Some Deductions from the Expression for the Pressure

Let us consider 1 gram molecule of a gas at absolute T occupying a volume V. according to kinetic theory, the pressure of the gas

$$P = \frac 13. \frac {mN}{V} \vec {v^2} \dots (1) $$

Where N = Avogadro number = number of molecules in 1 gram mole of a gas.

1. Kinetic interpretation of temperature:

   As we know,
   \begin{align*} PV &= \frac 13 mN\vec {v^2} \\ \text{According to gas equation} \\ PV &= RT \\ \text {Comparing equation}\: (2),\: \text {and}\:(3)\\ \frac 13 mN\vec {v^2} &= RT \\ \text{or,}\: \frac 12 m\vec {v^2} &= \frac 32\frac RN T = \frac 32 kT \dots (4)\\ \end{align*} $$ where\: k = \frac RN = \text {Boltzmann’s contant} = 1.38\times 10^{-23} \:\text {joule/K.}$$
   From equation (4), the average kinetic energy of translation of the molecule is directly proportional to the absolute temperature of the gas.

2. Boyle’s Law

   As we know,
   $$ PV = \frac 13 mN\vec {v^2} \dots (6) $$
   according to kinetic theory, the average kinetic energy of translation of molecules of a gas is directly proportional to its absolute temperature, i.e.,
   $$ \frac 12 m\vec {v^2} = \alpha T $$
   $$ PV = \frac 23 N.\frac 12 m\vec {v^2} = \frac 23N\alpha T$$

3. Charle’s Law

   As we know,
   \begin{align*} \text {From equation }\: (1) \\ PV &= \frac 13 mN\vec {v^2} \\ \text {Here the average K.E of a molecule,} \\ \frac12 m\vec{v^2} &= \alpha T \\ V\alpha T \: \text {for constant value of P}\\ \end{align*}
   Thus, at a given mass of gas, the volume is directly proportional to the absolute temperature, which is Charle’s Law.

4. Gas Equation:

   As we know,
   \begin{align*} PV &= \frac 13 mN\vec {v^2} \\ \text {Also} \frac 12 m\vec {v^2} &= \alpha T \\ \text {or}, PV &= \frac 23 N. \frac 12 m\vec {v^2} = \frac 23N\alpha T \\ \text {or,} PV &= RT \dots (7) \\\end{align*} $$ \text {where} \: R = \frac {2N\alpha }{3} =Nk = \text {universal gas constant for 1 gram mole.} $$
   Equation (7) represents well known gas equation for a perfect gas.

5. Avogadro Hypothesis

   Let us consider two gases A and B at pressure P and having volume V each. Let \(m_1\) and \(m_2\) be the masses of each molecule of A and B at pressure respectively and \(n_1\) and \(n_2\) be the number of molecules A and B respectively. If \(\vec {v_1^2}\) and \(\vec {v_2^2}\) be the mean square speed of A and B respectively, then
   \begin{align*} \text {Pressure for gas A,} \\ P &= \frac 13\frac {m_1n_1\vec {v_1^2}}{V}\dots (8) \\ \text {and pressure for gas B,} \\ P &= \frac 13\frac {m_2n_2\vec {v_2^2}}{V}\dots (9)\\ \text {From equations}\: (8)\: \text {and}\: (9)\\ P &= \frac 13. \frac {m_1n_1\vec {v_ 1^2}}{V} = \frac 13 \frac {m_2n_2\vec {v_2^2}}{V} \\ m_1n_1\vec {v_1^2} &= m_2n_2\vec {v_2^2} \dots (10)\\ \end{align*}
   If the two gases are at same temperature T, the average kinetic energy of both the gases is always same,
   \begin{align*} \frac 12 m_1\vec {v_1^2} &= \frac 12m_2 \vec {v_2^2} \dots (11)\\ \text {Comparing}\: (10)\: \text {and}\: (11) \\ n_1 &= n_2 \dots (12) \\ \end{align*}
   Thus according to Avogadro hypothesis equal volumes of all gases under similar conditions of temperature and pressure have the same number of molecules.

