The pressure inside an electric bulb is `10^(-3)mm` of Hg at a temperature of `25^(@)C`. If the volume of the bulb is `10^(-4)m^(3)`, find the number of molecules contained in it.

Explanation

To find the number of molecules contained in the electric bulb, we will use the ideal gas equation and Avogadro's number. Here’s a step-by-step solution:

### Step 1: Convert Pressure from mm of Hg to Pascals
The pressure inside the bulb is given as \(10^{-3}\) mm of Hg. We need to convert this to Pascals using the conversion factor \(1 \, \text{mm of Hg} = 133.3 \, \text{Pa}\).

\[ P = 10^{-3} \, \text{mm of Hg} \times 133.3 \, \text{Pa/mm of Hg} = 133.3 \times 10^{-3} \, \text{Pa} = 0.1333 \, \text{Pa} \]

### Step 2: Convert Temperature from Celsius to Kelvin
The temperature is given as \(25^\circ C\). We convert this to Kelvin using the formula \(T(K) = T(°C) + 273.15\).

\[ T = 25 + 273.15 = 298.15 \, K \approx 298 \, K \]

### Step 3: Use the Ideal Gas Equation
The ideal gas equation is given by:

\[ PV = nRT \]

Where:
- \(P\) = pressure in Pascals
- \(V\) = volume in cubic meters
- \(n\) = number of moles
- \(R\) = universal gas constant \(8.314 \, \text{J/(mol K)}\)
- \(T\) = temperature in Kelvin

Substituting the known values into the equation:

\[ (133.3 \times 10^{-3} \, \text{Pa}) \times (10^{-4} \, \text{m}^3) = n \times (8.314 \, \text{J/(mol K)}) \times (298 \, K) \]

### Step 4: Solve for the Number of Moles (n)
Rearranging the equation to solve for \(n\):

\[ n = \frac{(133.3 \times 10^{-3}) \times (10^{-4})}{8.314 \times 298} \]

Calculating \(n\):

\[ n = \frac{(133.3 \times 10^{-7})}{(8.314 \times 298)} \approx 5.45 \times 10^{-9} \, \text{moles} \]

### Step 5: Calculate the Number of Molecules
To find the number of molecules, we use Avogadro's number, which is \(6.022 \times 10^{23} \, \text{molecules/mol}\):

\[ \text{Number of molecules} = n \times N_A = (5.45 \times 10^{-9} \, \text{moles}) \times (6.022 \times 10^{23} \, \text{molecules/mol}) \]

Calculating the number of molecules:

\[ \text{Number of molecules} \approx 32.84 \times 10^{14} \, \text{molecules} \]

### Final Answer
The number of molecules contained in the electric bulb is approximately \(32.84 \times 10^{14}\) molecules.

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