## GAS LAWS

1. $\to$The volume of a given mass of a gas is inversely proportional to its pressure$\\$at constant temperature, i.e 𝑽∝$\frac{𝟏}{P}$ (if mass of gas and temperature of gasare constant)$\\$or, PV = constant Thus, $P_1 V_1$ = $P_2 V_2 \\$$\to$ Real gas obeys Boyle’s law at low pressure and high temperature.$\\$$\to$ Boyle’s law is represented graphically as follows:$\\$

2. $\textbf{Charle’s Law:}\\$$\to$The volume of given mass of a gas at constant pressure increases or$\\$decreases by a constant fraction 𝛼(= $\frac{1}{273}$ ) of its volume at $0^o$C for rise or$\\$fall of temperature by $1^o$C.$\\$i.e 𝑽 = $𝑽_𝟎$ [𝟏 + 𝜶𝒕]. This law is also known as Charle’s volume law. $\to$The volume of given mass of a gas at constant pressure is directly proportional to its absolute temperature i.e 𝑽∝𝑻 or, V/T = constant Thus, $𝑽_𝟏$ = $𝑽_𝟐$ $\to$ Charle’s law is represented graphically as follows:

3. $\textbf{Dalton’s law of partial pressure:}\\$$\to$The pressure exerted by a mixture of non-reacting gases at constant temperature$\\$is equal to sum of partial pressure of each components in the mixture.$\\$P=$P_1 + P_2 + P_3$ +..... $\to$Partial pressure = mole fraction x total pressure$\\$

4. $\textbf{Gay-Lussac’s law:}\\$$\to$The volume of given mass of a gas at constant volume increases or decreases$\\$by a constant fraction𝛼(= $\frac{1}{273}$ ) of its pressure at $0^o$C for rise or$\\$fall of temperature by $1^o$C. i.e 𝑷 = $𝑷_𝟎$ [𝟏 + 𝜶𝒕]. This law is also known as Charle’s pressure law.$\\$But, in Gay-Lussac’s law flexible container is used whereas in Charle’s law rigid container is used.$\\$$\to$The pressure of given mass of a gas at constant volume is directly proportional to its$\\$absolute temperature i.e 𝑷 ∝ 𝑻 or, P/T = constant$\\$Thus, $\frac{𝑷_𝟏}{𝑻_𝟏}$ = $\frac{𝑷_𝟐}{𝑻_𝟐} \\$

Combined Gas Equation:
Here, $\to$ $\frac{𝑷𝑽}{𝑻}$= 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕 $\to$ $\frac{𝑷_𝟏𝑽_𝟏}{𝑻_𝟏}$=$\frac{𝑷_𝟐𝑽_𝟐}{𝑻_𝟐}$

## Ideal Gas Equation

1. Here,$\\$$\to$ For n moles of gas, PV = nRT$\\$(R = 8.314 J$mol^{-1} K^{-1}$ is universal gas constant and is same for all gases)$\\$PV = nRT$\\$=> 𝑷𝑽 = 𝒎 𝑹𝑻 𝑴$\\$=> 𝑷𝑽 = 𝒎($\frac{𝑹}{𝑴}$)𝑻$\\$=> 𝑷𝑽 = 𝒎𝒓𝑻 (m is mass of gas, M is molecular mass, r = R/M is specific gas constant or$\\$gas constant per unit mass.)$\\$=>𝑷=𝝆𝒓𝑻 ( 𝜌 =$\frac{m}{v}$ is density of gas)$\\$Also,$\\$PV = nRT$\\$=> 𝑷𝑽 = $\frac{𝑵}{𝑵_a}$𝑹𝑻 (N = number of molecules, $N_a$ = avogadro’s number)$\\$=> 𝑷𝑽 = 𝑵𝒌𝑻 (K = R/$N_a$ = 1.3806 x 10-23 J/K is Boltzmann’s constant)

## Degree of freedom

Total number of independent quantities (or coordinates)$\\$required to define the state of a system is degree of freedom (f).$\\$It can also be defined as total number of independent ways in which gas molecule can have energy.$\\$f = 3N-R (N= no of atoms, R= no of restrictions or number of independent relation among the atom)$\\$$\underline{{For} {Monoatomic}}:$ N=1, R=0$\\$f= 3 i.e. only translation motion.$\\$$\underline{{For} {Diatomic}}:$ N=2, R=1$\\$f= 5 i.e. translation and rotational motion$\\$$\underline{{For} {Triatomic}}:$ Linear (Eg: C$O_2$): N=3, R=2, f= 7$\\$Non Linear (Eg: $H_2$O): N=3, R=3, f= 6

## Average Kinetic Energy

1. $\to \underline{{For} {Monoatomic}{Gases}}:$(f=3)$\\$Average KE per mole = $\frac{𝑓}{2}$𝑓 𝑅𝑇 = $\frac{3}{2}$ 𝑅𝑇$\\$Average KE per molecule = $\frac{𝑓}{2}$𝐾𝑇 = $\frac{3}{2}$ 𝐾𝑇$\\$$\to \underline{{For} {Diatomic} {Gases}}:$ (f=5) :$\\$Average KE per mole = $\frac{𝑓}{2}$𝑅𝑇 = $\frac{5}{2}$𝑅𝑇$\\$Average KE per molecule =$\frac{𝑓}{2}$𝐾𝑇 = $\frac{5}{2}$𝐾𝑇$\\$

$\to \underline{{For} {Triatomic} {Gases}}: \\$