The pressure of a gas in a `100 mL` container is `200 kPa` and the. Average translational kinetic energy of each gas particle is `6 xx 10^-21 J`. Find the number of gas particles in the container. How many moles are there in the container ?

Explanation

To solve the problem, we need to find the number of gas particles in a 100 mL container with a pressure of 200 kPa and an average translational kinetic energy of each gas particle of \(6 \times 10^{-21} \, \text{J}\). We will also calculate the number of moles of gas in the container.

### Step 1: Convert the Volume to Cubic Meters
The volume of the gas is given as 100 mL. We need to convert this to cubic meters (m³) for our calculations.

\[ \text{Volume} = 100 \, \text{mL} = 100 \times 10^{-6} \, \text{m}^3 = 1 \times 10^{-4} \, \text{m}^3 \]

### Step 2: Use the Ideal Gas Law
The ideal gas law is given by the equation:

\[ PV = nRT \]

Where:
- \(P\) = Pressure (in Pascals)
- \(V\) = Volume (in m³)
- \(n\) = Number of moles
- \(R\) = Universal gas constant (\(8.31 \, \text{J/(mol K)}\))
- \(T\) = Temperature (in Kelvin)

We can rearrange this equation to solve for \(n\):

\[ n = \frac{PV}{RT} \]

### Step 3: Find the Temperature (T)
The average translational kinetic energy of a gas particle is given by:

\[ KE = \frac{3}{2} k T \]

Where \(k\) is the Boltzmann constant (\(1.38 \times 10^{-23} \, \text{J/K}\)). We can rearrange this to find \(T\):

\[ T = \frac{2 \times KE}{3k} \]

Substituting the values:

\[ T = \frac{2 \times (6 \times 10^{-21})}{3 \times (1.38 \times 10^{-23})} \]

Calculating \(T\):

\[ T = \frac{12 \times 10^{-21}}{4.14 \times 10^{-23}} \approx 289.4 \, \text{K} \]

### Step 4: Substitute Values into the Ideal Gas Law
Now we can substitute the values into the ideal gas law equation:

- \(P = 200 \, \text{kPa} = 200 \times 10^{3} \, \text{Pa}\)
- \(V = 1 \times 10^{-4} \, \text{m}^3\)
- \(R = 8.31 \, \text{J/(mol K)}\)
- \(T \approx 289.4 \, \text{K}\)

Now substituting these values into the equation for \(n\):

\[ n = \frac{(200 \times 10^{3}) \times (1 \times 10^{-4})}{(8.31) \times (289.4)} \]

Calculating \(n\):

\[ n \approx \frac{20}{2406.694} \approx 0.00831 \, \text{mol} \]

### Step 5: Calculate the Number of Gas Particles
Using Avogadro's number (\(N_A = 6.022 \times 10^{23} \, \text{particles/mol}\)), we can find the number of gas particles:

\[ \text{Number of particles} = n \times N_A \]

Substituting the value of \(n\):

\[ \text{Number of particles} \approx 0.00831 \times (6.022 \times 10^{23}) \approx 5.00 \times 10^{21} \, \text{particles} \]

### Final Answers
- Number of gas particles: \(5.00 \times 10^{21}\)
- Number of moles: \(0.00831 \, \text{mol}\)