Fundamental Laws of Gases: Meaning, Formula, and Graphs

Fundamental Laws of Gases: Have you ever noticed that if you take a basketball outside on a cold day, the ball shrinks a bit as compared to the hot day because as the temperature decreases, the volume also decreases at constant pressure. During filing air in the tire of a bicycle, you may have observed the temperature of the pump increase. Both the above examples are related to the gas laws. This article will discuss the fundamental gas laws that discover the relationship between pressure, temperature, volume, and amount of gas. We will also discuss the ideal gas equation and its derivation.

Gas Laws

Under standard conditions, all gases exhibit similar behaviour. When the physical parameters associated with the gas are altered, such as temperature, pressure, and volume, then the variations arise in their behaviours that lead to the various fundamental laws given by the name after the scientists who discovered them. Anglo-Irish scientist Robert Boyle made the first reliable measurement of the properties of gases in 1662.

The law he gave is known as Boyle’s Law. Later on, attempts to fly in the air with the help of hot air balloons motivated Jacques Charles and Joseph Lewis Gay Lussac to discover additional gas laws. Contribution from Avogadro and others provided a lot of information about the gaseous state.

The fundamental laws of gas are given below:

1. Boyle’s Law (Pressure-Volume Relationship)

Based on his experiments, Robert Boyle states that at a constant temperature, the pressure of a fixed amount of gas varies inversely with its volume. This is known as Boyle’s law. Mathematically, we can write this as:
\(p \propto \frac{1}{V}{\rm{\;}}\left( {{\rm{at\;constant\;}}T{\rm{\;and\;}}n} \right)\)
\(p = {k_1}\;\frac{1}{V}\;\;\;\;\;\;\;\;\;\;……\left( 1 \right)\)
Where \({k_1}\) is the constant of proportionality and again on rearranging equation (1), we obtain:  \(pV = {k_1}\) It means that at a constant temperature, the product of pressure and volume of a fixed amount of gas is constant. Then we can write this as:
\({p_1}{V_1} = {p_2}{V_2} = {\rm{constant}}\;\)
\(\Rightarrow \,\,\,\,\;\frac{{{p_1}}}{{{p_2}}} = \frac{{{V_1}}}{{{V_2}}}\)

We can graphically represent Boyle’s law as in the figure shown above is the graph of equation \(\left( {pV = {k_1}} \right)\)at different temperatures. The value of  \({k_1}\) for each curve is different because, for a given mass of gas, it varies only with temperature. Corresponding to a different constant temperature, each curve is known as an isotherm. Higher curves correspond to a higher temperature.

2. Charles’s Law (Temperature – Volume Relationship)

According to Charles’s law, the volume of a fixed mass of a gas is directly proportional to its absolute temperature at the constant pressure. Hence, according to Charles’ law, we can write that :
\(V \propto T\)
\(\Rightarrow \,\,\frac{V}{T} = {\rm{Constant}} = \;{k_2}\)
Thus,
\(\Rightarrow \,\,\,\,V = {k_2}T\;\)

Charles found that for all gases, at any given pressure, the graph of volume vs temperature (in celsius), as shown in the above figure, is a straight line, and when it extends to zero volume, then each line intercepts the temperature axis at \(–{\rm{ }}273.15{\rm{ }}^\circ {\rm{C}}.\) Each line of the volume vs temperature graph is called isobar. We can see that the volume of the gas at \(–{\rm{ }}273.15{\rm{ }}^\circ {\rm{C}}\) will be zero, that gas will not exist. Absolute zero is the lowest imaginary temperature at which gases are supposed to occupy zero volume.

Learn About Charle’s Law

3. Gay Lussac’s Law (Pressure-Temperature Relationship)

The mathematical relationship between pressure and temperature was given by Joseph Gay Lussac’s that is known as Gay Lussac’s law. It states that at constant volume, the pressure of a fixed amount of a gas varies directly with the temperature. Now, according to this statement, we can write this as:
\(p \propto T\)
\(\Rightarrow \,\,\,\frac{p}{T} = {\rm{Constant}} = \;{k_3}\)

The pressure vs temperature (Kelvin) graph at constant molar volume is shown in the above figure. Each line of this graph is called isochore.

