The radioactive isotope `._(90)^(234)Th` undergoes two successive `beta-` decay followed by one `alpha-` decay. The atomic number and mass number respectively of the resulting atom is:

Options

A. 92 and 237
B. 94 and 230
C. 90 and 230
D. 92 and 230

Answer

To solve the problem, we need to analyze the decay process of the radioactive isotope \( _{90}^{234}\text{Th} \) (Thorium-234) step by step.

### Step 1: First Beta Decay
In the first beta decay, a neutron in the Thorium-234 nucleus is converted into a proton, resulting in the emission of a beta particle (electron) and an antineutrino. The atomic number increases by 1 while the mass number remains the same.

- **Initial Isotope**: \( _{90}^{234}\text{Th} \)
- **After First Beta Decay**:
\[ _{90}^{234}\text{Th} \rightarrow _{91}^{234}\text{Pa} + \beta^- + \bar{\nu} \]
Here, \( _{91}^{234}\text{Pa} \) is Protactinium-234.

### Step 2: Second Beta Decay
In the second beta decay, the newly formed Protactinium-234 undergoes the same process: a neutron is converted into a proton, resulting in another beta particle emission.

- **Initial Isotope**: \( _{91}^{234}\text{Pa} \)
- **After Second Beta Decay**:
\[ _{91}^{234}\text{Pa} \rightarrow _{92}^{234}\text{U} + \beta^- + \bar{\nu} \]
Here, \( _{92}^{234}\text{U} \) is Uranium-234.

### Step 3: Alpha Decay
In the final step, Uranium-234 undergoes alpha decay, which involves the emission of an alpha particle (which is essentially a helium nucleus, \( _{2}^{4}\text{He} \)). In this process, the atomic number decreases by 2 and the mass number decreases by 4.

- **Initial Isotope**: \( _{92}^{234}\text{U} \)
- **After Alpha Decay**:
\[ _{92}^{234}\text{U} \rightarrow _{90}^{230}\text{Th} + _{2}^{4}\text{He} \]
Here, \( _{90}^{230}\text{Th} \) is Thorium-230.

### Final Result
After completing all three decay processes, we find that the resulting atom is \( _{90}^{230}\text{Th} \).

- **Atomic Number**: 90
- **Mass Number**: 230

### Conclusion
The atomic number and mass number of the resulting atom are:
- **Atomic Number**: 90
- **Mass Number**: 230

### Answer
The atomic number and mass number respectively of the resulting atom is \( 90 \) and \( 230 \).

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