Empirical Formulation of the Ideal Gas

Have you ever wondered why a balloon expands when heated or why a tire’s pressure changes with altitude? In this lesson, we will review the laws governing these behaviors and how they lead to the ideal gas equation, along with its considerations and key points.

Learning Objectives
At the end of this lesson, the student will be able to:

1. Explain the empirical laws of ideal gases (Boyle–Mariotte, Charles, Gay-Lussac) and their synthesis in the equation of state (PV = nRT, PV = N k_B T).
2. Apply the ideal gas equation and the relation PV/T = cte to solve state changes with consistent units.
3. Analyze isothermal, isobaric, and isochoric processes and their trajectories in P–V, V–T, and P–T diagrams.
4. Recognize the range of validity of the ideal gas model and select alternative models (van der Waals, quantum, relativistic) when appropriate.

TABLE OF CONTENTS
Fundamental empirical laws
Combination of laws into the ideal gas equation
Process-based deductions
Comments and microscopic background
Range of validity and limitations
Practical notes

Fundamental Empirical Laws

Experiments with gases show a dependence between pressure P, volume V, and temperature T. Under controlled conditions, three fundamental empirical laws are observed:

1. Boyle–Mariotte’s Law (isothermal): In a process at constant temperature, the volume and pressure of a gas are inversely proportional; that is:
   P \propto \dfrac{1}{V}\quad\Leftrightarrow\quad PV=\text{cte.}
   Example: A gas that, at an initial pressure of P_1 = 15\ \mathrm{MPa}, expands isothermally from an initial volume V_1 = 1{,}00\ \mathrm{L} to a final volume V_2 = 2{,}00\ \mathrm{L} will have its pressure reduced by half. Since the product PV=\text{cte.}, it follows that P_1 V_1 = P_2 V_2, leading to:

   P_2 = \dfrac{P_1 V_1}{V_2} = 15\ \mathrm{MPa}\left(\dfrac{1{,}00\ \mathrm{L}}{2{,}00\ \mathrm{L}}\right) = 7{,}50\ \mathrm{MPa}

   
2. Charles’s Law (isobaric): In a process at constant pressure, the volume and temperature of a gas are directly proportional; that is:
   V \propto T \quad\Leftrightarrow\quad \dfrac{V}{T}=\text{cte.}
   Example: A gas that, at an initial temperature of T_1 = 300\ \mathrm{K}, is heated isobarically up to T_2 = 450\ \mathrm{K} starting from an initial volume V_1 = 2{,}00\ \mathrm{L} will see its volume increase by 50 % (a factor of \tfrac{3}{2}). Since \tfrac{V}{T}=\text{cte.}, it follows that \dfrac{V_1}{T_1}=\dfrac{V_2}{T_2}, leading to:

   V_2 = V_1 \cdot \dfrac{T_2}{T_1} = 2{,}00\ \mathrm{L}\left(\dfrac{450\ \mathrm{K}}{300\ \mathrm{K}}\right) = 3{,}00\ \mathrm{L}

   

3. Gay-Lussac’s Law (isochoric): In a process at constant volume, the pressure and temperature of a gas are directly proportional; that is:
   P \propto T \quad\Leftrightarrow\quad \dfrac{P}{T}=\text{cte.}
   Example: A gas that, at an initial temperature of T_1 = 300\ \mathrm{K}, is heated isochorically up to T_2 = 450\ \mathrm{K} starting from an initial pressure P_1 = 1{,}00\ \mathrm{MPa} will experience an increase in pressure in the same proportion. Since \tfrac{P}{T}=\text{cte.}, it follows that \dfrac{P_1}{T_1}=\dfrac{P_2}{T_2}, leading to:

   P_2 = P_1 \cdot \dfrac{T_2}{T_1} = 1{,}00\ \mathrm{MPa}\left(\dfrac{450\ \mathrm{K}}{300\ \mathrm{K}}\right) = 1{,}50\ \mathrm{MPa}

   

Combination of the Laws into the Ideal Gas Equation

These three laws can be synthesized into a single proportional relationship:

