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Question: A polystyrene, having formula \(\,B{r_3}{C_6}{H_3}{({C_8}{H_8})_n}\,\) was prepared by heating styre...

A polystyrene, having formula  Br3C6H3(C8H8)n \,B{r_3}{C_6}{H_3}{({C_8}{H_8})_n}\, was prepared by heating styrene with tribromo benzoyl peroxide in the absence of air. If it was found to contain  10.46% \,10.46\% \, bromine by weight, the value of  n \,n\, is:

Explanation

Solution

To solve this question, we have to use the mass percentage formula. One method of describing the concentration of an element in a compound or a part in a mixture is mass percentage. For that, we know the atomic weight of bromine is  80g \,80g\,.Here, in one molecule of polystyrene we have three bromine atoms. So, we can calculate  n \,n\, by deriving it from the equation.

Complete step by step answer:
The mass percentage is measured as the mass of a component divided by  100 \,100\, percent multiplied by the overall mass of the mixture.
Let us move into the calculations now;
 10.46% \,10.46\% \,Bromine by weight of polystyrene means  =mass of number of atoms of bromin⁡eTotal mass of the whole molecule \, = \dfrac{{mass\,of\,number\,of\,atoms\,of\,bro\min e}}{{Total\,mass\,of\,the\,whole\,molecule\,}}
Also, here  10.46% \,10.46\% \,means  10.46g \,10.46g\, of  Br \,Br\, in  100g \,100g\,So, mass of  Br=100g \,Br = 100g\,
 10.46%=10.46100 \,10.46\% = \dfrac{{10.46}}{{100}}\,
And mass of number of atoms of bromine  = number of atoms×atomic weight=3×80 \, = \,number\,of\,atoms \times atomic\,weight = 3 \times 80\,
And mass of the whole molecule  = \, = \,mass of bromine  + \, + \,the total mass of rest of the carbon and hydrogen  (H=1,C=12)\,(H = 1,C = 12)
 =3×80+75+44n \, = 3 \times 80 + 75 + 44n\,
Therefore, we get;
10.46100=3×80(3×80)+75+44n \dfrac{{10.46}}{{100}} = \dfrac{{3 \times 80}}{{(3 \times 80) + 75 + 44n}}\,
Where,  10.46(315=44n=3×80×100=24000 \,10.46(315 = 44n = 3 \times 80 \times 100 = 24000\,
 ⇒3294.9+460.24n=24000 \, \Rightarrow 3294.9 + 460.24n = 24000\,
From this we can calculate the value of  n \,n\,by subtracting the numerical values as follows;
460.24n=24000−3294.9 460.24n = 24000 - 3294.9\,
 ⇒n=24000−3294.9460.24=44.988 \, \Rightarrow n = \dfrac{{24000 - 3294.9}}{{460.24}} = 44.988\,
Which is approximately equal to  45 \,45\,.

Additional information:
Polystyrene is a composite of synthetic aromatic hydrocarbon made from a monomer called styrene. It may be solid or foamed polystyrene. Polystyrene for general use is smooth, hard, and very brittle. It is an economical resin. In medical equipment applications such as test tubes or petri dishes, or in day-to-day products such as the housing on your smoke alarms, and sometimes as a jar for foods such as yoghurt, the rigid plastic shape of polystyrene is widely used.

Note:  w/w ,w/v, v/v \,w/w{\text{ }},w/v,{\text{ }}v/v\,( w= \,w = \,weight and  v=volume \,v = volume\,) are another methods of expressing concentration. According to the type of mixture, we can choose the required method. Via elemental analysis, mass percentages can be calculated experimentally, and these values can be used to measure the empirical formula for unknown substances. However, without further knowledge on the compound's molecular weight, this knowledge is inadequate to determine the molecular formula.