Derive an expression for pressure exterted by an ideal gas?

Answer

To derive the expression for the pressure exerted by an ideal gas, we will follow a systematic approach based on the kinetic theory of gases.

### Step-by-Step Derivation:

1. **Understanding Pressure**:
Pressure (P) is defined as the force (F) exerted per unit area (A) on the walls of a container by gas molecules. Mathematically, it is expressed as:
\[ P = \frac{F}{A} \]

2. **Consider a Cubical Container**:
Let's consider a cubic container with side length \(L\). The area of one face of the cube is:
\[ A = L^2 \]

3. **Momentum Change During Collision**:
When a gas molecule collides with the wall of the container, it exerts a force due to the change in momentum. If a molecule with mass \(m\) moves with velocity \(V_x\) in the x-direction, upon colliding elastically with the wall, its velocity will reverse. The change in momentum (\(\Delta p\)) for one collision is:
\[ \Delta p = m(V_x - (-V_x)) = 2mV_x \]

4. **Time Between Collisions**:
The time (\(t\)) taken by a molecule to travel to the wall and back is given by the distance traveled (which is \(2L\)) divided by its velocity:
\[ t = \frac{2L}{V_x} \]

5. **Force Calculation**:
The average force exerted by the molecule on the wall can be calculated using the change in momentum over the time interval:
\[ F = \frac{\Delta p}{t} = \frac{2mV_x}{\frac{2L}{V_x}} = \frac{mV_x^2}{L} \]

6. **Total Force from Multiple Molecules**:
If there are \(N\) molecules in the container, the total force exerted on the wall by all molecules is:
\[ F_{\text{total}} = N \cdot \frac{mV_x^2}{L} \]

7. **Substituting into Pressure Formula**:
Now, substituting the total force into the pressure formula:
\[ P = \frac{F_{\text{total}}}{A} = \frac{N \cdot \frac{mV_x^2}{L}}{L^2} = \frac{N m V_x^2}{L^3} \]

8. **Relating to Volume**:
The volume \(V\) of the cubical container is:
\[ V = L^3 \]
Thus, we can rewrite the pressure as:
\[ P = \frac{N m V_x^2}{V} \]

9. **Using Average Velocity**:
Since the gas molecules move in three dimensions, we can relate \(V_x\) to the root mean square velocity (\(V_{\text{rms}}\)):
\[ V_x^2 = \frac{1}{3} V_{\text{rms}}^2 \]
Therefore, substituting this into the pressure equation gives:
\[ P = \frac{N m \left(\frac{1}{3} V_{\text{rms}}^2\right)}{V} = \frac{N m V_{\text{rms}}^2}{3V} \]

10. **Final Expression**:
The final expression for the pressure exerted by an ideal gas is:
\[ P = \frac{1}{3} \frac{N m V_{\text{rms}}^2}{V} \]

### Summary:
The pressure exerted by an ideal gas can be expressed in terms of the number of molecules, their mass, and their root mean square velocity. This derivation illustrates the relationship between microscopic motion and macroscopic properties of gases.