Question

below is the complete decay chain of uranium - 238. you need to work out the decay equations for each step and fill in the blanks. the first step is done for you.
alpha decay (α) ↓ uranium 238: $_{92}^{238}u\
ightarrow_{2}^{4}\alpha + _{90}^{234}th$
beta decay (β) + gamma (γ) ↓ thorium 234: th → po+β + γ
beta decay (β) + gamma (γ) ↓ protactinium 234: $_{91}^{234}pa\
ightarrow_{\\;\\;}^{\\;\\;} + β + γ$
alpha decay (α) + gamma ↓ uranium 234: $_{\\;\\;}^{\\;\\;}u\
ightarrow th+\alpha + γ$
alpha + gamma (γ) ↓ thorium 230: $_{90}^{230}th\
ightarrow ra+\alpha + γ$
alpha decay (α) + gamma (γ) ↓ radium 226: ra → r+α + γ
alpha ↓ radon 222: r → po + $_{2}^{4}he$
alpha decay (α) ↓ polonium 218: pa → pa+α
beta - alpha + gamma ↓ lead 214: $_{\\;\\;}^{\\;\\;}\
ightarrow_{\\;\\;}^{\\;\\;}+_{ - 1}^{0}e+γ$
beta decay (β) + gamma (γ) ↓ bismuth 214: $_{\\;\\;}^{\\;\\;}\
ightarrow_{\\;\\;}^{\\;\\;}+β + γ$
alpha decay (α) ↓ polonium: $_{\\;\\;}^{\\;\\;}\
ightarrow_{\\;\\;}^{\\;\\;}+α$
beta decay (β) + gamma (γ) ↓ $_{\\;\\;}^{\\;\\;}\
ightarrow_{83}^{\\;\\;}+β + γ$
beta decay (β) + gamma (γ) ↓ $_{\\;\\;}^{\\;\\;}\
ightarrow_{\\;\\;}^{\\;\\;}+β + γ$
alpha decay (α) ↓ $_{\\;\\;}^{\\;\\;}\
ightarrow_{\\;\\;}^{\\;\\;}+_{\\;\\;}^{\\;\\;}$
list of elements to help: atomic number: symbol: element.
80: hg: mercury, 81: tl: thallium, 82: pb: lead, 83: bi: bismuth, 84: po: polonium,
85: at: astatine, 86: r: radon, 87: fr: francium, 88: ra: radium, 89: ac: actinium,
90: th: thorium, 91: pa: protactinium, 92: u: uranium.
206 stable

Explanation:

Step1: Recall beta - decay rules

In beta - decay ($\beta$), a neutron in the nucleus is converted into a proton and an electron (beta particle $\beta^-$ or $_{- 1}^0e$). The atomic number increases by 1 while the mass number remains the same. In gamma - decay ($\gamma$), only high - energy photons are emitted and there is no change in the mass number or atomic number of the nucleus.

Step2: Recall alpha - decay rules

In alpha - decay ($\alpha$), the nucleus emits an alpha particle which is a helium nucleus $_{2}^4He$. The mass number decreases by 4 and the atomic number decreases by 2.

Step3: Analyze Thorium - 234 decay

For Thorium - 234 ($_{90}^{234}Th$) undergoing beta + gamma decay, a neutron turns into a proton. The resulting element is Protactinium - 234. The decay equation is $_{90}^{234}Th
ightarrow_{91}^{234}Pa + _{-1}^0e+\gamma$.

Step4: Analyze Protactinium - 234 decay

For Protactinium - 234 ($_{91}^{234}Pa$) undergoing beta + gamma decay, a neutron turns into a proton. The resulting element is Uranium - 234. The decay equation is $_{91}^{234}Pa
ightarrow_{92}^{234}U+_{-1}^0e + \gamma$.

Step5: Analyze Uranium - 234 decay

For Uranium - 234 ($_{92}^{234}U$) undergoing alpha + gamma decay, the mass number decreases by 4 and the atomic number decreases by 2. The resulting element is Thorium - 230. The decay equation is $_{92}^{234}U
ightarrow_{90}^{230}Th+_{2}^4He+\gamma$.

Step6: Analyze Thorium - 230 decay

For Thorium - 230 ($_{90}^{230}Th$) undergoing alpha + gamma decay, the mass number decreases by 4 and the atomic number decreases by 2. The resulting element is Radium - 226. The decay equation is $_{90}^{230}Th
ightarrow_{88}^{226}Ra+_{2}^4He+\gamma$.

