# Ideal Gas Model

Gases are made up of numerous molecules which move rapidly and interact both with each other and with various obstacles, such as the walls of their container. If we tried to explain their behavior considering all of these interactions, the calculations would be long and tedious due to complicated equations, even in the most primitive cases. That’s where the ideal gas model comes in. In physics, we create hypothetical systems which obey all the laws applied and are assumed to be as ideal so that we as scientists and engineers can verify the hypothesis and act accordingly. An ideal gas obeys all the gas laws which are agreed upon by scientists throughout the world to closely approximate the behavior of a real gas. In this article, we'll look at the behavior of ideal gases and various laws that govern their behavior.

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## How to Describe a Gas?

In total, there are four fundamental states of matter, gas is one of them. To understand the ideal gas model, we first must define what exactly is gas.

Gas is a collection of molecules (or atoms) in continuous random motion, with average speeds that increase as the temperature is raised.

A gas differs from a liquid in that, except during collisions, the molecules of a gas are widely separated and move in paths that are largely unaffected by intermolecular forces. Some everyday examples of gases include air, oxygen, and water vapor.

The three other fundamental states of matter are solid, liquid, and plasma!

## 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.

Boyle's law states that at constant temperature, the volume of a fixed amount of a gas is inversely proportional to its pressure.

Mathematically, Boyle's law may be expressed as follows,

$$V \propto \frac{1}{p}$$

where $$V$$ is the volume, and $$p$$ is the pressure. Now we can insert a constant of proportionality to make this an equation. Call this constant of proportionality $$k$$. Note that $$k$$ depends on the amount of gas and the temperature. The new expression becomes

$$V = \frac{k}{p}$$

or it can be rearranged as

$$p \, V = k$$

This means that the product of pressure and volume of a fixed amount of gas at a constant temperature is constant. An example of such behavior of gases is visible in the figure below.

Fig 1 - A practical example of the Boyle's law. As the volume of the gas in the container reduces, the presure increases proving the inverse proportionality of the two variables

Let $$V_1$$ be the volume of a given amount of gas at pressure $$p_1$$ and at a given temperature $$T$$. When the pressure is changed to $$p_2$$ at the same temperature, let the volume change to $$V_2$$. According to Boyle’s law,

$$p_1 \, V_1 = k.$$

Note we have not changed the amount of gas or the temperature, so our constant of proportionality is the same, thus

$$p_2 \, V_2 = k.$$

Combining these two yields

$$p_1 \, V_1 =k= p_2 \, V_2$$

or

$$p_1 \, V_1 = p_2 \, V_2.$$

This final equation is a somewhat more useful restatement of Boyle's law.

A vessel of $$120 \, \mathrm{mL}$$ capacity contains of gas at $$35 \, \mathrm{ ^\circ C}$$ and $$1.2 \, \mathrm{bar}$$ pressure. The gas is transferred to another vessel of volume $$180 \, \mathrm{mL}$$ at $$35 \mathrm{ ^\circ C}$$ and expands to fil the container completely. What is its pressure in this new vessel?

This problem is based on the equation of Boyle's law!

Solution

Because the temperature and amount of gas remain constant, we can apply Boyle's law. Let $$p_1 = 1.2 \, \mathrm{bar}$$, $$V_1=120 \, \mathrm{mL}$$, and $$V_1=180 \, \mathrm{mL}$$, then:

$$p_1 \, V_1 = p_2 \, V_2$$

$$1.2 \, \mathrm{bar} \cdot 120 \, \mathrm{mL} = p_2 \cdot 180 \, \mathrm{mL}$$

$$p_2 = \frac{1.2 \, \mathrm{bar} \cdot 120 \, \mathrm{\cancel{mL}}}{180 \, \mathrm{\cancel{mL}}}$$

$$p_2 = 0.8 \, \mathrm{bar}$$

Here we have used the $$\mathrm{bar}$$ as the unit of pressure, which is related to the SI unit of pressure Pascal $$(\mathrm{Pa})$$ as $$1\;\mathrm{bar}=100\;000 \; \mathrm{Pa}$$

### 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 states that at constant pressure, the volume of a given mass of a gas is directly proportional to the absolute temperature.

