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    Conservation of Mass

    Total mass is conserved within a system. Mass cannot be created or destroyed, and can only be transferred or converted between species.

    For mass conservation of a single species:

    $$ Input - Output + Generation = Accumulation $$

    One species of mass can be converted into another species through chemical reactions. This is demonstrated through the formulation of water.

    $$ 2H_{2} + O_{2} \rightarrow 2H_{2}O $$

    Hydrogen and oxygen gas are converted to water; however, 4 \( H \) atoms and 2 \( O \) atoms remain before and after the reaction.

    Need a review?

    These concepts have also been covered in Conservation of Mass and are analogous to Thermodynamic Laws: Conservation of Energy.

    Heads Up!

    Generation of mass and storage of mass are not the same thing!

    Generation is the transformation of mass from one species into another (i.e., the cause) whereas storage of mass is manifested by a change in concentration (i.e, the effect).

    Equilibrium

    Mass equilibrium is when the total mass entering a system is equal to the total mass leaving a system. In other words, the system is in a steady state where mass doesn’t change with time. Equilibrium does not mean equal concentrations. It depends on polarity, pH, and molecular crowding. The transfer of mass requires a departure from equilibrium.

    Equilibrium can occur between different phases:

    • Liquid-Gas
    • Gas-Solid
    • Solid-Liquid

    These interphases are described below!

    Heads Up!

    Mass concentration (\( \rho_{A} \)) and molecular concentration (\( c_{A} \)) are related by molecular weight (\( M_{A} \)):

    $$ \rho_{A} = c_{A} * M_{A} $$

    where

    • \( \rho_{A}=\frac{kgA}{m^3} \)
    • \( c_{A}=\frac{molA}{m^3} \)
    • \( M_{A}=\frac{kgA}{molA} \)

    Liquid-Gas

    Henry’s Law describes the pressure dependence of the solubility of a gas in a solution:

    $$ p_{A} = H x_{A} $$

    where

    • \( H \) = Henry's Constant \( [\frac{atm}{M}] \)
    • \( p_{A} \) = partial pressure of A in gas (at equilibrium) \( [atm] \)
    • \( x_{A} \) = concentration of A in liquid (at equilibrium) \( [M] \)

    The amount of gas dissolved in a liquid is proportional to the partial pressure of that gas above the liquid.

    Gas-Solid

    The equilibrium moisture curve describes the relationship between the concentration of moisture in air and the corresponding equilibrium moisture content of a solid:

    $$ 1 - RH = b_{1}e^{(-b_{2}\omega)} $$

    where

    • \( RH \) = Relative Humidity. Amount of moisture in air compared to maximum capacity [%]
    • \( b_{1},b_{2} \) = constants describing the curve
    • \( \omega \) = equilibrium moisture content [%]

    Hydroscopic: Attracts and holds water molecules from the surrounding environment and shrinks when drying. Examples: biomaterials, wood, soil, etc.

    Non-Hydroscopic: Does not hold onto water molecules. Water molecules behave as if the solid is not there. Examples: sand, ceramics, polymer particles, etc.

    Solid-Liquid

    Equilibrium between dissolved and surface-adsorbed represented mathematically by:

    $$ c_{A,adsorbed} = K^{*} c_{A}^{n} $$

    where

    • \( c_{A,adsorbed} \) = concentration of adsorbed solute A \( [\frac{mol}{kg}] \)
    • \( c_{A} \) = concentration of solute A in solution \( [M] \)
    • \( K^{*},n \) = empirical constants

    Chemical Kinetics

    Chemical kinetics is the study of rate and mechanism by which one species is chemically converted into another.

    Rate Laws

    Zeroth Order: reaction rate is independent of species concentration.

    $$ c = k^{''}t-c_{o} $$

    First Order: reaction rate is related to species concentration.

    $$ c = c_{o}e^{(-k^{''}t)} $$

    where

    • \( c \) = concentration \( [M] \)
    • \( c_{o} \) = initial concentration \( [M] \)
    • \( t \) = time \( [s] \)
    • \( k^{''} \) = reaction rate constant

    Heads Up!

    The units for the reaction rate constant (\( k^{''} \)) depend on the order of the reaction:

    First Order:

    \( k^{''} = \frac{M}{s} \)

    Second Order:

    \( k^{''} = \frac{1}{s} \)

    Half Life

    Half life is the time required for concentration to change by 50%. It is widely used in pharmacokinetics to describe reaction rate of drugs.

    $$ c = c_{o}e^{-(\frac{0.693}{t_{\frac{1}{2}}})} $$

    where

    • \( c \) = concentration \( [M] \)
    • \( c_{o} \) = initial concentration \( [M] \)
    • \( t_{\frac{1}{2}} \) = half life \( [s] \)

    In pharmacokinetics, half life is used to…

    • Determine dosing intervals of drugs
    • Predict time for drug to reach steady state
    • Assess total drug elimination

    Arrhenius’ Law

    Arrhenius Law: reaction rates increase with temperature

    $$ k^{''} = k^{''}_{o} e^{-(\frac{E_{a}}{R_{g}T})} $$

    where

    • \( k^{''} \) = reaction rate constant
    • \( k^{''}_{o} \) = initial reaction rate constant
    • \( E_{a} \) = activation energy \( [\frac{J}{mol}] \)
    • \( R_{g} \) = ideal gas constant \( [\frac{J}{molK}] \)
    • \( T \) = temperature \( [K] \)

    A biological example of Arrhenius’s Law is bacterial growth. As temperature increases, the reactions that control metabolism and cell division speed up, leading to faster growth. This is why bacteria are commonly incubated at around 37 degrees Celsius. At very low temperatures, such as freezing, these reactions slow dramatically or stop altogether, which is why freezing food helps prevent bacterial growth.