Show that "PV = Constant."

To show that PV = Constant, we can use the ideal gas law equation:

PV = nRT

Where:
P = pressure
V = volume
n = number of moles
R = gas constant
T = temperature

If we assume that the number of moles (n) and the gas constant (R) are constant, then the equation simplifies to:

PV = Constant

This means that the product of pressure and volume is constant as long as the number of moles and the gas constant remain constant. This relationship is known as Boyle's Law, which states that the pressure of a gas is inversely proportional to its volume at constant temperature.

Boyle's Law:

Boyle's law states that the volume of a gas at constant temperature is inversely proportional to its pressure. Mathematically, it can be expressed as:

```
P ∝ 1/V
```

or

```
P - V = Constant
```

Derivation:

Consider a gas confined to a cylinder fitted with a movable piston. The pressure of the gas on the piston is given by:

```
P = F/A
```

where:

- P is the pressure
- F is the force exerted by the gas
- A is the cross-sectional area of the piston

According to Boyle's law, the volume of the gas is inversely proportional to its pressure. Therefore, if we decrease the volume by moving the piston inward, the pressure increases. Conversely, if we increase the volume by moving the piston outward, the pressure decreases.

Let's assume we have a gas at a certain volume V1 and pressure P1. If we decrease the volume to V2, the pressure will increase to P2. According to Boyle's law, we have:

```
P1 - V1 = P2 - V2
```

This shows that the product of the pressure and volume remains constant as long as the temperature is kept constant. Therefore, we can write:

```
PV = Constant
```

Conclusion:

Boyle's law shows that under constant temperature, the pressure and volume of a gas are inversely related. The product of pressure and volume is a constant. This principle is essential for understanding the behavior of gases in various applications, such as pneumatics and gas compression systems.