Question

Number average molecular mass is given by the formula (a) \(\bar{M}_{n}=\frac{\Sigma M_{f}}{\Sigma N_{A}}\) (b) \(\bar{M}_{n}=\frac{\sum N_{1} M_{f}}{\Sigma N_{l}}\) (c) \(\bar{M}_{n}=\frac{\sum N_{i} M_{f}^{2}}{\sum M_{l}}\) (d) \(\bar{M}_{n}=\frac{\sum N_{1} M_{f}^{2}}{\Sigma M_{l}}\)

Short answer

None of the given options provide the correct formula for the number average molecular mass.

Step-by-step solution

Comparing Option (a)

Looking at formula a, the correct notation for the total mass of all molecules, \(\Sigma M_{f}\), is in the numerator. However, the denominator should be the total number of molecules not Avogadro's number, \(\Sigma N_{A}\). Hence option (a) is incorrect.

Comparing Option (b)

For formula b, the sum of all mole fractions multiplied by their respective molecular masses, \(\sum N_{1} M_{f}\), is in the numerator and the denominator is the sum of mole fractions \(\Sigma N_{l}\). This describes the weight average molecular weight, not the number average molecular weight. Hence, option (b) is incorrect.

Comparing Option (c)

Looking at formula c, the numerator \(\sum N_{i} M_{f}^{2}\) implies squaring of molecular mass which is not done in number average molecular mass. Hence this formula is not correct.

Comparing Option (d)

Considering formula d, similar to (c), squaring of molecular mass is not acceptable. Hence, option (d) is incorrect.

Key concepts

Molecular Weight Calculation

Molecular weight calculation is essential in understanding molecular chemistry, especially when determining different types of averages.
These calculations help to understand the collective mass of molecules within a given substance.
To achieve this, formulas are used to express the average molecular mass. There are different ways to calculate, such as the number average molecular weight and weight average molecular weight.

• Number average molecular weight is calculated by finding the total mass of all the molecules and then dividing it by the number of molecules.
• Weight average molecular weight, on the other hand, considers the proportion of each component, providing an average based on mass fractions.

In simple terms, molecular weight is an essential concept that aids chemists when dealing with large molecules, also called macromolecules.
These calculations can be complex, but they are vital in polymer chemistry to determine the physical properties of substances.

Avogadro's Number

Avogadro's number, a fundamental concept in chemistry, is a constant that defines the number of constituent particles (usually atoms or molecules) that are contained in one mole of any substance.
This number is approximately 6.022 x 10^{23}.
Avogadro's number helps relate the macroscopic scale to the atomic scale. Understanding this number is crucial when performing various chemical calculations.
For instance, it allows for calculating the number of molecules present in a sample of a gas given in moles.

• For example, if you have 2 moles of a substance, it contains approximately 2 x Avogadro's number of molecules.
• This relationship aids chemists in converting between the mass of a substance and the number of fundamental units it contains.

Thus, Avogadro's number is indispensable in stoichiometry and other calculations in chemistry.

Molecular Mass Theorem

The molecular mass theorem is a principle used to relate molecular mass distributions within a sample.
It is pivotal in understanding the properties of mixtures of molecules, especially in the realm of polymers and complex compounds.
The theorem helps distinguish between number average molecular weight and weight average molecular weight.

• These different averages help to describe the distribution of molecular sizes in complex systems.
• The molecular mass theorem outlines how these averages are calculated differently, with the weight average taking the mass of each molecule into account more than the number average.

This distinction is important because it affects the physical properties of materials.
For example, polymers with a high weight average molecular mass typically display higher mechanical strength. Understanding these averages helps in tailoring materials for specific applications by controlling the synthesis process.