Ideal Gas Model

The Ideal Gas Model Meaning

The ideal gas model allows us to understand the behavior of gases. Even though no gas is ideal in the real world, simplifying the concept results in good approximations of the behavior of the real gases in most conditions. From the study of the behavior of gases, certain generalizations were made. These generalizations are called gas laws. These gas laws give quantitative relationships between any two variables when the other two are kept constant. Let us discuss the various gas laws in detail.

The Ideal Gas Model Assumptions

Considering that the ideal gas model refers to a hypothetical system, certain assumptions must be followed to ensure consistency. The four main assumptions are:

• An ideal gas is made up of numerous identical point-like molecules, spread really far apart so that the intermolecular forces are negligible;

• Ideal gas molecules undergo random motion and obey Newton's laws of motion;

• Ideal gas molecules undergo elastic collisions with the walls of the container the gas is in;

• Ideal gas molecules experience only completely elastic collisions with one another.

The Ideal Gas Model First Law

The first law of the ideal gas model was defined by the Anglo-Irish physicist Robert Boyle. Boyle studied the relationship between the pressure and volume of a given mass of a gas at a constant temperature. The relationship is known as Boyle’s law.

The Ideal Gas Model Second Law

The second law of the ideal gas model is Charles' law. Charles studied the effect of temperature on the volume of gases at constant pressure. The following generalization about the relationship between the volume and the temperature of gas was observed, which is known as Charles’ law.

Charles' Law in Degrees Kelvin

In 1848 a British scientist, Lord Kelvin, suggested a new temperature scale known as the absolute scale of temperature. It is popularly known as the Kelvin temperature scale, and starts at absolute zero, \(0^\circ\mathrm{K}\), where particles have zero kinetic energy. The increment is the same as the Celsius scale, so \(1^\circ\mathrm{K}=1^\circ\mathrm{C}.\) Kelvin temperature is also called the thermodynamic scale of temperature and is used in all scientific measurements. This helped to simplify the relationship between temperature and gaseous volume.

Absolute zero, or \(0^\circ\mathrm{K} \), is equivalent to \(-273.15^\circ\mathrm{K} \). Therefore, to convert from a temperature in degrees Celcius, \(T_\mathrm{C}\) to degrees Kelvin \(T_\mathrm{K}\), we can use the formula:

$$T_K=T_C+273.15$$

Charles's law can be expressed using degrees Kelvin. Let's assume that

$$T_0=0^\circ\mathrm{C}=273.15^\circ\mathrm{K}$$

where \(T_0\) is the initial temperature, and therefore \(V_0\) is the initial volume of a given gas at this initial temperature. Based on the definition of Charles' law, if the temperature rises by \(1 \, ^\circ\mathrm{K} \) the volume will rise proportionally by:

$$ V_0 \cdot \frac{1}{T_0}$$

Consequently, if the temperature rises by \(t\), the volume will rise by

$$ \Delta V = V_0 \cdot \frac{1}{T_0} \cdot t.$$

Let's say that \(V_\mathrm{T}\) is the volume at any temperature \(T\), and \(t\) is the difference between the initial temperature \(T_0\) and final temperature \(T\), so:

$$t=T-T_0$$

The final volume \(V_\mathrm{T}\) is the sum of the initial volume and the increase in volume, so we obtain:

$$V_\mathrm{T} = V_0 + \frac{V_0 \cdot t}{T_0} = V_0 \left [ 1+\frac{t}{T_0} \right ] $$

where the term in square brackets can be expanded and expressed using degrees Kelvin

$$ \left [ 1+\frac{t}{T_0} \right ] = \frac{T_0 + t}{T_0}=\frac{T_0 + (T-T_0)}{T_0}= \frac{T}{T_0} $$

Now both of the volumes and temperatures can be combined into the following expression

$$V_\mathrm{T}=\frac{V_0 \, T}{T_0}$$

$$ \frac{V_\mathrm{T}}{V_0} = \frac{T}{T_0}$$

$$\frac{V_\mathrm{T}}{T} = \frac{V_0}{T_0}$$

Thus,

$$\frac{V}{T} = \mathrm{constant} = k_2$$

where \(k_2\) is a constant which depends upon the pressure and the mass of the gas, as well as the units of volume. The expression above can be rewritten as

$$ V = k_2 \, T $$

$$ V \propto T $$

and is visualized in the figure below.

