In a container of 1L there are 10 to the power 23 molecules each of mass 10 to the power - 22g if rms speed is 1000cm/s.calulate pressure in S. I unit

To find the pressure in SI units for the given container, we can use the ideal gas law and some fundamental principles of kinetic theory. Let's break this down step by step.

Understanding the Problem

We have a container with a volume of 1 liter, which is equivalent to 0.001 cubic meters (since 1 L = 0.001 m³). The number of molecules in the container is given as \(10^{23}\), and each molecule has a mass of \(10^{-22}\) grams. We also know the root mean square (rms) speed of the molecules is 1000 cm/s, which we need to convert to meters per second for our calculations.

Converting Units

First, let's convert the rms speed from centimeters per second to meters per second:

• 1000 cm/s = 1000 / 100 = 10 m/s

Calculating the Pressure

We can use the formula for pressure derived from the kinetic theory of gases, which is given by:

P = (1/3) * (N/V) * m * v²

Where:

• P = pressure in pascals (Pa)
• N = number of molecules
• V = volume of the container in cubic meters
• m = mass of a single molecule in kilograms
• v = rms speed in meters per second

Calculating Each Component

Now, let's calculate each component:

• N = \(10^{23}\)
• V = 0.001 m³
• m = \(10^{-22}\) grams = \(10^{-22} \times 10^{-3}\) kg = \(10^{-25}\) kg
• v = 10 m/s

Substituting Values

Now we can substitute these values into the pressure formula:

P = (1/3) * (10^{23} / 0.001) * (10^{-25}) * (10)²

Calculating this step by step:

• First, calculate \(N/V\): \(10^{23} / 0.001 = 10^{26}\)
• Next, calculate \(v²\): \(10² = 100\)
• Now, substitute these into the equation:

P = (1/3) * (10^{26}) * (10^{-25}) * 100

This simplifies to:

P = (1/3) * (10^{26} * 10^{-25} * 100) = (1/3) * (10^{1} * 100) = (1/3) * 10^{3} = \frac{1000}{3} \approx 333.33 \text{ Pa}

Final Result

The pressure in the container is approximately 333.33 pascals (Pa). This calculation illustrates how molecular properties and kinetic theory can be applied to determine macroscopic properties like pressure in a gas.