The radioactive decay `._(83)^(211)Birarr_(81)^(207)Ti` , takes place in 100L closed vessel at `27^(@)C` Starting with 2 mols of `_(83)^(211)Bi(t_(1//2)=130sec)` , the presuure development in the vessel after 520 sec will be:

Options

A. `1.875` atm
B. `0.2155atm`
C. `0.4618` atm
D. `4.618` atm

Answer

To solve the problem of radioactive decay and pressure development in a closed vessel, we will follow these steps:

### Step 1: Determine the number of half-lives that have passed
Given:
- Half-life of \( _{83}^{211}\text{Bi} \) = 130 seconds
- Total time = 520 seconds

To find the number of half-lives (\( n \)):
\[ n = \frac{\text{Total time}}{\text{Half-life}} = \frac{520 \text{ sec}}{130 \text{ sec}} = 4 \]

**Hint:** To find the number of half-lives, divide the total time by the half-life duration.

### Step 2: Calculate the remaining moles of \( _{83}^{211}\text{Bi} \)
Starting with 2 moles of \( _{83}^{211}\text{Bi} \):
Using the formula for remaining quantity after \( n \) half-lives:
\[ N_t = N_0 \left( \frac{1}{2} \right)^n \]
Where:
- \( N_0 = 2 \) moles (initial amount)
- \( n = 4 \) (number of half-lives)

Substituting the values:
\[ N_t = 2 \left( \frac{1}{2} \right)^4 = 2 \times \frac{1}{16} = 0.125 \text{ moles} \]

**Hint:** Use the formula for radioactive decay to find how much remains after several half-lives.

### Step 3: Calculate the amount of \( _{81}^{207}\text{Ti} \) produced
The amount of \( _{81}^{207}\text{Ti} \) produced is the difference between the initial moles and the remaining moles of \( _{83}^{211}\text{Bi} \):
\[ \text{Moles of } _{81}^{207}\text{Ti} = N_0 - N_t = 2 - 0.125 = 1.875 \text{ moles} \]

**Hint:** The amount of product formed is equal to the initial amount minus the remaining amount of the reactant.

### Step 4: Calculate the total moles in the vessel
The total moles in the vessel after decay:
\[ \text{Total moles} = \text{Remaining moles of } _{83}^{211}\text{Bi} + \text{Moles of } _{81}^{207}\text{Ti} = 0.125 + 1.875 = 2 \text{ moles} \]

**Hint:** Add the remaining reactant and the product to find the total moles present.

### Step 5: Calculate the pressure using the ideal gas law
Using the ideal gas law:
\[ PV = nRT \]
Where:
- \( P \) = pressure (atm)
- \( V \) = volume (L) = 100 L
- \( n \) = total moles = 2 moles
- \( R \) = ideal gas constant = 0.0821 L·atm/(K·mol)
- \( T \) = temperature in Kelvin = \( 27 + 273 = 300 \) K

Rearranging the equation to solve for \( P \):
\[ P = \frac{nRT}{V} \]
Substituting the values:
\[ P = \frac{2 \times 0.0821 \times 300}{100} = \frac{49.26}{100} = 0.4926 \text{ atm} \]

**Hint:** Use the ideal gas law to relate moles, pressure, volume, and temperature.

### Final Answer
The pressure development in the vessel after 520 seconds is approximately **0.4926 atm**.