Volume and pressure

Inverse Proportionality and Boyle's Law

For a fixed mass of gas held at a constant temperature, pressure and volume are inversely proportional. This means that if you double the volume of the gas, the pressure is halved. If you halve the volume, the pressure doubles.

Because they change at the exact same rate in opposite directions, multiplying the pressure by the volume always produces a constant value ($p \times V = \text{constant}$). This relationship is expressed in the equation:

Where:

• $p_1$ and $p_2$ = initial and final pressures in Pascal (Pa)
• $V_1$ and $V_2$ = initial and final volumes in cubic metres ($m^3$) or cubic centimetres ($cm^3$)

Worked Example

A sealed weather balloon holds a gas with a volume of $2.5\,m^3$ at a pressure of $150,000\,Pa$. As it rises slightly, the gas expands to a new volume of $5.0\,m^3$. Assuming the temperature inside the balloon remains constant, calculate the new pressure of the gas.

Step 1: Identify the known values.

• $p_1 = 150,000\,Pa$
• $V_1 = 2.5\,m^3$
• $V_2 = 5.0\,m^3$

Step 2: Substitute these values into the equation.

• $150,000 \times 2.5 = p_2 \times 5.0$

Step 3: Calculate the constant ($p_1 \times V_1$).

• $375,000 = p_2 \times 5.0$

Step 4: Rearrange to solve for the final pressure ($p_2$).

• $p_2 = \frac{375,000}{5.0}$
• $p_2 = 75,000\,Pa$