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Welcome to the forum Arya !!
Ka value of this acid is less than 1 and hence it is not a very strong acid
However, Ka vale is not very less so you can not use approximation also in calculations.
Because it's not a very weak acid so it means that the concentration of HIO3 will be very less as compared to the concentration of H3O+ and IO 3 - at the time of equilibrium.
Answer this question without doing calculations by using guesswork
Only in the (e) option, the concentration of HIO3 is less than that of the concentration of IO3 - and H3O+.
Rest all are not valid at all because either the concentration is the same as 0. 1 or in option a is even higher than 0.1 which is practically impossible.
You can do the calculations also because of all the information provided to us. You plug in the values in K expression and use the quadratic equation to get the value of x and compare it with the answers given in the question.
HIO3 IO3- H3O+
Initial 0.1 0 0
Change -X +X +X
Equilibrium 0.1-X X X
Ka = X2/(0.1-X)
No approximation is valid because Ka is comparable to the initial concentration. If the Ka value is 1000 times less than initial concentration only then approximation is valid.
Here X = 0.0694
Check other concentrations at the time of equilibrium and select the answer.