Question

The half life of a first order reaction varies with temperature according to

Answer 1

Explanation:

1. For a first order reaction, 2.t1/2-0.693/k (Here k is Rate constant). 3. Now from Arhenius Equation, k-Ae - Ea/RT) 4.From here you can find that ti/2 is proportional to e Ea/ RT)

Answer 2

Answer:

The half-life of a first-order reaction varies with temperature according to

$ln$ $t_{1/2}$  ∝  $\frac{1}{T}$.

Explanation:

For the first-order reaction, the half-life of a reaction can be expressed as:

                                          $t_{1/2} = \frac{0.693}{k}$

                                              $k = \frac{0.693}{t_{1/2} }$

Taking ln on both sides of the above expression and we get,

                                           $ln k = ln(0.693) - ln t_{1/2}$            ......(1)

From the Arrhenius equation, the rate constant can be expressed as,

                                              $k = A e^{\frac{-E_{a} }{RT} }$

Taking ln on both sides of the above expression:

                                            $ln k = ln A - \frac{E_{a} }{RT}$                    ......(2)

From equation (1) & (2) we get,                                              

                     $ln (0.693) - ln t_{1/2} = ln A - \frac{E_{a} }{RT}$

                                       $ln t_{1/2} = ln (0.693) - ln A + \frac{E_{a} }{RT}$

Thus,  $ln (0.693) - ln A + \frac{E_{a} }{RT}$ is constant. So,

                                       $ln t_{1/2}$ ∝ $\frac{1}{T}$

Therefore, the half-life of a first-order reaction decreases with increases in temperature.

Hence, the half-life of a first-order reaction varies with temperature as  $ln t_{1/2}$ ∝ $\frac{1}{T}$.

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