A sample of Uranium-238 has a half-life of 4.5 imes 109 years. (a) What is the half-life of Uranium-238? (b) Write the nuclear equation for its alpha decay. (c) If you start with 5 g of Uranium-238, h

Initial mass ($N_0$) = $5 \text{ g}$
Final mass ($N$) = $10\%$ of $5 \text{ g} = 0.10 \times 5 \text{ g} = 0.5 \text{ g}$
Half-life ($t_{1/2}$) = $4.5 \times 10^9 \text{ years}$
The radioactive decay formula is $N = N_0 \left(\frac{1}{2}\right)^{t/t_{1/2}}$.

a) iv) A. The same element can have different atomic masses because of the existence of isotopes. Isotopes are atoms of the same element that have the same number of protons (and thus the same atomic number) but different numbers of neutrons, leading to different mass numbers.

a) iv) B.

b) i) A rate law is an equation that expresses the relationship between the rate of a chemical reaction and the concentrations of the reactants, raised to appropriate powers (reaction orders), at a specific temperature.

b) ii)

• Doubling the concentration of $\text{A}_2$ doubles the reaction rate, so the reaction is first order with respect to $\text{A}_2$. ($\text{Rate} \propto [\text{A}_2]^1$)
• Tripling the concentration of $\text{B}$ triples the reaction rate, so the reaction is first order with respect to $\text{B}$. ($\text{Rate} \propto [\text{B}]^1$)
• Doubling the concentration of $\text{C}$ has no effect on the rate, so the reaction is zero order with respect to $\text{C}$. ($\text{Rate} \propto [\text{C}]^0$)

b) iii) A mechanism consistent with the rate law $\text{Rate} = k[\text{A}_2][\text{B}]$ would have the slowest (rate-determining) step involving one molecule of $\text{A}_2$ and one molecule of $\text{B}$. Reactant $\text{C}$ is not involved in the rate-determining step.

Proposed Mechanism:
Step 1 (Slow): $\text{A}_2 + \text{B} \to \text{AB} + \text{A}$ (Rate-determining step)
Step 2 (Fast): $\text{A} + \text{C} \to \text{AC}$
Overall Reaction: $\text{A}_2 + \text{B} + \text{C} \to \text{AB} + \text{AC}$