one carbon atom, four hydrogen atoms, and four oxygen atoms. The coefficients describe proportions of molecules or moles, not equal masses of the separate substances. The number of molecules need not remain constant: this equation represents three reactant molecules producing three product molecules, but other balanced reactions have different totals on the two sides. What must be conserved in ordinary chemical accounting is each element’s atom count and the total mass. (openstax.org)

The familiar relation

includes gases and all other participating materials. Comparing only a visible solid before and after a reaction can therefore be misleading. In Lavoisier’s investigations of metals heated in air, the increased mass of the solid was explained by its combination with part of the air, rather than by the creation of matter. (acs.org)

Historical development

Antoine-Laurent Lavoisier made quantitative weighing central to the development of modern chemistry in the late eighteenth century. His investigations of combustion and the heating of metals treated gases as material participants whose masses had to be included in an experiment. This approach helped replace the phlogiston explanation of combustion with an account based on combination with oxygen. (acs.org)

In his Traité élémentaire de chimie, published in 1789, Lavoisier explicitly formulated the principle that an equal quantity of matter exists before and after a chemical operation. Its importance lay not only in the statement itself but also in its use as a quantitative constraint: an explanation of a reaction had to account for all the material entering and leaving it. The principle became a foundation of modern chemical analysis and calculation. (acs.org)

Mass balances in open systems

A mass balance applies conservation to a chosen region, often called a control volume. For total mass in classical engineering calculations,

where $M$ is the mass within the region and $\dot m$ denotes mass flow rate. At steady state, $dM/dt=0$, so total inflow equals total outflow. Steady state is a condition of no accumulation, not a condition of no flow. (ocw.mit.edu)

A balance for an individual chemical substance may also require terms for its formation and consumption:

These internal terms describe changes in the amount of the selected substance, not creation or destruction of total mass. A model of dissolved phosphorus in a pond, for example, can distinguish transport through the pond boundary from removal from the dissolved pool by plant uptake. (ocw.mit.edu)

Mass balances also describe material transfers among organisms, soil, water, and air. In the carbon cycle, carbon moves between chemical compounds and environmental reservoirs. An ecosystem can accumulate carbon when its inputs exceed its outputs, or lose carbon when outputs exceed inputs; neither case violates conservation. (nature.com)

The continuity equation

For a continuous fluid, conservation of mass has a local mathematical form called the continuity equation:

where $\rho$ is mass density, $\mathbf{v}$ is the velocity field, and $\nabla\cdot$ denotes divergence. It states that a local increase in density must be balanced by a net inward mass flow. The equation applies to both compressible and incompressible flow; constant density is an additional assumption, not a consequence of mass conservation. (en.wikipedia.org)

For steady flow through a passage with uniform conditions across each section,

Here $A$ is cross-sectional area and $v$ is mean speed. If density is constant, $A_1v_1=A_2v_2$: flow accelerates through a narrower section. Together with equations for momentum and energy, continuity forms part of the governing equations used in fluid-flow calculations, including the Navier–Stokes equations. (www1.grc.nasa.gov)

Relativistic qualifications

In special relativity, a body’s rest energy is related to its mass by

where $c$ is the speed of light. Heating a body increases its internal energy and therefore its mass; energy leaving the body can reduce its mass even when no material particles leave. Thus, a system closed to matter is not necessarily closed to all transfers that affect its mass. (einstein-online.info)

The mass of a bound object is also not simply the sum of the masses of its separated constituents. A bound system has lower energy than those constituents at rest and separated, and the corresponding mass difference is

For an atomic nucleus, this difference is associated with nuclear binding energy. Chemical bonds produce analogous differences, but they are extremely small compared with the masses of the reacting materials. Classical mass conservation therefore remains an excellent approximation for ordinary chemical work, even though the sum of reactant and product rest masses is not exactly invariant. (einstein-online.info)

The distinction is between adding constituent rest masses and determining the mass of the complete system. The latter includes contributions from internal motion and interaction energies. Relativistic accounting must track energy as well as material particles; it cannot treat matter and energy as independently conserved quantities. (einstein-online.info)

References

1. Conservation of Masswww1.grc.nasa.gov
2. Antoine Laurent Lavoisier The Chemical Revolution - Landmarkacs.org
3. 1 Writing and Balancing Chemical Equations - Chemistryopenstax.org
4. Lectures 08_1 & 08_2 Outline: Introduction, Mass Balance, Evergladesocw.mit.edu
5. The Conservation of Massnature.com
6. Conservation of massen.wikipedia.org
7. Navier-Stokes Equationwww1.grc.nasa.gov
8. Is the whole the sum of its parts?einstein-online.info
9. equivalence between mass and energyeinstein-online.info