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QUESTION

# How do you solve the Arrhenius equation for T_2?

Here's how you could do that.

The Arrhenius equation does not include a T_2, it only includes a T, the absolute temperature at which the reaction is taking place.

However, you can use the Arrhenius equation to determine a T_2, provided that you know a T_1 and the rate constants that correspond to these temperatures.

The Arrhenius equation looks like this

color(blue)(|bar(ul(color(white)(a/a)k = A * "exp"(-E_a/(RT))color(white)(a/a)|)))" ", where

k - the rate constant for a given reaction A - the pre-exponential factor, specific to a given reaction E_a - the activation energy of the reaction R - the universal gas constant, useful here as 8.314"J mol"^(-1)"K"^(-1) T - the absolute temperature at which the reaction takes place

So, let's say that you know the activation energy of a chemical reaction you're studying.

You perform the reaction at an initial temperature T_1 and measure a rate constant k_1. Now let's say that you're interested in determining the temperature at which the rate constant changes to k_2.

You can use Arrhenius equation to write

k_1 = A * "exp"(-E_a/(R * T_1))

and

k_2 = A * "exp"(-E_a/(R * T_2))

To find T_2, divide these two equations

k_1/k_2 = color(red)(cancel(color(black)(A)))/color(red)(cancel(color(black)(A))) * ("exp"(-E_a/(R * T_1)))/("exp"(-E_a/(R * T_2)))

Use the property of exponents

color(purple)(|bar(ul(color(white)(a/a)color(black)(x^a/x^b = x^((a-b)), AA x !=0)color(white)(a/a)|)))

to rewrite the resulting equation as

k_1/k_2 = "exp"[E_a/R * (1/T_2 - 1/T_1)]

Next, take the natural log of both sides of the equation

ln(k_1/k_2) = ln("exp"[E_a/R * (1/T_2 - 1/T_1)])

This will be equivalent to

ln(k_1/k_2) = E_a/R * (1/T_2 - 1/T_1)

Finally, do some algebraic manipulation to isolate T_2 on one side of the equation

ln(k_1/k_2) = E_a/R * 1/T_2 - E_a/R * 1/T_1

1/T_2 = ln(k_1/k_2) + E_a/R * 1/T_1

Therefore,

color(green)(|bar(ul(color(white)(a/a)color(black)(T_2 = (R * T_1)/(R * T_1 * ln(k_1/k_2) + E_a))color(white)(a/a)|)))