4. Avogadro Law (Volume – Number of Moles Relationship)

Avogadro law states that equal volumes of all gases under the same conditions of temperature and pressure contain an equal number of molecules. This means that as long as the temperature and pressure remain constant, the volume depends upon the number of molecules of the gas or in other words amount of the gas. Hence,
\(\;V \propto n\)
\(\Rightarrow \,\,\,V = {k_4}n\;\)
Where \(n\) is the number of moles of gas. The number of molecules in one mole of a gas has been determined to be \(6.022 \times {10^{23}},\) this is known as the Avogadro number and is denoted by \(\left( {{N_A}} \right).\) The graph of volume vs the amount of substance is shown in the figure below:

Ideal Gas Equation

The three laws which we have learned till now can be combined in a single equation which is known as an ideal gas equation.
At constant \(T\) and \(n\) ;
\(V \propto \frac{1}{p}\;\;\;\;\;\left( {{\bf{Boyle’s}}\;{\bf{Law}}} \right){\rm{\;}}\)
At constant \(p\) and \(n\);
\(V \propto T{\rm{\;\;\;\;\;\;}}\left( {{\bf{Charles’sLaw}}} \right)\)
At constant \(p\) and \(T\);
\(V \propto n{\rm{\;\;\;\;\;\;\;}}\left( {{\bf{Avogadro}}\;{\bf{Law}}} \right)\)
Thus,
\(V \propto \frac{{nT}}{p}\;\)
\(\Rightarrow \;\;\,\,V = R\frac{{nT}}{p}\;\)
\(\Rightarrow \;\;\;pV = nRT\,\,\,\,\,\,\,…..\left( 3 \right)\)
\(R = \frac{{pV}}{{nT}}\;\)
Where,
\(n =\) number of moles of the gas \(= \frac{m}{M}\)
\(m = \) total mass of the gas
\(M = \) Molecular mass of the gas
\(R = \;\) Universal gas Constant
\( = 8.31\;\,\,{\rm{J\;mo}}{{\rm{l}}^{ – 1}}{{\rm{K}}^{ – 1}}\)
\( = 2.0\;{\rm{cal\;mo}}{{\rm{l}}^{ – 1}}{{\rm{K}}^{ – 1}}\)
Where \(R\) is the constant of proportionality known as the gas constant, it is also called Universal Gas Constant as it is the same for all gas. The ideal gas equation is a relation between four variables, and it describes the state of any gas. Therefore, it is also called the equation of state.

Study About Solubility of Gas In Liquid

Solved Examples on Fundamental Laws of Gases

Q.1. A gas at \(27\,^\circ {\rm{C}}\) in a cylinder has a volume of  \(4\,\;{\rm{litres\;}}\) and pressure of  \(0\;\,\,{\rm{N\;}}{{\rm{m}}^{ – 2}}.\)
(i) Calculate the change in volume of the gas at a constant temperature if the gas is first compressed to the pressure of \(\;150\;\,{\rm{N\;}}{{\rm{m}}^{ – 2}}.\)
(ii) Calculate the new pressure at a constant volume if it is then heated to the temperature of \({\rm{127}}\,{\rm{^\circ C}}{\rm{.}}\)
Ans: (i) At constant temperature, we can write
\({p_1}{V_1} = {p_2}{V_2}\)
\(\Rightarrow \,\,\,{V_2} = \frac{{{p_1}{V_1}}}{{{p_2}}}\)
\(\Rightarrow \,\,{V_2} = \frac{{100 \times 4}}{{150}} = 2.667\;{\rm{L}}\)
And,
the change in volume \( = {V_2} – {V_1} = 2.667 – 4\)
\(= – 1.333\;{\rm{L}}\)
(ii) Using Gay Lussac’s law for constant volume,
\(\frac{{{p_2}}}{{{p_1}}} = \frac{{{T_2}}}{{{T_1}}}\)
\(\Rightarrow \,\,\;{p_2} = \frac{{{T_2}}}{{{T_1}}}\; \times {p_1}\)
\(\Rightarrow \,\,\,\;{p_2} = \;\frac{{\left( {127 + 273} \right) \times 150}}{{\left( {27 + 273} \right)}}\)
\(\Rightarrow \,\,\,{p_{\rm{2}}}{\rm{ = }}\,{\rm{200\;N\;}}{{\rm{m}}^{{\rm{ – 2}}}}\)

