Step 1: Expressing the relation. \[ \overrightarrow{c} = \alpha \overrightarrow{a} + \beta \overrightarrow{b} \]
Step 2: Dot product condition. \[ \overrightarrow{a} \cdot \overrightarrow{c} = \alpha (\overrightarrow{a} \cdot \overrightarrow{a}) + \beta (\overrightarrow{b} \cdot \overrightarrow{a}) \] \[ 0 = \alpha + \beta \cos 15^\circ \] \[ \Rightarrow \alpha = -\beta \cos 15^\circ. \]
Step 3: Solving for \( \alpha \) and \( \beta \). \[ \cos 75^\circ = \alpha \cos 15^\circ + \beta \] \[ \beta = \frac{\cos 75^\circ}{\sin 15^\circ} = \frac{1}{\sqrt{3}-1} \frac{2\sqrt{2}}{\sqrt{3}-1} \]
Step 4: Computing \( \alpha + \sqrt{2} (\sqrt{3}-1) \beta \). \[ \alpha + \sqrt{2} (\sqrt{3}-1) \beta = (-\frac{\sqrt{3}+1}{\sqrt{3}-1}) + \frac{\sqrt{2} (\sqrt{3}-1) 2\sqrt{2}}{\sqrt{3}-1} \] \[ = -\frac{\sqrt{3}+1}{2} + 4 \] \[ = 2 - \sqrt{3}. \]
\([A]\) (mol/L) | \(t_{1/2}\) (min) |
---|---|
0.100 | 200 |
0.025 | 100 |