Gases are one of the basic types of systems we can study from a thermodynamic point of view. We don’t usually see gases around us (our atmosphere is made out of transparent gases), but if you look at a cloud of smoke, you can see that gases are made out of particles that move freely (which may seem random).
However, by imposing several restrictions on the type of interaction among particles (no loss of energy) and approximating the particles to an infinitesimally small body, we can obtain a simple model for the evolution of gases under certain thermodynamic conditions. This model is called the ideal gas model, and the laws that capture the relationship between the thermodynamic properties are called ideal gas laws. This model can accurately describe the behaviour of many gases under certain conditions.
The thermodynamic properties of gases
The thorough thermodynamic study of different systems involves many properties that have different meanings. However, it is enough to study a limited amount of properties to characterise the system’s features completely. In the case of ideal gases, these properties are temperature, pressure, and volume. Since thermodynamics is the statistical study of systems with many particles, all thermodynamic properties are statistical features that emerge from the microscopical structure.
Temperature as a thermodynamic property
The temperature is a measure of the average kinetic energy of particles in a system. It is denoted by the letter T. In thermodynamics, we use the unit Kelvin (K) for temperature.
The Kelvin scale uses absolute zero at its zero point. This means that 0K is the lowest possible temperature (where particles have no kinetic energy), which is equal to -273,15°C.
In general, all particles have different kinetic energies (associated with their state of movement). Due to the typical distributions of kinetic energy, the average temperature gives an important measure of the shape of this distribution. The typical distribution for ideal gases follows a law named after James Clerk Maxwell and Ludwig Boltzmann, namely the Maxwell-Boltzmann distribution.
Maxwell-Boltzmann distribution for different gases, commons.wikimedia.org
Volume as a thermodynamic property
Volume is denoted by the letter V. It is the sum of the volumes of all the particles that constitute a system or the total spatial volume occupied by the randomly moving particles. Unlike temperature, volume is an extensive property, which means that if the amount of matter that forms the system changes, the temperature would remain the same, but the volume would change.
Pressure as a thermodynamic property
Pressure, usually denoted by the letter P, is the measure of the average force per unit of area exerted by the particles on the boundaries of the volume it occupies. Like temperature, pressure is an intensive property, and it can also be interpreted as a measure of the density of energy of the system.
You might also see a lower-case p for pressure. Both p and P are used, but please always stick to what your teacher/textbook uses!
The ideal gas laws and formulae
In the case of ideal gases, three laws capture the relationships between temperature, pressure, and volume, namely Boyle’s law, Charles’ law, and Gay-Lussac’s law. Each law shows the relationship between two properties with a third that is kept constant.
Boyle’s law
Boyle’s law captures the relationship between pressure and volume for an isothermal process (constant temperature). The mathematical expression for this law is
\[P = \frac{k}{V} \space \text{or} \space P_1 \cdot V_1 = P_2 \cdot V_2\]
where k is a constant, and 1 and 2 indicate two different configurations of the system.
Boyle’s law indicates that whenever the temperature of an ideal gas is kept constant, the pressure depends inversely on the volume (and vice-versa).
Charles’ law
Charles’ law captures the relationship between temperature and volume for an isobaric process (constant pressure). The mathematical expression for this law is
\[V = k \cdot T \space \text{or} \space \frac{V_1}{T_1} = \frac{V_2}{T_2}\]
where k is a constant, and 1 and 2 indicate two different configurations of the system.
Charles’ law indicates that whenever the pressure of an ideal gas is kept constant, the volume is directly proportional to the temperature (and vice-versa).
Gay-Lussac’s law
Gay-Lussac’s law captures the relationship between pressure and temperature for an isochoric process (constant volume). The mathematical expression for this law is
\[P = k \cdot T \space \text{or} \space \frac{P_1}{T_1} = \frac{P_2}{T_2}\]
where k is a constant, and 1 and 2 indicate two different configurations of the system.
Gay-Lussac’s law indicates that whenever the volume of an ideal gas is kept constant, the pressure is directly proportional to the temperature (and vice-versa).
Check out our explanation on PV Diagrams, which are diagrams used to represent the thermodynamic stages of a process.
Combined gas law
The above three laws were discovered experimentally in laboratories. It was only later that they were theoretically understood as parts of a general combined law for ideal gases. The mathematical expression for this general combined law is
\[P \cdot V = n \cdot R \cdot T\]
where n is the amount of substance that forms the system and R is the ideal gas constant (with an approximate value of 8.314J/K·mol). Since R is a constant, and if we keep the number of particles constant, we can re-write the equation as
\[P \cdot V = k \cdot T\]
Here we can see that if we fix the pressure, the volume, or the temperature, we can derive the three laws from this expression.
Gas law equations
Here are some examples of using each law in calculations. Note that temperature is measured in K, pressure is measured in N/m2, and volume is measured in m3. Helpful tip: label each property in the example as V1, P2, T1, etc. This will help you plug in the values easily into the correct equation.
Example 1
Consider an ideal gas with a temperature of 100K. We start with the gas at 50N/m2 and 10m3. If we increase the volume to 50m3, what is the final pressure of the gas?
Solution
If we use Boyle’s law, the final volume will be
\[V_1 \cdot P_1 = V_2 \cdot P_2 \rightarrow P_2 = \frac{V_1 \cdot P_1}{V_2} = \frac{10 m^3 \cdot 50 N/m^2}{50 m^3} = 10 N/m^2\]
Example 2
Consider an ideal gas occupying a volume of 10m3. We start with the gas at 50N/m2 and 100K. If we increase the pressure to 100N/m2, what is the final temperature of the gas?
Solution
If we use Gay-Lussac’s law, the final temperature will be
\[\frac{P_1}{T_1} = \frac{P_2}{T_2} \rightarrow T_2 = \frac{P_2 \cdot T_1}{P_1} = \frac{100 N/m^2 \cdot 100 K}{50 N/m^2} = 200 K\]
Example 3
Consider an ideal gas at 50N/m2 of pressure. We start with the gas at 100K and 10m3. If we decrease the temperature to 10K, what is the final volume occupied by the gas?
If we use Charles’ law, the final volume will be
\[\frac{V_1}{T_1} = \frac{V_2}{T_2} \rightarrow V_2 = \frac{T_2 \cdot V_1}{T_1} = \frac{10 K \cdot 10 m^3}{100 K} = 1 m^3\]
Gas Laws - Key Takeaways
- Thermodynamics is the statistical study of many-particle systems. Gases are some of the more widely studied systems because of the freedom of their particles.
- An approximation of gases allows us to model their properties simply. This approximation is called the approximation of ideal gases.
- Usually, it is enough to use three thermodynamic properties: temperature, pressure, and volume. We have three laws that capture the relationship between these quantities for ideal gases, namely Boyle’s law, Charles’ law, and Gay-Lussac’s law.
- There is a general law for ideal gases that expresses the relationship between the three quantities and the amount of substance of the system.
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