To solve the problem of calculating the number of molecules of oxygen in a 1-liter flask at 0°C and under a pressure of \(10^{-12}\) bar, we can use the Ideal Gas Law, which is expressed as:
\[
PV = nRT
\]
Where:
- \(P\) = pressure (in bar)
- \(V\) = volume (in liters)
- \(n\) = number of moles
- \(R\) = ideal gas constant
- \(T\) = temperature (in Kelvin)
### Step-by-step Solution:
**Step 1: Convert the temperature to Kelvin.**
- The temperature given is \(0^\circ C\).
- To convert Celsius to Kelvin, use the formula:
\[
T(K) = T(°C) + 273
\]
- Therefore,
\[
T = 0 + 273 = 273 \, K
\]
**Step 2: Identify the values for the Ideal Gas Law.**
- Given:
- Pressure, \(P = 10^{-12} \, \text{bar}\)
- Volume, \(V = 1 \, \text{liter}\)
- Gas constant, \(R = 0.083 \, \text{L bar K}^{-1} \text{mol}^{-1}\)
- Temperature, \(T = 273 \, K\)
**Step 3: Rearrange the Ideal Gas Law to solve for \(n\) (number of moles).**
- Rearranging gives:
\[
n = \frac{PV}{RT}
\]
**Step 4: Substitute the values into the equation.**
- Now substituting the known values:
\[
n = \frac{(10^{-12} \, \text{bar}) \times (1 \, \text{liter})}{(0.083 \, \text{L bar K}^{-1} \text{mol}^{-1}) \times (273 \, K)}
\]
**Step 5: Calculate \(n\).**
- Performing the calculation:
\[
n = \frac{10^{-12}}{0.083 \times 273} = \frac{10^{-12}}{22.659} \approx 4.41 \times 10^{-14} \, \text{moles}
\]
**Step 6: Calculate the number of molecules.**
- To find the number of molecules, use Avogadro's number (\(N_A = 6.022 \times 10^{23} \, \text{molecules/mol}\)):
\[
\text{Number of molecules} = n \times N_A
\]
- Substituting the values:
\[
\text{Number of molecules} = (4.41 \times 10^{-14} \, \text{moles}) \times (6.022 \times 10^{23} \, \text{molecules/mol})
\]
**Step 7: Perform the final calculation.**
- Calculating gives:
\[
\text{Number of molecules} \approx 2.65 \times 10^{10} \, \text{molecules}
\]
### Final Answer:
The number of molecules of oxygen in the flask is approximately \(2.65 \times 10^{10}\) molecules.
