Derivation of Boyle's Law from Kinetic Gas Equation

Derivation of Boyle's Law

Boyle's Law states that the volume ($V$) of a given mass of a gas is inversely proportional to its pressure ($P$) at a constant temperature ($T$).

$$\quad V \propto \frac{1}{P} \implies V = \frac{K}{P} \quad \implies P_1V_1 = P_2V_2$$

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1. The Kinetic Gas Equation

According to the kinetic theory of gases, the fundamental kinetic gas equation is given by:

$$PV = \frac{1}{3} m N v_{rms}^2$$

Where:

• $P$ = Pressure of the gas
• $V$ = Volume of the gas
• $m$ = Mass of a single gas molecule
• $N$ = Total number of molecules in volume $V$
• $v_{rms}$ = Root mean square velocity of the gas molecules

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2. Introducing Kinetic Energy

To relate this equation to temperature, we can manipulate the right side of the equation to feature the average translation kinetic energy expression ($\frac{1}{2}mv^2$). Multiply and divide the right side by 2:

$$PV = \frac{2}{3} \cdot N \left( \frac{1}{2} m v_{rms}^2 \right)$$

Here, $\frac{1}{2} m v_{rms}^2$ represents the average kinetic energy ($K.E.$) of a single molecule.

$$PV = \frac{2}{3} N (K.E.)$$

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3. Applying the Temperature Condition

According to the kinetic theory of gases, the average kinetic energy of gas molecules is directly proportional to its absolute temperature ($T$):

$$K.E. \propto T \quad \implies \quad K.E. = k T$$

(where $k$ is a proportionality constant)

Substituting this back into our pressure-volume equation:

$$PV = \frac{2}{3} N k T$$

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4. Final Derivation (At Constant Temperature)

If the temperature ($T$) is kept constant (as specified by Boyle's Law) and the mass of the gas is fixed (meaning the number of molecules, $N$, is constant):

• $\frac{2}{3}$ is a constant.
• $N$ is a constant.
• $k$ is a constant.
• $T$ is a constant.

Therefore, the entire right side of the equation becomes a single constant value, which we can call $K$:

$$PV = K \quad (\text{at constant } T)$$

This can be rewritten as:

$$V = \frac{K}{P} \quad \implies \quad V \propto \frac{1}{P}$$

See Diagram

This is Boyle's Law, proving that at a constant temperature, the volume of a fixed mass of gas is inversely proportional to its pressure.