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Molar Mass

25th Aug 2019 @ 32 min read

Physical Chemistry

In chemistry, the molar mass is an important quantity. It measures the mass of a mole of a given substance.

Definition

The molar mass of a given substance is defined as the mass of a sample divided by the moles of that substance in the sample. In other words, it is the mass per mole of a substance.

It is usually denoted by the symbol M.

M =\frac{\text{Mass in grams of a Sample}}{\text{Moles of the Sample}} =\frac{m}{n}

Unit

The standard unit is g mol−1. The SI unit is kg mol−1, however, it is very uncommon.

Mole

We know that one mole of a substance consists of 6.022 140 76 × 1023 elementary particles. This number (aka Avogadro’s constant) is mostly approximated to 6.022 × 1023. Thus, one mole of carbon contains 6.022 × 1023 atoms of carbon.

When we say the molar mass of carbon is 12.0 g mol−1, it means one mole of carbon weighs 12.01 g. In other words, 6.022 × 1023 atoms of carbon weigh 12.01 g.

Important of Molar Mass

In chemistry, calculations are related to chemical reactions and stoichiometry. Calculations often include quantities like molarity, molality, mole fraction, molar volume. All these involve the number of moles. Although there is no direct way to measure the number of moles of any substance, the number of moles can be calculated by knowing the molar mass of the substance. The molar mass links the mass of a substance to its moles. Thus, by knowing the molar mass, we can determine the number of moles contained in a given mass of a sample.

Let me make it more clear with an example of sodium chloride. The molar mass of sodium chloride is known; it is 58.44 g mol−1. If we have to measure one mole of sodium chloride, there is no instrument that can directly measures it. But we know 58.44 g of sodium chloride is equivalent to one mole of it. So, we measure 58.44 g of NaCl with a weight scale, which is equivalent to one of 58.44 g. Hence, we have one mole of NaCl measured.

analytical balance weighing one mole (58.44 g) of sodium chloride, NaCl, which is the molar mass.
Figure 1: One mole of sodium chloride NaCl weighs 58.44 g
[Image Source: Wikimedia]

Some of the important points regarding the molar mass are as follows:

Atomic Mass and Molar Mass

The atomic mass and the molar mass are confused with one another. But they are two different quantities with a definition. The table below describes the differences between the two.

Table 1: Difference between Atomic Mass and Molar Mass
Atomic MassMolar Mass
The atomic mass is the sum of the mass of protons, neutrons, and electrons.It is the mass of a mole of a substance.
It is denoted by ma.The symbol used for it is M.
It has a unit of the unified mass unit (u) or the atomic mass unit (amu).g mol−1 is the standard unit for the molar mass.
The atomic mass is an atomic property.It is a bulk property.
It does not account for isotopes.The existence of isotopes is accounted for since it is a bulk property.
The atomic mass is constant for a particular isotopic element. It does not vary from sample to sample. For example, the atomic mass of carbon-12 is 12 u. This is true for all atoms of carbon-12 in the universe.It could vary from sample to sample depending on the percentages of constituents in the sample.

Average Atomic Mass to Molar mass

If the average atomic mass (ma) or average molecular mass is known, we can convert it into the molar mass (M) with the help of the molar mass constant (Mu).

The average atomic mass is measured in the unified mass unit (u). The unified mass unit is related to the molar mass constant (Mu) by the Avogadro constant (NA).

1 \,\text{u} =\frac{M_\text{u}}{N_\text{A}}

ma is the average atomic mass or the average mass of an atom. So, the mass of a mole of the atom is ma × NA. But this quantity is in the unified mass unit (u). The above equation can convert u to g mol−1.

M &=m_\text{a} \times N_\text{A} \times \frac{M_\text{u} \big/ N_\text{A}}{1 \,\text{u}} \\ &=m_\text{a} \times \frac{M_\text{u}}{1 \,\text{u}} \\ &=\frac{m_\text{a}}{1 \,\text{u}} \times M_\text{u}

Therefore, the average atomic mass divided by one unified mass unit times the molar mass constant results in the molar mass.

The precise value of Mu is 0.999 999 999 65(30) g mol−1. But this value approximately equals 1 g mol−1.

M \approx \frac{m_\text{a}}{1 \,\text{u}} \times 1 \,\text{g} \,\text{mol}^{-1}

From the above equation, we can say the numeric value of the molar mass and the average atomic mass is approximately equal. This is also true for molecules. The table below lists the elements (and the compounds) with the average atomic mass and the average molar mass.

