A vessel of `831 cc` contains `._(1)^(3)H` atm and `27^(@)C`. If half life of `._(1)^(3)H` is `12.3` years then the activity of the gas is

Options

A. `3.04 xx 10^(13)dps`
B. `582 Ci`
C. `2.15 xx 10^(13)dps`
D. `823 Ci`

Explanation

To solve the problem, we need to calculate the activity of the gas in the vessel containing tritium (³H) at a given pressure and temperature. Here's the step-by-step solution:

### Step 1: Convert Temperature to Kelvin
The temperature is given as \(27^\circ C\). To convert this to Kelvin:
\[ T(K) = T(°C) + 273.15 = 27 + 273.15 = 300.15 \, K \approx 300 \, K \]

**Hint:** Always convert Celsius to Kelvin when using gas laws.

### Step 2: Use the Ideal Gas Law to Find the Number of Moles
The ideal gas law is given by:
\[ PV = nRT \]
Where:
- \(P\) = pressure in atm (given as \(0.6 \, atm\))
- \(V\) = volume in liters (convert \(831 \, cc\) to liters: \(831 \, cc = 0.831 \, L\))
- \(R\) = ideal gas constant \(= 0.0821 \, L \cdot atm/(mol \cdot K)\)
- \(T\) = temperature in Kelvin (calculated above)

Rearranging the equation to find \(n\):
\[ n = \frac{PV}{RT} \]
Substituting the values:
\[ n = \frac{(0.6 \, atm)(0.831 \, L)}{(0.0821 \, L \cdot atm/(mol \cdot K))(300 \, K)} \]
Calculating \(n\):
\[ n = \frac{0.4986}{24.63} \approx 0.0202 \, mol \]

**Hint:** Ensure units are consistent when using the ideal gas law.

### Step 3: Calculate the Number of Molecules
To find the number of molecules \(N\), use Avogadro's number (\(N_A = 6.022 \times 10^{23} \, molecules/mol\)):
\[ N = n \times N_A = 0.0202 \, mol \times 6.022 \times 10^{23} \, molecules/mol \approx 1.216 \times 10^{22} \, molecules \]

**Hint:** Avogadro's number helps convert moles to molecules.

### Step 4: Calculate the Decay Constant
The decay constant \(\lambda\) is given by:
\[ \lambda = \frac{\ln(2)}{t_{1/2}} \]
Where \(t_{1/2} = 12.3 \, years\). First, convert years to seconds:
\[ t_{1/2} = 12.3 \times 365 \times 24 \times 60 \times 60 \approx 3.88 \times 10^{11} \, seconds \]
Now calculate \(\lambda\):
\[ \lambda = \frac{0.693}{3.88 \times 10^{11}} \approx 1.79 \times 10^{-12} \, s^{-1} \]

**Hint:** Remember to convert half-life into the same time units as your decay constant.

### Step 5: Calculate the Activity
The activity \(A\) is given by:
\[ A = \lambda N \]
Substituting the values:
\[ A = (1.79 \times 10^{-12} \, s^{-1})(1.216 \times 10^{22} \, molecules) \approx 2.18 \times 10^{10} \, decays/s \]

**Hint:** Activity is a measure of the number of decays per second.

### Step 6: Convert Activity to Curie
To convert the activity from decays per second to Curie:
\[ 1 \, Curie = 3.7 \times 10^{10} \, decays/s \]
Thus,
\[ A_{Curie} = \frac{2.18 \times 10^{10}}{3.7 \times 10^{10}} \approx 0.59 \, Curie \]

**Hint:** Use the conversion factor to switch between different units of activity.

### Final Answer
The activity of the gas is approximately \(0.59 \, Curie\).