# Gas Laws and Ideal Gas Equation and Kinetic Molecular Theory of Gases

Gas Laws:
\tt V\propto\frac{1}{P} (Boyle's Law)
PV = constant Charles' Law:
V ∝ T (or) \tt \frac{V_{1}}{V_{2}}=\frac{T_{1}}{T_{2}}

Gay-Lussac Law:
\tt \frac{P_{1}}{T_{1}}=\frac{P_{2}}{T_{2}} V ∝ n

Ideal Gas Equation:
\tt V \propto \frac{1}{P}
V ∝ T
V ∝ n
\tt \therefore V \propto \frac{nT}{P}
PV ∝ nT
PV = nRT

Graham Law of Diffusion of Gases:
\tt r\propto\frac{1}{\sqrt{d}}\propto\frac{1}{\sqrt{M.wt}}\propto\frac{1}{\sqrt{V.D}} (Constant T and P)
\tt \frac{r_{1}}{r_{2}}=\frac{\sqrt{d_{2}}}{\sqrt{d_{1}}}=\sqrt{\frac{M_{2}}{M_{1}}}=\sqrt{\frac{V.D_{2}}{V.D_{1}}}=\frac{V_{1}t_{2}}{V_{2}t_{1}}
\tt \frac{r_{H_{2}}}{r_{CH_{4}}}=\frac{\sqrt{M_{CH_{4}}}}{\sqrt{M_{H_2}}}=\sqrt{\frac{16}{2}}=2\sqrt{2}

Dalton's Law of Partial Pressure:
Ptotal = P1 + P2 + P3 + ....
P.P = M.F × T.P = Xi × T.P [Xi = mole fraction]
\tt Xi=\frac{P.P}{T.P}
Ptotal = PA + PB + PC = (n1 + n2 + n3) \tt \frac{RT}{V}

Kinetic Gas Equation:
Urms = \tt \sqrt{\frac{u_1^2+u_2^2+u_3^2...+u_n^2}{n}}
K.E equation = \tt PV=\frac{1}{3}MC^2
\tt K.E=\frac{3}{2}RT
\tt K.E=\frac{3}{2}kT (PV=RT)

### Part3: View the Topic in this Video from 0:56 to 1:00:00

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1. Boyle's law
PV = K or \tt P_{1}V_{1} = P_{2}V_{2} = K (For two or more gases)

2. Boyle's law can also be expressed as \tt \left(\frac{dP}{dV}\right)_{T} = \frac{-k}{V^{2}} \ or \ \frac{dV}{V} = \frac{-dP}{P}

3. \tt K = \frac{V}{T} \ or \ \frac{V_{1}}{T_{1}} = \frac{V_{2}}{T_{2}} = K \left(For \ two \ or \ more \ gases\right)

4. Charle's law can be represented as \tt \left(\frac{dV}{dt}\right)_{P} = K

5. \tt K = \frac{P}{T} \ or \ \frac{P_{1}}{T_{1}} = \frac{P_{2}}{T_{2}} = K \left(For \ two \ or \ more \ gases\right)

6. Avogadro's Law: \tt \frac{V_{1}}{n_{1}} = \frac{V_{2}}{n_{2}} = ............ = K

7. PV = nRT, This is called ideal gas equation.

8. If a number of gases having volume V1, V2, V3 ...... at pressure P1, P2, P3 ....... are mixed together in container of volume V, then,
\tt P_{Total} = \frac{P_{1}V_{1} + P_{2}V_{2} + P_{3}V_{3}......}{V}
or = \tt \left(n_{1} + n_{2} + n_{3}.......\right)\frac{RT}{V} \ \left(\because PV = nRT\right)
or = \tt n\frac{RT}{V} \ \left(\because n = n_{1} + n_{2} + n_{3}.......\right)

9. \tt \frac{r_{1}}{r_{2}} = \sqrt{\frac{d_{2}}{d_{1}}} = \sqrt{\frac{d_{2} \times 2}{d_{1} \times 2}} = \sqrt{\frac{M_{2}}{M_{1}}}
Where, M1 and M2 are the molecular weights of the two gases.

10. For gases \tt \frac{V_{1}}{V_{2}} = {\frac{n_{1}}{n_{2}}}
\tt {\frac{n_{1}}{n_{2}}}\times{\frac{t_{2}}{t_{1}}}= \sqrt{\frac{M_{2}}{M_{1}}}

11. When equal volume of the two gases diffuse, i.e. V1 = V2 then, \tt {\frac{r_{1}}{r_{2}}} = {\frac{t_{2}}{t_{1}}}= \sqrt{\frac{d_{2}}{d_{1}}}

12. When volumes of the two gases diffuse in the same time, i.e. t1 = t2 then, \tt {\frac{r_{1}}{r_{2}}} = {\frac{V_{1}}{V_{2}}}= \sqrt{\frac{d_{2}}{d_{1}}}

13. Kinetic gas equation : On the basis of above postulates, the following gas equation was derived,
\tt PV = \frac{1}{3}mnu_{rms}^2
where, P = pressure exerted by the gas
V = Volume of the gas
m = average mass of a molecule
n = number of molecules
urms = root mean square (RMS) velocity of the gas.