Fig 2 - A practical example of Charles' law. As the temperature of the gas in the container increases, so does the volume, proving the direct proportionality, Wikimedia Commons

Charles' law can also be expressed as the volume of a fixed mass of a gas being directly proportional to the absolute temperature, while pressure remains constant. The working formula for Charles' law is:

$$ \frac{V_1}{T_1} = \frac{V_2}{T_2} = \mathrm{constant}. $$

The Ideal Gas Model Third Law

The third law of the ideal gas model is Gay-Lussac's law. It defines the relationship between pressure and temperature and was discovered by Joseph Gay-Lussac.

Mathematically, it may be expressed as,

$$p \propto T $$

where pressure \(p\) is directly proportional to the temperature \(T\). From the proportionality conditions, we can arrive at a conclusion that,

$$ \frac{p}{T} = \mathrm{constant} = k_3 $$

$$ \Rightarrow \frac{p_1}{T_1} = \frac{p_2}{T_2}. $$

The above equation has been derived by combining Boyle's Law and Charles's Law.

The Ideal Gas Model Fourth Law

The fourth law of the ideal gas model is Avogadro's law. Amadeo Avogadro studied the relationship between the volume of a gas to the number of molecules at constant temperature and pressure. This has been accepted as a law and is known as Avogadro's law.

Mathematically, Avogadro’s law may be expressed as follows, where \((V)\) represents volume and \((n)\) the number of moles.

$$V \propto n. $$

In other words, volume is directly proportional to the number of moles while the temperature and pressure remain constant, which implies:

$$V = k_4 \cdot n$$

$$\frac{V_1}{n_1} = \frac{V_2}{n_2} = \mathrm{constant} = k_4.$$

From this expression we obtain the final equation

$$ \frac{V_1}{n_1} = \frac{V_2}{n_2} $$

The Ideal Gas Law

All the laws mentioned above can be combined in the form of one singular law. The macroscopic properties of an ideal gas are related by the ideal gas law.

Mathematically, it can be expressed as follows

$$ pV = nRT $$

where the volume \(V\) of a gas depends upon the number of moles \(n\), pressure \(p\), and temperature \(T\). Here \(R\) is the universal gas constant equal to \(8.314 \, \frac{\mathrm{J}}{\mathrm{mol \cdot K}}\).

The Ideal Gas Model Thermodynamics

The ideal gas law defines the behavior of ideal gases in terms of their pressure, volume and temperature. You might know that thermodynamics is the study of the relationship between heat and other forms of energy. One of the founders of thermodynamics, german physicist Julius von Mayer, discovered that heat added to a gas has an equivalence in mechanical work. In other words, a relationship can be defined between the amount of energy transferred to a gas and the corresponding change in temperature using the specific heat capacity of the gas.

Depending on if the gas is held at a constant volume, or if the container is allowed to expand and the gas remains at constant pressure, the specific heat capacity has different values:

$$C_p=\mathrm{Specific}\;\mathrm{heat}\;\mathrm{capacity}\;\mathrm{at}\;\mathrm{constant}\;\mathrm{pressure}$$

$$C_v=\mathrm{Specific}\;\mathrm{heat}\;\mathrm{capacity}\;\mathrm{at}\;\mathrm{constant}\;\mathrm{volume}$$

The amount of heat energy transferred to the gas in joules \((\mathrm{J})\) is then given by \(Q_p\) (at constant pressure) or \(Q_v\) (at constant volume).

$$Q_p=mC_p\Delta T$$

$$Q_v=mC_v\Delta T$$

Where \(m\) is the mass of gas in \(\mathrm{kg}\) and \(\Delta T\) is the change in temperature in \(^\circ \mathrm{K}.\)

The ideal gas law can be used to represent the change in temperature in terms of changes in pressure or volume, which is useful for analyzing thermodynamic cycles such as the Otto Cycle or Diesel Cycle.

The Ideal Gas Model Equations

The ideal gas model has a set of four major equations which are widely used for any type of calculations and measurements. All of these equations together make up the ideal gas law. All of these equations are derived and explained above and are compiled here for easy access and understanding.

1. Ideal Gas Law \(pV=nRT\)
2. Boyle's Law (Pressure-Volume Relationship): \(p_1 \, V_1 = p_2 \, V_2 \)
3. Charles's Law (Volume-Temperature Relationship): \(\frac{V_1}{T_1} = \frac{V_2}{T_2}\)
4. Gay Lussac's Law (Pressure-Temperature Relationship): \(\frac{p_1}{T_1} = \frac{p_2}{T_2}\)
5. Avogadro's Law (Volume-Amount of Relationship): \(\frac{V_1}{n_1} = \frac{V_2}{n_2}\)

Fig 3 - All of the ideal gas model relationships are interrelated, considering they all describe the behavior of gases based on varying parameters.