A flask contains `10^(-3)m^(3)` gas. At a temperature, the number of molecules of oxygen at `3.0xx10^(22)`. The mass of an oxygen molecule is `5.3xx10^(-26)` kg and at that temperature the rms velocity of molecules is 400m/s. The pressure in `N//m^(2)` of the gas in the flask is ..............

Answer

To find the pressure of the gas in the flask, we can use the formula derived from the kinetic theory of gases:

\[ P = \frac{1}{3} \frac{m \cdot n \cdot (V_{rms})^2}{V} \]

Where:
- \( P \) = pressure in \( N/m^2 \)
- \( m \) = mass of a single molecule of gas (in kg)
- \( n \) = number of molecules of gas
- \( V_{rms} \) = root mean square velocity of the gas molecules (in m/s)
- \( V \) = volume of the gas (in \( m^3 \))

### Step 1: Identify the given values
- Volume \( V = 10^{-3} \, m^3 \)
- Number of molecules \( n = 3.0 \times 10^{22} \)
- Mass of an oxygen molecule \( m = 5.3 \times 10^{-26} \, kg \)
- Root mean square velocity \( V_{rms} = 400 \, m/s \)

### Step 2: Substitute the values into the formula
Substituting the known values into the pressure formula:

\[ P = \frac{1}{3} \cdot \frac{(5.3 \times 10^{-26} \, kg) \cdot (3.0 \times 10^{22}) \cdot (400 \, m/s)^2}{10^{-3} \, m^3} \]

### Step 3: Calculate \( (V_{rms})^2 \)
First, calculate \( (V_{rms})^2 \):

\[ (V_{rms})^2 = (400)^2 = 160000 \, m^2/s^2 \]

### Step 4: Substitute \( (V_{rms})^2 \) back into the formula
Now substitute \( (V_{rms})^2 \) back into the pressure equation:

\[ P = \frac{1}{3} \cdot \frac{(5.3 \times 10^{-26}) \cdot (3.0 \times 10^{22}) \cdot (160000)}{10^{-3}} \]

### Step 5: Simplify the equation
Now, simplify the expression:

1. Calculate \( 5.3 \times 3.0 = 15.9 \)
2. Now multiply \( 15.9 \times 160000 = 2544000 \)
3. Now, divide by \( 10^{-3} \) which is equivalent to multiplying by \( 10^3 \):

\[ P = \frac{1}{3} \cdot \frac{2544000}{10^{-3}} = \frac{2544000 \cdot 10^3}{3} \]

### Step 6: Calculate the final pressure
Now calculate:

\[ P = \frac{2544000000}{3} \approx 848000000 \, N/m^2 \]

### Step 7: Convert to scientific notation
Convert \( 848000000 \) to scientific notation:

\[ P \approx 8.48 \times 10^{8} \, N/m^2 \]

### Final Answer
The pressure of the gas in the flask is approximately:

\[ P \approx 8.48 \times 10^4 \, N/m^2 \]