PV \propto T

Where, based on experimental and microscopic considerations, it is possible to infer the proportionality constant as the product of the number of particles N and the Boltzmann constant k_B = 1{,}380\,649\times10^{-23}\ \mathrm{J\,K^{-1}}, obtaining the microscopic relation:

\boxed{PV = N\,k_B\,T}

Similarly, in molar terms, the proportionality constant is obtained as the product of the number of moles n and the universal gas constant R=8{,}314\,462\,6\ \mathrm{J\,mol^{-1}\,K^{-1}}=0{,}082\,057\ \mathrm{L\,atm\,mol^{-1}\,K^{-1}}, leading to the molar relation:

\boxed{PV = n\,R\,T}

Regardless of the case, the fact remains that there is a direct proportionality between the product PV and T, which is equivalent to stating that if an ideal gas moves between two states, one with initial values (P_\alpha, V_\alpha, T_\alpha) and another with final values (P_\omega, V_\omega, T_\omega), then they satisfy the relation

\dfrac{P_\alpha V_\alpha}{T_\alpha} = \dfrac{P_\omega V_\omega}{T_\omega}

and therefore, PV/T = cte.

This relationship, which can serve as an experimental foundation to formulate both the microscopic and molar expressions, can be directly inferred from the experimental laws of Boyle-Mariotte, Charles, and Gay-Lussac. The reasoning is as follows:

This can be demonstrated through three paths:

1. A change in volume through an isothermal and an isobaric process
2. A change in pressure through an isothermal and an isochoric process
3. A change in temperature through an isobaric and an isochoric process

For the development of these three cases, we will need an intermediate state with values (P_i,V_i,T_i)

Deductions by Processes

Deduction by Change of Volume

If the initial state (P_\alpha,V_\alpha,T_\alpha) is connected to the intermediate state (P_i,V_i,T_i) through an isothermal process, and then the intermediate state is connected to the final state (P_\omega,V_\omega,T_\omega) through an isobaric process, then:

\begin{array}{rclcl} & P_\alpha V_\alpha= P_i V_i & & V_i/T_i = V_\omega/T_\omega & \\ &\text{isothermal}& &\text{isobaric} & \\ P_\alpha & \longrightarrow & P_i = \dfrac{P_\alpha V_\alpha}{V_i} & \longrightarrow & P_\omega = P_i \\ \\ V_\alpha & \longrightarrow & V_i = \dfrac{P_\alpha V_\alpha}{P_i} & \longrightarrow & V_\omega = \dfrac{V_i T_\omega}{T_i} \\ \\ T_\alpha & \longrightarrow & T_i = T_\alpha & \longrightarrow & T_\omega = \dfrac{V_\omega T_i}{V_i} \end{array}

From this, it follows that:

\begin{array}{rl} & V_\omega = \left(\dfrac{T_\omega}{T_i}\right) V_i = \left(\dfrac{T_\omega}{T_i}\right) \left(\dfrac{P_\alpha}{P_i} \right) V_\alpha = \dfrac{T_\omega P_\alpha V_\alpha}{T_\alpha P_\omega} \\ \\ \equiv & \dfrac{P_\alpha V_\alpha}{T_\alpha} = \dfrac{P_\omega V_\omega}{T_\omega} \end{array}

Deduction by Change of Pressure

If the initial state (P_\alpha,V_\alpha,T_\alpha) is connected to the intermediate state (P_i,V_i,T_i) through an isothermal process, and then the intermediate state is connected to the final state (P_\omega,V_\omega,T_\omega) through an isochoric process, then:

\begin{array}{rclcl} & P_\alpha V_\alpha= P_i V_i & & P_i/T_i = P_\omega/T_\omega & \\ &\text{isothermal}& &\text{isochoric} & \\ P_\alpha & \longrightarrow & P_i = \dfrac{P_\alpha V_\alpha}{V_i} & \longrightarrow & P_\omega = \dfrac{P_i T_\omega}{T_i} \\ \\ V_\alpha & \longrightarrow & V_i = \dfrac{P_\alpha V_\alpha}{P_i} & \longrightarrow & V_\omega = V_i \\ \\ T_\alpha & \longrightarrow & T_i = T_\alpha & \longrightarrow & T_\omega = \dfrac{V_\omega T_i}{V_i} \end{array}