#### 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:

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

A sample of helium has a volume of $$520 \, \mathrm{mL}$$ at $$100 \mathrm{ ^\circ C}$$. Calculate the temperature at which the volume will become $$260 \, \mathrm{mL}$$. Assume that pressure is constant.

This problem is based on Charles law and the application of Charles law formula!

Solution

The values we are given

$$V_1 = 520 \, \mathrm{mL}, V_2= 260 \, \mathrm{mL},$$

$$T_1 = 100 + 273 = 373 \, \mathrm{K},$$

we need to find the value of $$T_2$$.

Pressure remains constant, therefore, by applying Charle's law:

$$\frac{V_1}{T_1} = \frac{V_2}{T_2}$$

$$T_2 = \frac{V_2 \, T_1}{V_1}$$

$$T_2 = \frac{260 \, \mathrm{\cancel{mL}} \cdot 373 \, \mathrm{K}}{520 \, \mathrm{\cancel{mL}}}$$

$$T_2 = 186.5 \, \mathrm{K}$$

In degree centigrade it's

$$t = 186.5 -273.15 = -86.65 \mathrm{ ^\circ C}.$$

### 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.

At constant volume, the pressure of a fixed amount of gas varies directly with temperature, this is known as Gay-Lussac's Law.

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.

Equal volumes of all gases under the same temperature and pressure conditions contain an equal number of molecules- - this is known as Avogadro's Law.

This implies that as long as the temperature and pressure remain constant, the volume of the gas depends upon the number of molecules of the gas.

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 number of molecules in one mole of a gas has been determined to be $$6.022 \cdot 10^{23}$$ and is known as the Avogadro constant $$N_\mathrm{A}$$.

## 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.

The ideal gas law is an approximation used to solve problems involving the temperature, pressure and volume of gases. It’s a combination of the laws created by Boyle, Charles, Gay Lussac and Avogadro.

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.

## The Ideal Gas Model - Key takeaways

• Gas is a phase of matter that does not have a shape, but can occupy the entire space in which it is kept.
• Four primary laws govern idealized gases; Boyle's law, Charle's Law, Gay Lussac's Law, and Avogadro's Law.
• Boyle's law deals with the Pressure-Volume relationship
• Charle's law deals with the Volume-Temperature relationship
• Gay Lussac's Law deals with the Pressure-Temperature relationship
• Avogadro's Law deals with Volume- Amount relationship.
• The ideal gas law is a combination of the four primary laws that govern ideal gases.
• The ideal gas law is an approximation used to solve problems involving the temperature, pressure and volume of gases.

## References

1. Figure 1 - Boyle's law final.gif from (https://commons.wikimedia.org/wiki/File:Boyle%27s_law_final.gif) licensed by (Public Domain)
2. Figure 2 - Charles and Gay-Lussac's Law animated.gif from (https://commons.wikimedia.org/wiki/File:Charles_and_Gay-Lussac%27s_Law_animated.gif) licensed by (Public Domain)
3. Figure 3 - Ideal gas law relationships.svg from (https://commons.wikimedia.org/wiki/File:Ideal_gas_law_relationships.svg) licensed by CC BY_SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/deed.en)

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##### Frequently Asked Questions about Ideal Gas Model

What is an ideal gas model?

An ideal gas model is a set of generalizations made about gases, allowing us to better understand the behavior of non-ideal gases.

What are the assumptions of the ideal gas model?

The assumptions of the ideal gas model state that it's made up of numerous point like molecules spread really far apart and moving randomly. These molecules undergo elastic collisions with one another and the container walls.

What describes the behavior of an ideal gas?

The behavior of an ideal gas is described by identical particles moving randomly and experiencing no intermolecular forces.

What is the formula for ideal gas model?

The formulae describing the ideal gas model are pV=nRT, p1V1=p2V2, V1T2=V2T1, p1T2=p2T1, V1n2=V2n1.

What are the properties of the ideal gas model?

The ideal gas model is based on the properties of an imaginary, ideal gas - these are that the gas is made up of numerous point-like particles spread really far apart and moving randomly. These molecules undergo elastic collisions with one another and the container walls, and are far enough apart that they are unaffected by inter-molecular forces.

## Test your knowledge with multiple choice flashcards

If the pressure of a gas is held constant, how is the volume related to the temperature?

If the volume of a gas is held constant, how is the pressure related to the temperature?

If the temperature of a gas is held constant, how is the pressure related to the volume?

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