Final answer:
The nuclear equation for radon-222 decay is $$\ce{^222_86Rn -> ^4_2He + ^218_84Po}$$. The half-life of radon-222 is 3.823 days. Using the half-life, one can calculate the time required for a sample of radon-222 to decay to a specific mass.
Step-by-step explanation:
After 5 days, a sample of radon-222 has decayed to 50% of its original amount. The half-life of radon-222 is 3.823 days. Radon-222, denoted as 222 Rn, undergoes radioactive decay by emitting an alpha particle, which can be represented by a nuclear equation.
The nuclear equation for the decay of radon-222 is as follows:
$$\ce{^222_86Rn -> ^4_2He + ^218_84Po}$$
Where:
- ^222_86Rn represents radon-222.
- ^4_2He represents the alpha particle (a helium nucleus).
- ^218_84Po represents the daughter isotope, polonium-218.
This decay drops the atomic number by 2 (from 86 to 84) and the mass number by 4 (from 222 to 218), transforming radon-222 into polonium-218.
To calculate the time it would take for a certain mass of radon-222 to decay to another specified mass, we can use the concept of half-life. For instance, if we start with a 0.750 g sample of radon-222 and want to know how long it would take to decay to 0.100 g, we can use the following formula:
$$N=N_0(\frac{1}{2})^{\frac{t}{t_{1/2}}}$$
Where:
- N is the final amount of the substance.
- N_0 is the initial amount of the substance.
- t is the time elapsed.
- t_{1/2} is the half-life of the substance.
By applying this formula and solving for t, we can determine the time needed for radon-222 to decay from 0.750 g to 0.100 g given its half-life.