Calculate the number of molecules in a sample of an ideal gas whose volume is 0.45 L at `67^(@)C` and 0.76 bar pressure.

Explanation

To calculate the number of molecules in a sample of an ideal gas, we will follow these steps:

### Step 1: Convert the temperature from Celsius to Kelvin
The temperature in Celsius is given as \(67^\circ C\). To convert this to Kelvin, we use the formula:
\[ T(K) = T(°C) + 273.15 \]
So,
\[ T = 67 + 273.15 = 340.15 \, K \]

### Step 2: Identify the given values
- Volume \(V = 0.45 \, L\)
- Pressure \(P = 0.76 \, bar\)
- Temperature \(T = 340.15 \, K\)

### Step 3: Use the Ideal Gas Law to find the number of moles
The Ideal Gas Law is given by the equation:
\[ PV = nRT \]
Where:
- \(P\) = pressure in bar
- \(V\) = volume in liters
- \(n\) = number of moles
- \(R\) = ideal gas constant (0.0831 L·bar/(K·mol))
- \(T\) = temperature in Kelvin

Rearranging the equation to solve for \(n\):
\[ n = \frac{PV}{RT} \]

### Step 4: Substitute the values into the equation
Substituting the known values into the equation:
\[ n = \frac{(0.76 \, bar)(0.45 \, L)}{(0.0831 \, L·bar/(K·mol))(340.15 \, K)} \]

### Step 5: Calculate the number of moles
Calculating the numerator:
\[ 0.76 \times 0.45 = 0.342 \, bar·L \]
Calculating the denominator:
\[ 0.0831 \times 340.15 \approx 28.287 \, L·bar/(K·mol) \]
Now substituting these values back into the equation for \(n\):
\[ n = \frac{0.342}{28.287} \approx 0.0121 \, mol \]

### Step 6: Calculate the number of molecules
To find the number of molecules, we use Avogadro's number:
\[ N = n \times N_A \]
Where \(N_A = 6.022 \times 10^{23} \, molecules/mol\).

Substituting the value of \(n\):
\[ N = 0.0121 \, mol \times 6.022 \times 10^{23} \, molecules/mol \]
Calculating:
\[ N \approx 7.29 \times 10^{21} \, molecules \]

### Final Answer
The number of molecules in the sample of the ideal gas is approximately \(7.29 \times 10^{21}\) molecules.

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