Chemistry - Calculations with significant figures

Calculations with significant figures : 

When performing calculations with measured quantities the rule is that the accuracy of the final result is limited to the accuracy of the least accurate measurement. In other words, the final result can not be more accurate than the least accurate number involved in the calculation.

Rounding off : 

The final result of a calculation often contains figures that are not significant. When this occurs the final result is rounded off. The following rules are used to round off a number to the required number of significant figures :
If the digit following the last digit to be kept is less than five, the last digit is left unchanged.
e.g. 46.32 rounded off to two significant
figures is 46.
If the digit following the last digit to be kept is five or more, the last digit to be kept is increased by one.
e.g. 52.87 rounded to three significant figures is 52.9.

Determination of molecular formula : 

Molecular formula of a compound is the formula which indicates the actual number of atoms of the constituent elements in a molecule. It can be obtained from the experimentally determined values of percent
elemental composition and molar mass of that
compound.

Percent composition and empirical formula: 

Compounds are formed by chemical combination of different elements. Quantitative determination of the constituent 

element by suitable methods provides the
percent elemental composition of a compound. If the percent total is not 100, the difference
is considered as percent oxygen. From the the
per cent composition, the ratio of the atoms
of the constituent elements in the molecule is
calculated. The simplest ratio of atoms of the constituent elements in a molecule is called the empirical formula of that compound. Molecular formula can be obtained from the empirical formula if the molar mass is known. The molar mass of the substance under examination is determined by some convenient method. The following example illustrates this sequence.

Chemical reactions and stoichiometric calculations :

Calculation based on a balanced chemical equations are known as stoichiometric calculations. Balanced chemical equation is symbolic representation of a chemical reaction. It supplies the following information which is useful in solving problems based on chemical equations.

i. It indicates the number of moles of the reactants involved in a chemical reaction and the number of moles of the products formed.

ii. It indicates the relative masses of the
reactants and products linked with a
chemical change.

iii. it indicates the relationship between the
volume/s of the gaseous reactants and
products, at STP.

Stoichiometric problems :

Generally problems based on stoichiometry
are of the following types :

a. Problems based on mass-mass
relationship;

b. Problems based on mass-volume
relationship and

c. Problems based on volume-volume
relationship.

Steps involved in problems based on 
stoichiometric calculations : 

1. Write down the balanced chemical
equation representing the chemical
reaction.

2. Write the number of moles and the relative
masses or volumes of the reactants and
products below the respective formulae.

3. Relative masses or volumes should be
calculated from the respective formula
mass referring to the condition of STP.

4. Apply the unitary method to calculate
the unknown factor/s as required by the
problem.

Limiting reagent :

When a chemist carries out a reaction, the reactants are not usually present in exact stoichiometric amounts, that is, in the
proportions indicated by the balanced equation. Because the goal of a reaction is to produce the maximum quantity of a useful compound from the starting materials, frequently, a large excess of one reactant is supplied to ensure that the more expensive reactant is completely converted into the desired product. The reactant which is present in lesser amount gets consumed after some time and subsequently, no further reaction takes place, whatever be the amount left of the other reactant present. Hence, the reactant which gets consumed, limits the amount of product formed and is therefore, called the limiting reagent.

Concentration of solution : 

A majority of reactions in the laboratory are carried out in solutions. Therefore, it is important to understand how the amount of substance is expressed when it is present in the form of a solution. The concerntration of a solution or the amount of substance present in given volume of a solution can be expressed in any of the following ways :

1. Mass percent or weight percent (w/w %)
2. Mole fraction
3. Molarity (M)
4. Molality (m)

Mass percent : 

It is obtained by using following relation:
Mass percent = Mass of solute/Mass of solution × 100 %

Mole fraction :

It is the ratio of number of moles of a particular component of a solution to the total
number of moles of the solution. If a substance
‘A’ dissolves in substance ‘B’ and their number
of moles are nA and nB , repsectively, then the
mole fraction of A and B are given as :

Mole fraction of A= No. of moles of A / No. of moles of solution
∴ Mole fraction of A = nA / nA + nB
Mole fraction of B = No. of moles of B / No. of moles of solution
= nB / nA + nB

Molarity :

It is the most wideley used unit and is
denoted by M. It is defined as the number of
moles of the solute present in 1 litre of the
solution . Thus,

Molarity (M) = No. of moles of solute / Volume of solution in litres

Molality :

It is defined as the number of moles of
solute present in 1 kg of solvent. It is denoted
by m.

Note that molality of a solution does not change with temperature since mass remains unaffected with temperature. Often in chemistry laboratory, a solution of desired
concentration is prepared by diluting a solution of known higher concentration. The solution of higher concentration is also known as stock solution.

Use of graph in analysis :

Analytical chemistry also involves deducing some relation, if any, between two or more properties of matter under study. One of the classic example in the relation between temperature and volume of a given amount of gas. A set of experimentally measured values of volume and temperature of a definite mass of a gas upon plotting on a graph paper appeared as in the figure. When the points are
directly connected, a zig zag pattern results, From this pattern no meaningful result can be deduced. A zig zag pattern results due to many types of errors that incur in many measurements involved an experiment. Figure shows a smooth curve which may be called an average curve passing through these points. In the above example it happens to be straight line and the inference is that V ∝ T.

Drawing an average curve through the points on graph

While fitting it to a smooth curve, care
is taken that the plotted points are evenly
distributed about it. Mathematically ‘even
distribution’ is understood as follows :

From each point draw a perpendicular
to the curve. The perpendicular represents
deviation of each point from the curve. The positive deviations are shown in red and negative deviations are shown in blue. Take sum of all the red perpendiculars and all the blue perpendiculars separately. If the two sums are equal (or nearly equal) the curve drawn shows the experimental points in the best possible representation.