The solutions provided cover nuclear equations for decay, defining half-life, and calculating remaining…

Question

The solutions provided cover nuclear equations for decay, defining half-life, and calculating remaining radioactive material after a certain time.

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Here are the solutions to the questions:

4.

a) Write a nuclear equation for the decay process.

Identify the parent nucleus (X), daughter nucleus ( $\frac{226}{88}Ra$ ), and emitted particle (alpha, $\alpha$ ).
An alpha particle is represented as $\frac{4}{2}He$ .
Let the parent nucleus be $\frac{A}{Z}X$ .
Write the general decay equation.
$\frac{A}{Z}X \longrightarrow \frac{226}{88}Ra + \frac{4}{2}He$
Conserve the mass number (A) and atomic number (Z).
For mass number: $A = 226 + 4 = 230$
For atomic number: $Z = 88 + 2 = 90$
Write the complete nuclear equation.
$\boxed{\frac{230}{90}Th \longrightarrow \frac{226}{88}Ra + \frac{4}{2}He}$
b) Determine the mass and atomic number of X.

From the nuclear equation in part (a):
- Mass number of X is 230.
- Atomic number of X is 90.
5.

a) Define the term half-life.

Half-life is the time taken for half of the radioactive nuclei in a sample to decay, or for the activity of a radioactive sample to fall to half its original value.

b) The half-life of $\frac{235}{92}U$ is 4500 years. Work out the fraction of radioactive material that would be remaining after 18000 years.
Calculate the number of half-lives ( $n$ ).
$n = \frac{\text{Total time}}{\text{Half-life}} = \frac{18000 \text{ years}}{4500 \text{ years}} = 4$
The substance undergoes 4 half-lives.
Calculate the fraction remaining.
The fraction remaining is given by the formula $\left(\frac{1}{2}\right)^n$ .
$\text{Fraction remaining} = \left(\frac{1}{2}\right)^4 = \frac{1}{2 \times 2 \times 2 \times 2} = \frac{1}{16}$
The fraction of radioactive material remaining is $\boxed{\frac{1}{16}}$ .

6.

a) Determine the value of m and n in the nuclear equation:
$\frac{216}{84}Po \longrightarrow \frac{208}{82}Pb + m\alpha + n\beta$
Identify the particles.
An alpha particle ( $\alpha$ ) is $\frac{4}{2}He$ .
A beta particle ( $\beta$ ) is $\frac{0}{-1}e$ .
Substitute the particles into the equation.
$\frac{216}{84}Po \longrightarrow \frac{208}{82}Pb + m\frac{4}{2}He + n\frac{0}{-1}e$
Conserve the mass number.
$216 = 208 + m(4) + n(0)$ $216 = 208 + 4m$ $4m = 216 - 208$ $4m = 8$ $m = \frac{8}{4} = 2$
Conserve the atomic number.
$84 = 82 + m(2) + n(-1)$
Substitute the value of $m=2$ :
$84 = 82 + 2(2) - n$ $84 = 82 + 4 - n$ $84 = 86 - n$ $n = 86 - 84$ $n = 2$
The values are $\boxed{m=2, n=2}$ .

b) The table below gives the rate of decay of radioactive element Y. Calculate the half-life of element Y.
Identify the initial mass ( $N_0$ ) and final mass ( $N_t$ ).
At 0 days, $N_0 = 48 \text{ grams}$ .
At 270 days, $N_t = 1.5 \text{ grams}$ .
Use the decay formula $N_t = N_0 \left(\frac{1}{2}\right)^n$ to find the number of half-lives ( $n$ ).
$1.5 = 48 \left(\frac{1}{2}\right)^n$ $\frac{1.5}{48} = \left(\frac{1}{2}\right)^n$ $\frac{15}{480} = \left(\frac{1}{2}\right)^n$ $\frac{1}{32} = \left(\frac{1}{2}\right)^n$
Since $2^5 = 32$ , then $\left(\frac{1}{2}\right)^5 = \frac{1}{32}$ .
Therefore, $n = 5$ .
Calculate the half-life ( $T_{1/2}$ ).
$T_{1/2} = \frac{\text{Total time}}{n} = \frac{270 \text{ days}}{5} = 54 \text{ days}$
The half-life of element Y is $\boxed{\text{54 days}}$ .

7. Distinguish between nuclear fission and nuclear fusion.
- Nuclear fission is the process where a heavy atomic nucleus splits into two or more lighter nuclei, releasing a large amount of energy.
- Nuclear fusion is the process where two or more light atomic nuclei combine to form a heavier nucleus, also releasing a large amount of energy.
8.

a) State one use of radioisotopes in agriculture.

One use is pest control (e.g., sterile insect technique) or tracing nutrient uptake by plants.

b) State one danger associated with exposure of human beings to radioisotopes.

One danger is causing cancer (e.g., leukemia, thyroid cancer) or genetic mutations.

The solutions provided cover nuclear equations for decay, defining half-life, and calculating remaining radioactive material after a certain time.

Step-by-step solution

Identify the parent nucleus (X), daughter nucleus ( $\frac{226}{88}Ra$ ), and emitted particle (alpha, $\alpha$ ).

Write the general decay equation.

Conserve the mass number (A) and atomic number (Z).

Write the complete nuclear equation.

Calculate the number of half-lives ( $n$ ).

Calculate the fraction remaining.

Identify the particles.

Substitute the particles into the equation.

Conserve the mass number.

Conserve the atomic number.

Identify the initial mass ( $N_0$ ) and final mass ( $N_t$ ).

Use the decay formula $N_t = N_0 \left(\frac{1}{2}\right)^n$ to find the number of half-lives ( $n$ ).

Calculate the half-life ( $T_{1/2}$ ).

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More Chemistry Questions

The solutions provided cover nuclear equations for decay, defining half-life, and calculating remaining radioactive material after a certain time.

Step-by-step solution

Identify the parent nucleus (X), daughter nucleus (22688Ra\frac{226}{88}Ra88226​Ra), and emitted particle (alpha, α\alphaα).

Write the general decay equation.

Conserve the mass number (A) and atomic number (Z).

Write the complete nuclear equation.

Calculate the number of half-lives (nnn).

Calculate the fraction remaining.

Identify the particles.

Substitute the particles into the equation.

Conserve the mass number.

Conserve the atomic number.

Identify the initial mass (N0N_0N0​) and final mass (NtN_tNt​).

Use the decay formula Nt=N0(12)nN_t = N_0 \left(\frac{1}{2}\right)^nNt​=N0​(21​)n to find the number of half-lives (nnn).

Calculate the half-life (T1/2T_{1/2}T1/2​).

Still stuck on this one?

More Chemistry Questions

Identify the parent nucleus (X), daughter nucleus ( $\frac{226}{88}Ra$ ), and emitted particle (alpha, $\alpha$ ).

Calculate the number of half-lives ( $n$ ).

Identify the initial mass ( $N_0$ ) and final mass ( $N_t$ ).

Use the decay formula $N_t = N_0 \left(\frac{1}{2}\right)^n$ to find the number of half-lives ( $n$ ).

Calculate the half-life ( $T_{1/2}$ ).