Table 2: Average Atomic Mass to Molar Mass

Element (or Compund)

Average Atomic Mass (or Average Molecular Mass) (u)

Divide (÷)

Unified Mass Unit (u)

Multiple (×)

Molar Mass Constant (Mu)

Equal to (=)

Molar Mass (1 g mol−1)

Hydrogen (H)

1.008 u

÷

1 u

×

1 g mol−1

=

1.008 g mol−1

Carbon (C)

12.011 u

÷

1 u

×

1 g mol−1

=

12.011 g mol−1

Oxygen (O)

15.999 u

÷

1 u

×

1 g mol−1

=

15.999 g mol−1

Nitrogen (N)

14.007 u

÷

1 u

×

1 g mol−1

=

14.007 g mol−1

Potassium (K)

39.098 u

÷

1 u

×

1 g mol−1

=

39.098 g mol−1

Calcium (Ca)

40.078 u

÷

1 u

×

1 g mol−1

=

40.078 g mol−1

Chlorine (Cl)

35.45 u

÷

1 u

×

1 g mol−1

=

35.45 g mol−1

Hydrogen Gas (H2)

2.016 u

÷

1 u

×

1 g mol−1

=

2.016 g mol−1

Water (H2O)

18.015 u

÷

1 u

×

1 g mol−1

=

18.015 g mol−1

Methane (CH4)

16.04 u

÷

1 u

×

1 g mol−1

=

16.04 g mol−1

Nitrogen Dioxide (NO2)

46.006 u

÷

1 u

×

1 g mol−1

=

46.006 g mol−1

Ammonia (NH3)

17.031 u

÷

1 u

×

1 g mol−1

=

17.031 g mol−1

Glucose (C6H12O6)

180.156 u

÷

1 u

×

1 g mol−1

=

180.156 g mol−1

Atomic Weight to Molar Mass

The atomic weight (aka relative atomic mass) (Ar) is a dimensionless quantity having the numeric value of the average atomic mass. The average atomic mass and atomic weight are related by the following equation.

A_\text{r} =\frac{m_\text{a}}{1 \,\text{u}}

From the above two equations,

M \approx A_\text{r} \times 1 \,\text{g} \,\text{mol}^{-1}

Therefore, the atomic weight times the molar mass constant results in the molar mass.

So, we can calculate the molar mass of any compound in the same way we calculate the atomic weight from its constituent elements.

Consider an example of glucose. Glucose molecular formula is C6H12O6; it has six carbons, twelve hydrogens, and six oxygens. The molar mass of glucose is the sum of the relative atomic mass of all the atoms in the molecular formula.

A structure of glucose
Figure 2: A structure of glucose
Table 3: Molar Mass of Glucose (C6H12O6)
ElementAtomic WeightMolar Mass (g mol−1)
Carbon (C)12.01112.011
Hydrogen (H)1.0081.008
Oxygen (O)15.99915.999
Molar Mass of Glucose (C6H12O6)180.156

Molar Mass of Mixtures

The molar mass of a mixture is determined using the mole fraction (xi).

M_\text{mix} =\sum_i^n {x_i \,M_i}

where: Mmix is the molar mass of a mixture of n components, and xi and Mi are the mole fraction and the molar mass of ith component.

The above formula can also be expressed in terms of the mass fraction wi.

M_\text{mix} =\frac{1}{\sum_i^n {\frac{w_i}{M_i}}}

or

\frac{1}{M_\text{mix}} =\sum_i^n {\frac{w_i}{M_i}}

Consider a liquid-liquid mixture of water (H2O) and ethanol (C2H6O). Water is 20 % and ethanol, 80 %.

The molar mass of pure water is MH2O.

M_{\text{H}_2 \text{O}} &=2 \times 1.008 + 1 \times 15.999 \\ &=18.015 \,\text{g} \,\text{mol}^{-1}

Similarly, for ethanol,

M_{\text{C}_2 \text{H}_6 \text{O}} &=2 \times 12.011 +6 \times 1.008 + 1 \times 15.999 \\ &=46.069 \,\text{g} \,\text{mol}^{-1}

The molar mass of the mixture is Mmix.

Thus,

\frac{1}{M_\text{mix}} &=\sum_i^n {\frac{w_i}{M_i}} \\ &=\frac{w_{\text{H}_2 \text{O}}}{M_{\text{H}_2 \text{O}}}+\frac{w_{\text{C}_2 \text{H}_6 \text{O}}}{M_{\text{C}_2 \text{H}_6 \text{O}}} \\ &=\frac{0.20}{18.015}+\frac{0.80}{46.069} M_\text{mix} &=\frac{1}{\frac{0.20}{18.015}+\frac{0.80}{46.069}} \\ &\approx 35 \,\text{g} \,\text{mol}^{-1}

Measurement of Molar Mass

From Atomic Weight

The measurement from the atomic weight is the most reliable and precise method in comparison to others. The atomic weight is determined from the atomic mass and distribution of isotopes. The precision in the molar mass depends upon the precision in the measurement of the atomic weight, which depends on the atomic mass and the percentages of isotopes. Today, with the help of mass spectrometry, we are able to achieve high precision in the atomic mass.

The value of molar mass calculated from the atomic weight are reliable for all practical calculations.

From Vapour Density

The vapour density (ρ) is the mass of vapour to its volume. We are familiar with the ideal gas equation.

PV=nRT

Here, P is the pressure of a gas occupying the volume V at the temperature T. n and R are the moles of the gas and the ideal gas constant.

Now, the number of moles (n) is the mass of the gas divided by the molar mass (M).

n =\frac{m}{M}

Using the above two equations,

PV =\frac{m}{M} RT M =\frac{m}{V} \frac{RT}{P} =\rho \frac{RT}{P}

If we are able to determine the vapour density of a sample of gas for a known pressure and temperature, we can estimate the molar mass of the vapour from the above equation.

