Chemistry core problem solutions

Solution to Problem 1 – Reaction resonance rate#

The rate constant is

$$ k = X e^{-1/(ΛΘ)}. $$

If $$Λ$$ doubles, then the exponent becomes

$$ \frac{1}{(2Λ)Θ} = -\frac{1}{2(ΛΘ)} $$

This is less negative than before, so the exponential factor increases.

Answer: $$k$$ increases (nonlinearly), because the exponential suppression becomes weaker.

Solution to Problem 2 – Molecular vibration energy#

The vibrational energy is

$$ E = D_3 T_f^2. $$

If $$T_f$$ increases by 5%, then

$$ T_f' = 1.05 T_f. $$

Thus,

$$ E' = D_3 (1.05 T_f)^2 = 1.1025 E. $$

Answer: $$E$$ increases by 10.25%.

Solution to Problem 3 – pH resonance drift#

We have

$$ ΔpH = \frac{F_3}{τ_r}. $$

To reduce $$ΔpH$$ by 30%:

$$ ΔpH' = 0.7 ΔpH = \frac{F_3}{τ_r'}. $$

Solve for $$τ_r'$$:

$$ τ_r' = \frac{τ_r}{0.7}. $$

Answer: $$τ_r$$ must be increased by approximately 43%.