Solve for k natural log of k = natural log of A-E_a/(RT)

$\ln (k) = \ln (A) - \frac{E_{a}}{R T}$

Step 1

Move all the terms containing a logarithm to the left side of the equation.

$\ln (k) - \ln (A) = - \frac{𝐄_{a}}{R T}$

Step 2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{k}{A}) = - \frac{𝐄_{a}}{R T}$

Step 3

To solve for $k$ , rewrite the equation using properties of logarithms.

$e^{\ln (\frac{k}{A})} = e^{- \frac{𝐄_{a}}{R T}}$

Step 4

Rewrite $\ln (\frac{k}{A}) = - \frac{𝐄_{a}}{R T}$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

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

Step 5

Solve for $k$ .

Rewrite the equation as $\frac{k}{A} = e^{- \frac{𝐄_{a}}{R T}}$ .

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

Multiply both sides by $A$ .

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

Simplify.

Simplify the left side.

Cancel the common factor of $A$ .

Cancel the common factor.

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

Rewrite the expression.

$k = e^{- \frac{𝐄_{a}}{R T}} A$

Simplify the right side.

Reorder factors in $e^{- \frac{𝐄_{a}}{R T}} A$ .

$k = A e^{- \frac{𝐄_{a}}{R T}}$

Solve for k natural log of k = natural log of A-E_a/(RT)

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