Molar concentration

"Molarity" redirects here. It is not to be confused with Molality.
In chemistry, the molar concentration, c_i is defined as the amount of a constituent n_i (usually measured in moles – hence the name) divided by the volume of the mixture V:
c_i = \frac {n_i}{V}
It is also called molarity, amount-of-substance concentration, amount concentration, substance concentration, or simply concentration. Whereas mole fraction is a ratio of moles to moles, molar concentration is a ratio of moles to volume. The volume V in the definition c_i = n_i/V refers to the volume of the solution, not the volume of the solvent. One litre of a solution usually contains either slightly more or slightly less than 1 liter of solvent because the process of dissolution causes volume of liquid to increase or decrease (discussed further at volume fraction).
The reciprocal quantity represents the dilution (volume) which can appear in Ostwald's law of dilution.

Contents

  
  • 1 Units and notation
  • 2 Related quantities
    • 2.1 Number concentration
    • 2.2 Mass concentration
    • 2.3 Mole fraction
    • 2.4 Mass fraction
    • 2.5 Molality
  • 3 Properties
    • 3.1 Sum of molar concentrations – normalizing relation
    • 3.2 Sum of products molar concentrations-partial molar volumes
    • 3.3 Dependence on volume
  • 4 Spatial variation and diffusion
  • 5 Examples

    Units and notation

    In addition to the notation c_i there is a notation using brackets and the formula of a compound like [A]. This notation is encountered especially in equilibrium constants and reaction quotients.
    The SI unit is mol/m3. However, more commonly the unit mol/L is used. A solution of concentration 1 mol/L is also denoted as "1 molar" (1M). In many publication styles, the "M" symbol (as well as mM, µM, and so on) is like the degree (°) and percent (%) symbols in that it is closed up to the number, whereas most unit symbols (for example, cm, mm, L, mL, g, kg, s) take an intervening space. Some styles deprecate the M, mM, µM notation and replace it with mol/L, mmol/L, or µmol/L.
    1 mol/L = 1 mol/dm3 = 1 mol dm−3 = 1 M = 1000 mol/m3.
    An SI prefix is often used to denote concentrations. Commonly used units are listed in the table hereafter:
    NameAbbreviationConcentrationConcentration (SI unit)
    millimolarmM10−3 mol/dm3100 mol/m3
    micromolarμM10−6 mol/dm310−3 mol/m3
    nanomolarnM10−9 mol/dm310−6 mol/m3
    picomolarpM10−12 mol/dm310−9 mol/m3
    femtomolarfM10−15 mol/dm310−12 mol/m3
    attomolaraM10−18 mol/dm310−15 mol/m3
    zeptomolarzM10−21 mol/dm310−18 mol/m3
    yoctomolaryM10−24 mol/dm3
    (1 particle per 1.6 L)    		10−21 mol/m3

    Related quantities

    Number concentration

    Main article: Number concentration
    The conversion to number concentration C_i is given by:
    C_i = c_i \cdot N_{\rm A}
    where N_{\rm A} is the Avogadro constant, approximately 6.022×1023 mol−1.

    Mass concentration

    Main article: Mass concentration (chemistry)
    The conversion to mass concentration \rho_i is given by:
    \rho_i = c_i \cdot M_i
    where M_i is the molar mass of constituent i.

    Mole fraction

    Main article: Mole fraction
    The conversion to mole fraction x_i is given by:
    x_i = c_i \cdot \frac{M}{\rho} = c_i \cdot \frac{\sum_i x_i M_i}{\rho}
    x_i= c_i \cdot \frac{\sum x_j M_j}{\rho - c_i M_i}
    where M is the average molar mass of the solution, \rho is the density of the solution and j is the index of other solutes.
    A simpler relation can be obtained by considering the total molar concentration namely the sum of molar concentrations of all the components of the mixture.
    x_i = \frac{c_i}{c} = \frac{c_i}{\sum c_i}

    Mass fraction

    Main article: Mass fraction (chemistry)
    The conversion to mass fraction w_i is given by:
    w_i = c_i \cdot \frac{M_i}{\rho}

    Molality

    Main article: Molality
    The conversion to molality (for binary mixtures) is:
    b_2 = \frac{{c_2}}{{\rho - c_2 \cdot M_2}} \,
    where the solute is assigned the subscript 2.
    For solutions with more than one solute, the conversion is:
    b_i = \frac{{c_i}}{{\rho - \sum c_i \cdot M_i}} \,

    Properties

    Sum of molar concentrations – normalizing relation

    The sum of molar concentrations gives the total molar concentration, namely the density of the mixture divided by the molar mass of the mixture or by another name the reciprocal of the molar volume of the mixture. In an ionic solution ionic strength is proportional to the sum of molar concentration of salts.

    Sum of products molar concentrations-partial molar volumes

    The sum of products between these quantities equals one.
    \sum_i c_i \cdot \bar{V_i} = 1
    Dependence on volume
    Molar concentration depends on the variation of the volume of the solution due mainly to thermal expansion. On small intervals of temperature the dependence is :
    c_i = \frac {{c_{i,T_0}}}{{(1 + \alpha \cdot \Delta T)}}
    where c_{i,T_0} is the molar concentration at a reference temperature, \alpha is the thermal expansion coefficient of the mixture.

    Spatial variation and diffusion

    Molar and mass concentration have different values in space where diffusion happens.

    Examples

    Example 1: Consider 11.6 g of NaCl dissolved in 100 g of water. The final mass concentration \rho(NaCl) will be:
    \rho(NaCl) = 11.6 g / (11.6 g + 100 g) = 0.104 g/g = 10.4 %
    The density of such a solution is 1.07 g/mL, thus its volume will be:
    V = (11.6 g + 100 g) / (1.07 g/mL) = 104.3 mL
    The molar concentration of NaCl in the solution is therefore:
    c(NaCl) = (11.6 g / 58 g/mol) / 104.3 mL = 0.00192 mol/mL = 1.92 mol/L
    Here, 58 g/mol is the molar mass of NaCl.
    Example 2: Another typical task in chemistry is the preparation of 100 mL (= 0.1 L) of a 2 mol/L solution of NaCl in water. The mass of salt needed is:
    m(NaCl) = 2 mol/L x 0.1 L x 58 g/mol = 11.6 g
    To create the solution, 11.6 g NaCl are placed in a volumetric flask, dissolved in some water, then followed by the addition of more water until the total volume reaches 100 mL.
    Example 3: The density of water is approximately 1000 g/L and its molar mass is 18.02 g/mol (or 1/18.02=0.055 mol/g). Therefore, the molar concentration of water is:
    c(H2O) = 1000 g/L / (18.02 g/mol) = 55.5 mol/L
    Likewise, the concentration of solid hydrogen (molar mass = 2.02 g/mol) is:
    c(H2) = 88 g/L / (2.02 g/mol) = 43.7 mol/L
    The concentration of pure osmium tetroxide (molar mass = 254.23 g/mol) is:
    c(OsO4) = 5.1 kg/L / (254.23 g/mol) = 20.1 mol/L.
    Example 4: A typical protein in bacteria, such as E. coli, may have about 60 copies, and the volume of a bacterium is about 10^{-15} L. Thus, the number concentration C is:
    C = 60 / (10−15 L)= 6×1016 L−1
    The molar concentration is:
    c = C / N_A = 6×1016 L−1 / (6×1023 mol−1) = 10−7 mol/L = 100 nmol/L
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