Q.2. At the earth’s surface, where pressure is \(76\;\,\,{\rm{cm\;of\;Hg}}\) and temperature is \(27\,{\rm{^\circ C}}\) a balloon is partially filled with helium gas has a volume of \(30\;\,{{\rm{m}}^3}.\) Find the increase in the volume of a gas if the balloon rises to a height, where pressure is \(\;7.6\;{\rm{cm\;of\;Hg}}\) and temperature is \(- 54\,{\rm{^\circ C}}.\)
Ans: As from the given equation,
\(\frac{{{P_1}{V_1}}}{{{T_1}}} = \frac{{{P_2}{V_2}}}{{{T_2}}}\)
\(\; \Rightarrow \,\,\,{V_2} = \frac{{{P_1}{V_1}{T_2}}}{{{T_1}{P_2}}}\;\;\)
\(\; \Rightarrow \,\,{V_2} = \frac{{76 \times 30 \times \left( { – 54 + 273} \right)}}{{\left( {27 + 273} \right) \times 7.6}}\;\)
\(\; \Rightarrow \,\,{V_2} = 219\;{{\rm{m}}^3}\)
Therefore, an increase in the volume of gas will be
\(\; \Rightarrow \,\,\,{V_2} – {V_1} = 219 – 30\)
\(= 189\;{{\rm{m}}^3}\;\)

Summary

Boyle’s Law states that for a given mass of a gas, the volume of a gas at constant temperature is inversely proportional to its pressure. According to Charles’ Law, “for a given mass of gas, the volume of a gas at constant pressure is directly proportional to its absolute temperature”.

Gay Lussac’s Law states that for a given mass of gas at constant volume, the pressure of a gas is directly proportional to its absolute temperature. Avogadro’s Law states that at the same temperature and pressure, equal volumes of all gases contain an equal number of molecules. All these four laws can be written in one single equation known as an ideal gas equation.
According to this equation: \(pV = nRT.\)

FAQs

Q.1. Give some examples of gas law applied in everyday life?
Ans: Gas law is applicable in our daily life in various ways, such as when a scuba diver exhales under the water, then these released bubbles become larger as it reaches the surface of the water. During winter, the football got shrinks. We check the pressure of the car tire before heading to a drive.

Q.2. What are the different gas laws?
Ans: The gas laws consist of some primary laws that are: Charles’s Law \(V \propto T.\) Boyle’s Law \(\left( {p \propto \frac{1}{V}} \right),\) Gay Lussac’s Law \(\left( {p \propto T} \right)\) and Avogadro’s Law \(\left( {V \propto n} \right).\) From these gas laws, we can also get the Ideal gas law.

Q.3. What gas laws apply to airbags?
Ans: Ideal gas law is used in vehicle airbags that can save a life during a car crash. A large amount of Nitrogen gas is produced by reacting Sodium Azide, with excess heat that fills up fast in the airbag during a crash.

Q.4. What is an ideal gas and non-ideal gas?
Ans: Two types of gases exist. Real gas and Ideal gas. As the particle size of an ideal gas is extremely small, and the mass is almost zero, and it has no volume. An ideal gas is also considered as a point mass. The molecules of real gas are small particles and also have volume, so they occupy space.

Q.5. What is the ideal gas equation?
Ans: After combining all three laws that is Charles’ Law, Boyle’s Law, and Avogadro’s Law, it can be written in one single equation that equation is known as an ideal gas equation. According to this equation,we can write \(pV = nRT.\)

We hope this article on Fundamental Laws of Gases helps you in your preparation. Do drop in your queries in the comments section if you get stuck and we will get back to you at the earliest.