$P = \frac{1}{3} \frac{m N}{L^3} v_{r m s}^2$ $v_{r m s}^2$

$P = \frac{1}{3} \rho v_{r m s}^2$

$P V=\frac{1}{3} M v_{r m s}^2=\frac{1}{3} m N v_{r_{m s}}^2$ $=\frac{1}{N} \sum_{i=1}\left(v_i^x\right)^2$ $+\left(v_i^y \right)^2$

How do you relate it temp.

Ideal gas eqn, of state PV=nRT

PV=$Nk_BT$

$\frac{1}{2} m v_{\text {rms }}^2=\frac{3}{2} k_B T$ $\frac{1}{2} m N v_{\text {rms }}^2=\frac{3}{2} N k_B T$

$P V=\frac{2}{3} E_{\text {translational}}$

$E_p=\frac{p^2}{2 m}$

$\frac{1}{2} m\left(3 v_x^2\right)=\frac{3}{2} k_BT$

$\frac{1}{2} m v_x^2=\frac{1}{2} k_BT$

$\frac{1}{2} N{m v_2^2} =\frac{N}{2} K_i$

$\frac{1}{2} m v_y^2=\frac{1}{2} k_BT$

$v^2 =v_x^2+v_y^2+v_z^2 =3 v_x^2$

Three components of velocity

E = $\frac{1}{2}k_BT$

$N \longrightarrow 3 \text { dimensions } \longrightarrow v_i^x, v_i^y, v_i^z$

$\frac{1}{2} N k_B T+\frac{1}{2} N k_B T+\frac{1}{2} N k_B T=\frac{3}{2} N k_B T$

$E=\frac{p^2}{2 x}+\frac{1}{2} k x^2$

$\langle E\rangle=k_B T$

$\langle E\rangle=3Nk_B T$

$f = - kx$

$V =\frac{1}{2}kx^2$

$C_V = \frac{d E}{d T}$

$C_p=Lt_{\Delta T \rightarrow 0} \frac{\Delta E}{\Delta T}$

For ideal Gas

$C_p-C_r=R$

$C_V=\frac{3}{2} N k_{ B}$

Harmonic Oscillator :

$C_V = 3 N k_B$

Ideal Gas : Monoatomic

Diatomic molecule

Rotational Kinetic Energy:

$\frac{3}{2} k_B T$ = $\frac{1}{2} I_1 \omega_1^2 + \frac{1}{2} I_2 \omega_2$

Rigid body

6 degrees of freedom

$3 \rightarrow$ translational

$3 \rightarrow$ rotational

Rigid body approximation

$10^{23}$ particles

$\rightarrow N 6\frac{1}{2}kT = 3NKT$

$\left(3 k T+\frac{f}{2} k T\right) N$

$P=\frac{1}{3} m N v^2$

$v^2 \propto k_{B } T$

$T \propto f\left(\frac{1}{2} m v^2\right)$

$P V=\frac{1}{3} m N v^2$

$P_0 V=\frac{1}{3} m N v_0^2$

$\frac{P}{P_0}=\frac{v^2}{v_0^2}$

$v^2=\left(\frac{P}{P_0}\right) v_0^2$

$v^2=\left(\frac{v_0^2}{T_0}\right) T$

$v^2 \propto T$

$\frac{1}{2} m v^2 =a k_B T$, Where a is any number

Mix two Gases

Thermal equation

$\frac{1}{2} m_1 v_1^2=\frac{1}{2} m_2 v_2^2$

$P V=\frac{1}{3} M n V^2 \propto$ Temperature

PV = Constant if T is constant

Boyle's law: $P \propto T$

Given T, p, equal volumes of all Gases have equal number of particles

• $p V=\frac{1}{3} N_1 m_1 \overline {v_1^2}$

• $p V=\frac{1}{3} N_2 m_2 v_2^2$

T is fixed :

• $\frac{1}{2} m_1 v_1^2=\frac{1}{2} m_2 v_2^2$

• $N_1 = N_2$

$p_1=p_1+p_2+\cdots,$

p = Momentum transfered

$\Delta F_1 \longrightarrow F_1 \quad F_2 \rightarrow \sum_iF_i$

Pressure: $\quad \frac{F_{n e t}}{L^2}=p_1+p_2 \cdots,$

Two gases P,t

allowed to diffuse

$\frac{r_1}{r_2}=\frac{v_{1, r m s}}{v_{2, r m s}}$

• $V_{\text {rms }}=\sqrt{\frac{3 p}{\rho}}=\sqrt{\frac{\rho_2}{\rho_1}}$

Collisions between the molecules

Average distance that a gas molecule travels between two successes.

$l = \frac{1}{cn\pi d^2}$

$n \rightarrow 0$

$d \rightarrow 0$