A flask contains 10⁻³ m³ gas. At a temperature, the number of molecules of oxygen are 3.0x10²² . The mass of an oxygen molecule is 5.3 10⁻²⁶ kg and at that temperature the rms velocity of molecules is 400 m/s. The pressure in N /m² of the gas in the flask is

To find the pressure of the gas in the flask, we can use the kinetic theory of gases. The pressure $P$ exerted by an ideal gas is given by the formula:
$P=\frac{1}{3}\frac{Nm{v}_{\text{rms}}^2}{V}$
where:
- $N$ is the total number of molecules,
- $m$ is the mass of a single molecule,
- $v_{\text{rms}}$ is the root-mean-square (rms) velocity of the molecules, and
- $V$ is the volume of the flask.

From the given problem, we have the following values:
- Volume of the flask, $V={10}^{-3}{\text{m}}^3$
- Number of oxygen molecules, $N=3.0\times {10}^{22}$
- Mass of an oxygen molecule, $m=5.3\times {10}^{-26}\text{kg}$
- Root-mean-square velocity, $v_{\text{rms}}=400\text{m/s}$

Substitute these values into the pressure formula:
$P=\frac{1}{3}\times \frac{(3.0\times {10}^{22})\times (5.3\times {10}^{-26}\text{kg})\times {\left(400\text{m/s}\right)}^2}{{10}^{-3}{\text{m}}^3}$

First, simplify the numerator term by term:
- The coefficient $\frac{1}{3}\times 3.0=1.0$
- Now we have:
$P=\frac{1.0\times {10}^{22}\times 5.3\times {10}^{-26}\times 160000}{{10}^{-3}}$
- Multiply the power of tens and coefficients:
$5.3\times 160000=848000=8.48\times {10}^5$
- Combine the powers of 10 in the numerator:
${10}^{22}\times {10}^{-26}={10}^{-4}$
- Thus, the numerator becomes:
$8.48\times {10}^5\times {10}^{-4}=8.48\times 10=84.8$

Now, divide by the volume in the denominator:
$P=\frac{84.8}{{10}^{-3}}=84.8\times {10}^3=8.48\times {10}^4{\text{N/m}}^2$

Therefore, the pressure of the gas in the flask is $8.48\times {10}^4{\text{N/m}}^2$.