From this, it follows that:

\begin{array}{rl} & P_\omega = \left(\dfrac{T_\omega}{T_i}\right) P_i = \left(\dfrac{T_\omega}{T_i}\right) \left(\dfrac{V_\alpha}{V_i}\right)P_\alpha = \dfrac{T_\omega V_\alpha P_\alpha}{T_\alpha V_\omega} \\ \\ \equiv & \dfrac{P_\alpha V_\alpha}{T_\alpha} = \dfrac{P_\omega V_\omega}{T_\omega} \end{array}

Deduction by Change of Temperature

If the initial state (P_\alpha,V_\alpha,T_\alpha) is connected to the intermediate state (P_i,V_i,T_i) through an isobaric process, and then the intermediate state is connected to the final state (P_\omega,V_\omega,T_\omega) through an isochoric process, then:

\begin{array}{rclcl} & V_\alpha/ T_\alpha= V_i / T_i & & P_i/T_i = P_\omega/T_\omega & \\ &\text{isobaric}& &\text{isochoric} & \\ P_\alpha & \longrightarrow & P_i = P_\alpha & \longrightarrow & P_\omega = \dfrac{P_i T_\omega}{T_i} \\ \\ V_\alpha & \longrightarrow & V_i = \dfrac{V_\alpha T_i}{T_\alpha} & \longrightarrow & V_\omega = V_i \\ \\ T_\alpha & \longrightarrow & T_i = \dfrac{V_i T_\alpha}{V_\alpha} & \longrightarrow & T_\omega = \dfrac{P_\omega T_i}{P_i} \end{array}

From this, it follows that:

\begin{array}{rl} & T_\omega = \left(\dfrac{P_\omega}{P_i}\right) T_i = \left(\dfrac{P_\omega}{P_i}\right) \left(\dfrac{V_i}{V_\alpha}\right)T_\alpha = \dfrac{P_\omega V_\omega T_\alpha}{P_\alpha V_\alpha} \\ \\ \equiv & \dfrac{P_\alpha V_\alpha}{T_\alpha} = \dfrac{P_\omega V_\omega}{T_\omega} \end{array}

Comments and Microscopic Background

Although the previous formulation is empirical, it can be derived from first principles through the Kinetic Theory of Gases. In this model, the gas is a collection of particles that move and collide with each other and with the walls of the container. It is idealized with assumptions such as:

1. Absence of long-range attractive or repulsive forces between particles.
2. Point-like or negligibly small spherical particles.
3. Perfectly elastic collisions between particles and with the walls.

These idealizations simplify the analysis and, although no real gas perfectly satisfies them, they describe many gases well across a wide range of conditions and provide a foundation for Classical Thermodynamics, with applications ranging from heat engines to atmospheric and astrophysical physics.

Scope of Validity and Limitations

The ideal gas law is not universal. It deviates when the above assumptions cease to be reasonable or when effects beyond classical physics emerge.

• High pressures and low temperatures: interactions between molecules are no longer negligible and the finite size of the particles becomes significant. A common correction is the van der Waals equation:
  \left(P + a\left(\dfrac{n}{V}\right)^2\right)\,(V - nb)=nRT

  with parameters a and b characteristic of each gas.

• Quantum regime: at very low temperatures or high densities, Bose–Einstein or Fermi–Dirac statistics appear, requiring quantum gas models.
• Relativistic regime: if particles move at speeds close to the speed of light, relativistic corrections are required.

Practical Notes

• Always use temperature in Kelvin in the formulas: T(\mathrm{K}) = T(^{\circ}\mathrm{C}) + 273{,}15.
• Ensure unit consistency: if working with \mathrm{atm} and \mathrm{L}, use R=0{,}082\,057\ \mathrm{L\,atm\,mol^{-1}\,K^{-1}}; if using \mathrm{Pa} and \mathrm{m^3}, use R=8{,}314\,462\,6\ \mathrm{J\,mol^{-1}\,K^{-1}}.
• Remember that each empirical law was obtained by keeping one variable constant. Combining results requires understanding which thermodynamic process is being performed at each stage.