6. Grahm’s law of diffusion

   According to Kinetic theory
   \begin{align*} P &= \frac 13 \rho \vec {v^2} \\ \text {or,}\: V_{r.m.s} &= \sqrt {\vec {v^2}} \\ &= \sqrt {\left (\frac {3P}{\rho}\right)}\dots (13) \end{align*}
   As rate of diffusion of gas is proportional to the root mean square speed of molecules. The rate of diffusion of a gas is inversely proportional to the square root of the density.

7. Dalton’s Law of Partial Pressure

   Let \(P_1\) and \(P_2\) be the partial pressures of the individual gases in a mixture of volume V. let the gas of partial pressure \(p_1\) contain \(n_1\) molecules each of mass \(m_1\) and mean-square velocity \(\vec {v_1^2}\) and the gas of partial pressure \(p_2\) contain \(n_2\) molecules each of mass \(m_2\) and mean-square velocity \(\vec {v_2^2}\) and so on. According to Kinetic theory,
   \begin{align*} p_1V = \frac 13m_1n_1\vec {v_1^2},\\ p_2V = \frac 13m_2n_2\vec {v_2^2}, \dots \\ (p_1 + p_2 + p_3+ \dots)\: V &= \frac 13 (m_1n_1\vec {v_1^2} + m_2n_2\vec {v_2^2} + \dots )\dots (14) \end{align*}
   As whole of the mixture is at the same temperature, the average kinetic energy of translation per molecule of each gas is the same
   \begin{align*} \frac 12m_1\vec {v_1^2} = \frac 12m_2\vec {v_2^2} \dots = \frac 12m\vec {v^2} \\ (p_1 + p_2+\dots )V &= \frac 13(n_1 + n_2+ \dots )m\vec {v^2} \dots (15) \end{align*}
   But \((n_1 + n_2+\dots )\) is the total number of molecules in the mixture. Net pressure P exerted by the mixture according to kinetic theory, is given by
   \begin{align*} PV &= \frac 13(n_1 + n_2+\dots )m\vec {v^2} \dots (16) \\ \text {from equation} (15) \text {and} (16) P &= p_1 + p_2+ \dots \\ \end{align*}

Some useful standard definite integrals

If \(I_n = \int _0^{\infty} v^ne^{-av^2}\: dv\), then the value of \(I_n\) for various values of n

\begin{align*} I_0 &= \int _0^{\infty} e^{-av^2}\: dv = \frac 12 \left ( \frac {\pi}{\alpha }\right )^{1/2} \dots (1) \\ I_1 &= \int _0^{\infty} ve^{-av^2}\: dv = \frac {1}{2\alpha} \dots (2) \\ I_2 &= \int _0^{\infty} v^2 e^{-av^2}\: dv = \frac 14 \left (\frac{\pi}{\alpha ^3}\right )^{1/2} \dots (3) \\ I_3 &= \int _0^{\infty} v^3e^{-av^2}\: dv = \left (\frac {1}{2\alpha ^2}\right ) \dots (4) \\ I_4 &= \int _0^{\infty} v^4e^{-av^2}\: dv = \frac 38 \sqrt {\left (\frac {\pi}{\alpha^5}\right )}\dots {5} \\ I_5 &=\int _0^{\infty} v^5e^{-av^2}\: dv = \frac {1}{\alpha ^3}\dots (6) \end{align*}

If the limits of integration are \(-\infty \) to \(\infty \) in place of 0 to \(\infty \) then the integrals \(I_0’,I_2’, I_4’\) will be double while the integrals \(I_1’, I_3’, I_5’\) will vanish.

Bibliography

S.S. Singhal, J.P. Agarwal, Satya Prakash. heat and thermodynamics and statistical physics. pragati prakashan, 2010.

—. Heat and Thermodynamics and Statistical Physics. Pragati Prakashan, 2010.

Vatsyayan, Dr. Rakesh Ranjan. Refresher Course in Physics. kathmandu: Surya Book Traders, 2015.