Question

Write the mathematical equation associated with the following laws-

• Boyles law
• Charles law
• Gay Lussac law
• Ideal gas equation

Answer 1

Boyles law- Boyle's law states that-

“Temperature remaining constant, the volume of a given mass of a gas is inversely proportional to its pressure.”

If T= constant

$\mathrm{V}\mathrm{is}\mathrm{inversaly}\mathrm{propotional}\mathrm{to}\frac{1}{\mathrm{P}}$

which means,

$\mathrm{PV}=\mathrm{K}$

Charles law- According to Charles's law,

The volume occupied by a fixed amount of gas is directly proportional to its absolute temperature if the pressure remains constant.

$$\begin{gathered} \mathrm{If} \\ \mathrm{P}=\mathrm{constant} \\ \mathrm{V}\mathrm{is}\mathrm{directly}\mathrm{propotional}\mathrm{to}\mathrm{the}\mathrm{T} \\ \mathrm{that}\mathrm{means} \\ \frac{\mathrm{V}}{\mathrm{T}}=\mathrm{constant} \end{gathered}$$

Gay Lussac law- This law is also known as Amonton's law which tells us about the variation of pressure with the temperature at constant volume.

According to Gay Lussac law-

Volume remaining constant, the pressure of a given mass of a gas increases or decreases by 1/273 of its pressure $0°\mathrm{C}$ for every $1°\mathrm{C}$ rise or fall in temperature.

Mathematically $$\begin{gathered} {\mathrm{P}}_{\mathrm{t}}={\mathrm{P}}_{\mathrm{o}}\left(1+\frac{\mathrm{t}}{273}\right)={\mathrm{P}}_{\mathrm{o}}\left(\frac{273+\mathrm{t}}{273}\right)={\mathrm{P}}_{\mathrm{o}}\frac{\mathrm{T}}{273} \\ \mathrm{Where}; \\ {\mathrm{P}}_1\mathrm{and}{\mathrm{P}}_{\mathrm{o}}\mathrm{are}\mathrm{the}\mathrm{pressure}\mathrm{of}\mathrm{a}\mathrm{certain}\mathrm{amount}\mathrm{of}\mathrm{gas}\mathrm{at}{\mathrm{t}}^{\mathrm{o}}\mathrm{C}\mathrm{and} \\ 0°\mathrm{C}\mathrm{respectively}. \\ \mathrm{As}{\mathrm{P}}_{\mathrm{o}}\mathrm{and}273\mathrm{are}\mathrm{constant},\mathrm{hence} \\ {\mathrm{P}}_{\mathrm{t}}\mathrm{is}\mathrm{directly}\mathrm{propotional}\mathrm{to}\mathrm{T} \\ \mathrm{or}\mathrm{simply} \\ \mathrm{P}\mathrm{is}\mathrm{directly}\mathrm{propotional}\mathrm{to}\mathrm{T} \\ \mathrm{i}.\mathrm{e}. \\ \mathrm{P}=\mathrm{KT} \end{gathered}$$

Thus the above law may also be defined as follows-

Volume remaining constant, the pressure of a given mass of a gas is directly proportional to its temperature in degree kelvin.

Ideal gas equation- The equation which gives the simultaneous effect of pressure and temperature on the volume of a gas is known as ideal gas equation of state for an ideal gas.

$$\begin{gathered} \mathrm{PV}=\mathrm{nRT} \\ \mathrm{where}. \\ \mathrm{P}=\mathrm{pressure}\mathrm{of}\mathrm{a}\mathrm{gas} \\ \mathrm{V}=\mathrm{volume}\mathrm{of}\mathrm{a}\mathrm{gas} \\ \mathrm{n}=\mathrm{moles}\mathrm{of}\mathrm{a}\mathrm{gas} \\ \mathrm{R}=\mathrm{gas}\mathrm{constant} \\ \mathrm{T}=\mathrm{temperature}\mathrm{of}\mathrm{a}\mathrm{gas} \end{gathered}$$