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**Hint:**The dissociation constant is determined as the product of the square of the degree of dissociation and concentration. The degree of dissociation tells how much an electrolyte dissociates in solution. The degree of dissociation is calculated for the weak electrolyte.

**Complete Step by step answer:**The formula which relates the degree of dissociation with dissociation constant is as follows:

${K_a} = \,\dfrac{{C{\alpha ^2}}}{{1 - \alpha }}$

Where,

${K_a}$ is the acid dissociation constant.

$\,C$ is the concentration.

$\alpha $ is the degree of dissociation

Mostly weak electrolyte dissociates very less so, for weak electrolyte the degree of dissociation is very-very less than one,$\alpha < < 1$ so, the value of $1 - \alpha $ can be taken to equal to$1$so, the formula of the dissociation constant can be reduced as, ${K_a} = \,C{\alpha ^2}$.

Rearrange the formula for degree of dissociation as follows:

\[{\alpha ^2} = \dfrac{{{{\text{K}}_{\text{a}}}}}{{\text{C}}}\]

\[\Rightarrow \alpha = \sqrt {\dfrac{{{K_a}}}{C}} \]

Substitute $0.1\,{\text{mol}}\,\,{{\text{L}}^{ - 1}}$for the concentration and ${10^{ - 5}}$ for dissociation constant.

\[\alpha = \sqrt {\dfrac{{{{10}^{ - 5}}}}{{0.1}}} \]

\[\Rightarrow \alpha = \sqrt {1\, \times {{10}^{ - 4}}} \]

$\Rightarrow \alpha = \,0.01$

So, the degree of dissociation for acetic acid in water is $\,0.01$.

**Therefore, option (A) $\,0.01$, is correct.**

**Note:**The degree of dissociation tells the dissociated amount of the weak electrolyte. The strong electrolyte dissociates completely, so it is not calculated for the strong electrolyte. A similar formula is used for the determination of the degree of dissociation for a weak base only the ${{\text{K}}_{\text{a}}}$ is replaced with ${{\text{K}}_{\text{b}}}$ where, ${{\text{K}}_{\text{b}}}$ is the base dissociation constant. The degree of dissociation can also be calculated by using equivalent conductance at a concentration and equivalent conductance at infinite dilution. The formula to calculate the degree of dissociation is $\alpha = \,\dfrac{{{\lambda _m}}}{{\lambda _m^ \circ }}$. Where, ${\lambda _m}$is the equivalent conductance at a concentration and $\lambda _m^ \circ $is the equivalent conductance at infinite dilution.

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