Kinetic Theory of Gases

GAS LAWS

$Boyle’s Law:$ $\to$ 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: $\\$

$\textbf{Charle’s Law:}\\$ $\to$ 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:

$\textbf{Dalton’s law of partial pressure:}\\$ $\to$ 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 $\\$

$\textbf{Gay-Lussac’s law:}\\$ $\to$ 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

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).

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It can also be defined as total number of independent ways in which gas molecule can have energy.

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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

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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

$For Different gas:$ $\to \underline{{For} {Monoatomic}{Gases}}:$ (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 $\\$

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