Step7: Analyze Radium - 226 decay

For Radium - 226 ($_{88}^{226}Ra$) undergoing alpha + gamma decay, the mass number decreases by 4 and the atomic number decreases by 2. The resulting element is Radon - 222. The decay equation is $_{88}^{226}Ra
ightarrow_{86}^{222}Rn+_{2}^4He+\gamma$.

Step8: Analyze Radon - 222 decay

For Radon - 222 ($_{86}^{222}Rn$) undergoing alpha decay, the mass number decreases by 4 and the atomic number decreases by 2. The resulting element is Polonium - 218. The decay equation is $_{86}^{222}Rn
ightarrow_{84}^{218}Po+_{2}^4He$.

Step9: Analyze Polonium - 218 decay

For Polonium - 218 ($_{84}^{218}Po$) undergoing alpha decay, the mass number decreases by 4 and the atomic number decreases by 2. The resulting element is Lead - 214. The decay equation is $_{84}^{218}Po
ightarrow_{82}^{214}Pb+_{2}^4He$.

Step10: Analyze Lead - 214 decay

For Lead - 214 ($_{82}^{214}Pb$) undergoing beta + gamma decay, a neutron turns into a proton. The resulting element is Bismuth - 214. The decay equation is $_{82}^{214}Pb
ightarrow_{83}^{214}Bi+_{-1}^0e+\gamma$.

Step11: Analyze Bismuth - 214 decay

For Bismuth - 214 ($_{83}^{214}Bi$) undergoing beta + gamma decay, a neutron turns into a proton. The resulting element is Polonium - 214. The decay equation is $_{83}^{214}Bi
ightarrow_{84}^{214}Po+_{-1}^0e+\gamma$.

Step12: Analyze Polonium - 214 decay

For Polonium - 214 ($_{84}^{214}Po$) undergoing alpha decay, the mass number decreases by 4 and the atomic number decreases by 2. The resulting element is Lead - 210. The decay equation is $_{84}^{214}Po
ightarrow_{82}^{210}Pb+_{2}^4He$.

Step13: Analyze Lead - 210 decay

For Lead - 210 ($_{82}^{210}Pb$) undergoing beta + gamma decay, a neutron turns into a proton. The resulting element is Bismuth - 210. The decay equation is $_{82}^{210}Pb
ightarrow_{83}^{210}Bi+_{-1}^0e+\gamma$.

Step14: Analyze Bismuth - 210 decay

For Bismuth - 210 ($_{83}^{210}Bi$) undergoing beta + gamma decay, a neutron turns into a proton. The resulting element is Polonium - 210. The decay equation is $_{83}^{210}Bi
ightarrow_{84}^{210}Po+_{-1}^0e+\gamma$.

Step15: Analyze Polonium - 210 decay

For Polonium - 210 ($_{84}^{210}Po$) undergoing alpha decay, the mass number decreases by 4 and the atomic number decreases by 2. The resulting element is Lead - 206 which is stable. The decay equation is $_{84}^{210}Po
ightarrow_{82}^{206}Pb+_{2}^4He$.

Answer:

$_{90}^{234}Th
ightarrow_{91}^{234}Pa + _{-1}^0e+\gamma$; $_{91}^{234}Pa
ightarrow_{92}^{234}U+_{-1}^0e + \gamma$; $_{92}^{234}U
ightarrow_{90}^{230}Th+_{2}^4He+\gamma$; $_{90}^{230}Th
ightarrow_{88}^{226}Ra+_{2}^4He+\gamma$; $_{88}^{226}Ra
ightarrow_{86}^{222}Rn+_{2}^4He+\gamma$; $_{86}^{222}Rn
ightarrow_{84}^{218}Po+_{2}^4He$; $_{84}^{218}Po
ightarrow_{82}^{214}Pb+_{2}^4He$; $_{82}^{214}Pb
ightarrow_{83}^{214}Bi+_{-1}^0e+\gamma$; $_{83}^{214}Bi
ightarrow_{84}^{214}Po+_{-1}^0e+\gamma$; $_{84}^{214}Po
ightarrow_{82}^{210}Pb+_{2}^4He$; $_{82}^{210}Pb
ightarrow_{83}^{210}Bi+_{-1}^0e+\gamma$; $_{83}^{210}Bi
ightarrow_{84}^{210}Po+_{-1}^0e+\gamma$; $_{84}^{210}Po
ightarrow_{82}^{206}Pb+_{2}^4He$