The compound that will react most readily with gaseous bromine has the formula:

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1. **Identify the Compounds:**
   - $C_4H_{10}$: Butane (Alkane, saturated).
   - $C_2H_2$: Ethyne (Alkyne, unsaturated, triple bond).
   - $C_2H_4$: Ethene (Alkene, unsaturated, double bond).
   - $C_3H_6$: Propene (Alkene, unsaturated, double bond).

2. **Reactivity Principle:**
   - **Alkanes** react with halogens via free radical substitution, which typically requires UV light or high heat and is slower than addition reactions.
   - **Unsaturated hydrocarbons (Alkenes/Alkynes)** react with bromine via electrophilic addition. This reaction is generally fast and characteristic of unsaturation.
   - **Alkenes vs. Alkynes:** Alkenes are generally more reactive than alkynes towards electrophilic addition reactions [NCERT 11th, Ch 13, Sec 13.4.4]. This is because the $\pi$-electron cloud in alkynes is more tightly held by the $sp$-hybridised carbons compared to the $sp^2$-hybridised carbons in alkenes.

3. **Compare Alkenes (Ethene vs. Propene):**
   - Both Ethene ($CH_2=CH_2$) and Propene ($CH_3-CH=CH_2$) undergo electrophilic addition.
   - Propene contains an electron-donating methyl group ($-CH_3$) which exerts a **+I (Inductive) effect**. This increases the electron density on the double bond, making it more nucleophilic and thus more reactive towards the electrophile ($Br^+$) compared to ethene.
   - Therefore, Propene ($C_3H_6$) reacts most readily.

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