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Secondary and tertiary alcohols react with strong acid (and bases) such $HX$$\ce{HX}$ following $S_N1$SN1 mechanism.

$$\ce{ [1] \space \space ROH <=> HX + ROH_2^+ + X^-}$$$$\ce{ROH <=> HX + ROH_2^+ + X^-} \tag{1}$$ $$\ce{[2] \space \space ROH_2^+ <=> R^+ +H_2O}$$$$\ce{ROH_2^+ <=> R^+ +H_2O} \tag{2} $$ $$\ce{[3] \space \space R+ +X- \rightarrow RX}$$$$\ce{R+ +X- \rightarrow RX} \tag{3} $$

In these reactions, the rate determining step is the formation of the carbocation [2] that determine(2). So the successive attack of the nucleophilic chlorides is dependent on that.  

enter image description here

Here is wherehow the inductive effect do the differencecomes into play. The inductive effect promote byof $R$$\ce{R}$ groups of the alcohols stabilize the carbocation (figure show the stability of carbocations from the more stable to the less) and then increase the chances of a nucleophilic attack!

Secondary and tertiary alcohols react with strong acid (and bases) such $HX$ following $S_N1$ mechanism.

$$\ce{ [1] \space \space ROH <=> HX + ROH_2^+ + X^-}$$ $$\ce{[2] \space \space ROH_2^+ <=> R^+ +H_2O}$$ $$\ce{[3] \space \space R+ +X- \rightarrow RX}$$

In these reactions the rate determining step is the formation of the carbocation [2] that determine the successive attack of the nucleophilic chlorides.  

enter image description here

Here is where inductive effect do the difference. The inductive effect promote by $R$ groups of the alcohols stabilize the carbocation (figure show the stability of carbocations from the more stable to the less) and then increase the chances of a nucleophilic attack!

Secondary and tertiary alcohols react with strong acid (and bases) such $\ce{HX}$ following SN1 mechanism.

$$\ce{ROH <=> HX + ROH_2^+ + X^-} \tag{1}$$ $$\ce{ROH_2^+ <=> R^+ +H_2O} \tag{2} $$ $$\ce{R+ +X- \rightarrow RX} \tag{3} $$

In these reactions, the rate determining step is the formation of the carbocation (2). So the successive attack of the nucleophilic chlorides is dependent on that.

enter image description here

Here is how the inductive effect comes into play. The inductive effect of $\ce{R}$ groups of the alcohols stabilize the carbocation (figure show the stability of carbocations from the more stable to the less) and then increase the chances of a nucleophilic attack!

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Secondary and tertiary alcohols react with strong acid (and bases) such $HX$ following $S_N1$ mechanism.

$$\ce{ [1] \space \space ROH <=> HX + ROH_2^+ + X^-}$$ $$\ce{[2] \space \space ROH_2^+ <=> R^+ +H_2O}$$ $$\ce{[3] \space \space R+ +X- \rightarrow RX}$$

In these reactions the rate determining step is the formation of the carbocation [2] that determine the successive attack of the nucleophilic chlorides.

enter image description here

Here is where inductive effect do the difference. The inductive effect promote by $R$ groups of the alcohols stabilize the carbocation (figure show the stability of carbocations from the more stable to the less) and then increase the chances of a nucleophilic attack!

Secondary and tertiary alcohols react with strong acid (and bases) such $HX$ following $S_N1$ mechanism. In these reactions the rate determining step is the formation of the carbocation that determine the successive attack of the nucleophilic chlorides.

enter image description here

Here is where inductive effect do the difference. The inductive effect promote by $R$ groups of the alcohols stabilize the carbocation (figure show the stability of carbocations from the more stable to the less) and then increase the chances of a nucleophilic attack!

Secondary and tertiary alcohols react with strong acid (and bases) such $HX$ following $S_N1$ mechanism.

$$\ce{ [1] \space \space ROH <=> HX + ROH_2^+ + X^-}$$ $$\ce{[2] \space \space ROH_2^+ <=> R^+ +H_2O}$$ $$\ce{[3] \space \space R+ +X- \rightarrow RX}$$

In these reactions the rate determining step is the formation of the carbocation [2] that determine the successive attack of the nucleophilic chlorides.

enter image description here

Here is where inductive effect do the difference. The inductive effect promote by $R$ groups of the alcohols stabilize the carbocation (figure show the stability of carbocations from the more stable to the less) and then increase the chances of a nucleophilic attack!

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Secondary and tertiary alcohols react with strong acid (and bases) such $HX$ following $S_N1$ mechanism. In these reactions the rate determining step is the formation of the carbocation that determine the successive attack of the nucleophilic chlorides.

enter image description here

Here is where inductive effect do the difference. The inductive effect promote by $R$ groups of the alcohols stabilize the carbocation (figure show the stability of carbocations from the more stable to the less) and then increase the chances of a nucleophilic attack!