From Freezing-Point Depression

When a non-volatile solute is added into a solvent, the freezing point of the solvent decreases. This decrease is called freezing-point depression. Freezing-point depression ∆TF = TFsolvent − TFsolution depends on the concentration (molality) of the solute and is governed by the equation below.

\Delta T_\text{F} =K_\text{F} \,b \,i

Here, ∆T is freezing-point depression, KF is the solvent dependent cryoscopic constant, b is the molality, and i is the van ‘Hoff factor (number of ions formed after the dissociation of the solute).

For a very dilute solution, b = \frac{w_\text{solute}}{M_\text{solute}}.

\Delta T_\text{F}=K_\text{F} \frac{w_\text{solute}}{M_\text{solute}} \,i M_\text{solute} =\frac{K_\text{F} \,w_\text{solute} \,i}{\Delta T_\text{F}}

By knowing ∆TF, KF, wsolute, and i, we can determine the molar mass, Msolute.

From Boiling-Point Elevation

Boiling-point elevation is the rise in the boiling point of a solvent due to the presence of a non-volatile solute. Similar to freezing-point depression, the rise in the boiling point depends upon the molality.

\Delta T_\text{B} =K_\text{B} \,b \,i

Here, ∆TB = Tsolution − Tsolute is boiling-point elevation, KB is the ebullioscopic constant, b is the molality, and i is the van ‘t Hoff factor.

For a very dilute solution, b = \frac{w_\text{solute}}{M_\text{solute}}

\Delta T_\text{B}=K_\text{B} \frac{w_\text{solute}}{M_\text{solute}} \,i M_\text{solute}=\frac{K_\text{B} \,w_\text{solute} \,i}{\Delta T_\text{B}}

By knowing ∆TB, KB, wsolute, and i, we can determine the molar mass, Msolute.

Examples and Calculations

Example 1: To Determine Molar Mass of Oleic Acid

Oleic acid is a odourless fatty acid having a molecular formula of C18H34O2. It has eighteen carbons, thirty-four hydrogens, and two oxygens.

Oleic acid structure
Figure 3: Oleic Acid
Table 4: Molar Mass of Oleic Acid, C18H34O2
ElementNumber of AtomMolar Mass (g mol−1)Total
Carbon (C)1812.011216.198
Hydrogen (H)341.00834.272
Oxygen (O)215.99931.998
Molar Mass of Oleic Acid, (C18H34O2)282.468 g mol −1

Therefore, the molar of oleic acid is 282.468 g mol−1.

Example 2: To Determine Molar Mass of Reinecke’s Salt

Reinecke’s salt is a red crystalline salt with chromium in the centre as shown in the above figure. The chromium atom is surrounded by six nitrogens, four carbons, and four sulphur. The molecular formula is C4H12N7OCrS4.

Reinecke's Salt
Figure 4: Reinecke's Salt
Table 5: Molar Mass of Reinecke's Salt, C4H12N7OCrS4
ElementNumber of AtomMolar Mass (g mol−1)Total
Carbon (C)412.01148.044
Hydrogen (H)121.00812.096
Nitrogen (N)714.00798.049
Oxygen (O)115.99915.999
Chromium (Cr)151.99651.996
Sulphur (S)432.065128.26
Molar Mass of Reinecke’s Salt, C4H12N7OCrS4354.444 g mol−1

Example 3: To Determine Molar Mass of Air

Air is a mixture. It mainly consists 79 % nitrogen (N2) and (21 %) oxygen (O2). Other components are argon (Ar), carbon dioxide, (CO2), water (H2O), helium (He) etc., but these components are in small fraction and can be ignored.

The molar mass of nitrogen (N2) and oxygen (O2) are 28.014 g mol−1 and 31.998 g mol−1 respectively.

Table 6: Molar Mass of Air
MoleculesFractionMolar Mass (g mol−1)Total
Nitrogen (N2)0.7928.01422.13
Oxygen (O2)0.2131.9985.88
Molar Mass of Air≈ 28 g mol−1

Therefore, the molar of air is 28 g mol−1.

Example 4: To Determine Molar Mass of Sodium Chloride Solution

Sodium chloride a common salt. It is white and odourless powder. Consider a solution of 50 % NaCl. 100 g of the solution consists of 50 g of NaCl and 50 g of H2O.

The molar mass of water is 2 × 1.008 + 15.999 = 18.015 g mol−1 and of sodium chloride is 22.99 + 35.45 = 58.44 g mol−1.

The molar of the solution is calculated as follows:

\frac{1}{M_\text{mix}} =\sum_i^n {\frac{w_i}{M_i}} =\frac{0.50}{58.44} + \frac{0.50}{18.015} M_\text{mix} =\frac{1}{\frac{0.50}{58.44} + \frac{0.50}{18.015}} =27.54 \approx 28 \,\text{g} \,\text{mol}^{-1}

Thus, the molar mass of 50 % sodium chloride solution is 28 g mol−1.

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