1a) Define half-life. 1b) If 20 g of a radioactive isotope decays to 2.5 g in 100 years, what is its half-life? 2i) State Gay-Lussac's Law and write the balanced equation for the synthesis of ammonia.

The formula for radioactive decay is $N_t = N_0 \left(\frac{1}{2}\right)^n$, where $N_t$ is the final mass, $N_0$ is the initial mass, and $n$ is the number of half-lives.

$$2.5 \text{ g} = 20 \text{ g} \left(\frac{1}{2}\right)^n$$ $$\frac{2.5}{20} = \left(\frac{1}{2}\right)^n$$ $$0.125 = (0.5)^n$$

To find $n$, we recognize that $0.5^3 = 0.125$.

$$n = 3$$

The total time ($T$) is related to the number of half-lives ($n$) and the half-life ($t_{1/2}$) by the formula $T = n \times t_{1/2}$.

$$100 \text{ years} = 3 \times t_{1/2}$$ $$t_{1/2} = \frac{100 \text{ years}}{3}$$ $$t_{1/2} = \boxed{33.33 \text{ years}}$$

2i)
The balanced chemical equation is:

$$3\text{H}_2(\text{g}) + \text{N}_2(\text{g}) \longrightarrow 2\text{NH}_3(\text{g})$$

According to Gay-Lussac's Law of Combining Volumes, the ratio of the volumes of reacting gases and gaseous products is given by their stoichiometric coefficients, assuming constant temperature and pressure.

From the equation, 3 volumes of H$_2$ react with 1 volume of N$_2$.

$$\frac{\text{Volume of N}_2}{\text{Volume of H}_2} = \frac{1}{3}$$ $$\text{Volume of N}_2 = \frac{1}{3} \times 800 \text{ cm}^3$$ $$\text{Volume of N}_2 = \boxed{266.67 \text{ cm}^3}$$

From the equation, 3 volumes of H$_2$ produce 2 volumes of NH$_3$.

$$\frac{\text{Volume of NH}_3}{\text{Volume of H}_2} = \frac{2}{3}$$ $$\text{Volume of NH}_3 = \frac{2}{3} \times 800 \text{ cm}^3$$ $$\text{Volume of NH}_3 = \boxed{533.33 \text{ cm}^3}$$

2ii)
The reaction is exothermic ($\Delta H = -192 \text{ kJ/mol}$).
If the temperature is lowered from $200^\circ\text{C}$ to $150^\circ\text{C}$:

• According to Le Chatelier's principle, lowering the temperature for an exothermic reaction will shift the equilibrium position to the right, favoring the formation of more ammonia (products).
• However, lowering the temperature also decreases the kinetic energy of the reactant molecules, which leads to fewer effective collisions and thus a slower reaction rate.