Kinetic Theory Of Gases Equation Of Ideal Gas L-01 Kinetic Theory of Gases Equation of Ideal Gas
→ \rightarrow → → \rightarrow → Kinetic Theory of Gases Equation of Ideal Gas → \rightarrow → → \rightarrow →
Kinetic Theory Of Gases Equation Of Ideal Gas L-01 Probability distribution
→ \rightarrow → → \rightarrow → Probability distribution → \rightarrow → → \rightarrow →
Kinetic Theory Of Gases Equation Of Ideal Gas L-01 What did We Learn?
P ( v x ) d v x = A e − α v x 2 d v x P(v_x) d v_x=A e^{-\alpha v_x^2} d v_x P ( v x ) d v x = A e − α v x 2 d v x
v x → v x + d v x v_x \rightarrow v_x+d v_x v x → v x + d v x
v x ˉ = 0 \bar{v_x} = 0 v x ˉ = 0
Speed distribution v = v x 2 + v y 2 + v z 2 v=\sqrt{v_x^2+v_y^2+v_z^2} v = v x 2 + v y 2 + v z 2
v ⟹ v + d v v \Longrightarrow v+d v v ⟹ v + d v
P ( v ) d v = B v 2 e − a v 2 d v P(v) d v=B v^2 e^{-a v^2} d v P ( v ) d v = B v 2 e − a v 2 d v
Kinetic Theory of Gases Equation of Ideal Gas → \rightarrow → → \rightarrow → What did We Learn? → \rightarrow → → \rightarrow →
Kinetic Theory Of Gases Equation Of Ideal Gas L-01 Ideal Gas
Probability distribution → \rightarrow → → \rightarrow → Ideal Gas → \rightarrow → → \rightarrow →
Kinetic Theory Of Gases Equation Of Ideal Gas L-01 Confined a Volume
Ideal Gas : Confined in a Volume.
Microscopic approach → \rightarrow →
Volume → \rightarrow → L 3 L^3 L 3
Momentum transferred
What did We Learn? → \rightarrow → → \rightarrow → Confined a Volume → \rightarrow → → \rightarrow →
Kinetic Theory Of Gases Equation Of Ideal Gas L-01 Elastic Collision
What is momentum change of i'th particle: − m i v i x − m i v i x - m_i v_i^x-m_i v_i^x − m i v i x − m i v i x
Magnitude of change in momentum = − 2 m v i x - 2 m v_i^x − 2 m v i x
Dilute limit "Mean free path"
Δ t = 2 L v i x \Delta t=\frac{2 L}{v_i^x} Δ t = v i x 2 L
per unit time: M = 1 Δ t = v i x 2 L M=\frac{1}{\Delta t}=\frac{v_i^x}{2 L} M = Δ t 1 = 2 L v i x
Ideal Gas → \rightarrow → → \rightarrow → Elastic Collision → \rightarrow → → \rightarrow →
Kinetic Theory Of Gases Equation Of Ideal Gas L-01 Momentum Transferred
Total momentum transferred per unit time
Δ F = n × 2 m v i x \Delta F =n \times 2 m v_i^x Δ F = n × 2 m v i x
= v i x 2 L × 2 m v i x =\frac{v_i^x}{2 L} \times 2 m v_i^x = 2 L v i x × 2 m v i x
= v i x L × m v i x =\frac{v_i^x}{ L} \times m v_i^x = L v i x × m v i x
Confined a Volume → \rightarrow → → \rightarrow → Momentum Transferred → \rightarrow → → \rightarrow →
Kinetic Theory Of Gases Equation Of Ideal Gas L-01 Isotropy
F = m L ∑ i ( v i x ) 2 = m N L ⋅ ( 1 N ∑ ( v i x ) 2 ) F =\frac{m}{L} \sum_i(v_i^x)^2 = \frac{m N}{L} \cdot(\frac{1}{N} \sum(v_i^x)^2) F = L m ∑ i ( v i x ) 2 = L m N ⋅ ( N 1 ∑ ( v i x ) 2 )
∑ ( v i x ) 2 = ∑ ( v i y ) 2 = ∑ i ( v i z ) 2 \sum (v_i^x)^2=\sum(v_i^y)^2=\sum_i(v_i^z)^2 ∑ ( v i x ) 2 = ∑ ( v i y ) 2 = ∑ i ( v i z ) 2
∑ i ( v i x ) 2 = 1 3 ∑ i [ ( v i x ) 2 + ( v i y ) 2 + ( v i z ) 2 ] \sum_i(v_i^x)^2 = \frac{1}{3}\sum_i[(v_i^x)^2+(v_i^y)^2+(v_i^z)^2] ∑ i ( v i x ) 2 = 3 1 ∑ i [( v i x ) 2 + ( v i y ) 2 + ( v i z ) 2 ]
Elastic Collision → \rightarrow → → \rightarrow → Isotropy → \rightarrow → → \rightarrow →
Kinetic Theory Of Gases Equation Of Ideal Gas L-01 Momentum Transferred
F = 1 3 m L ∑ i = 1 N v ⃗ i 2 {F=\frac{1}{3} \frac{m}{L} \sum_{i=1}^N \vec{v}_i^2} F = 3 1 L m ∑ i = 1 N v i 2
v ⃗ i 2 = ( v i x ) 2 + ( v i y ) 2 + ( v i z ) 2 \vec{v}_i^2=(v_i^x)^2+(v_i^y)^2+(v_i^z)^2 v i 2 = ( v i x ) 2 + ( v i y ) 2 + ( v i z ) 2
= 1 3 m N L ( 1 N ∑ i = 1 N v i 2 ) =\frac{1}{3} \frac{mN}{L}(\frac{1}{N} \sum_{i=1}^N v_i^2) = 3 1 L m N ( N 1 ∑ i = 1 N v i 2 )
Mean Square
v rms 2 = 1 3 m N L v r m s 2 v_{\text {rms }}^2=\frac{1}{3} \frac{m N}{L} v_{r m s}^2 v rms 2 = 3 1 L m N v r m s 2
Momentum Transferred → \rightarrow → → \rightarrow → Momentum Transferred → \rightarrow → → \rightarrow →
Kinetic Theory Of Gases Equation Of Ideal Gas L-01 Pressure
Isotropy → \rightarrow → → \rightarrow → Pressure → \rightarrow → → \rightarrow →
Kinetic Theory Of Gases Equation Of Ideal Gas L-01 Momentum Transferred
Momentum Transferred → \rightarrow → → \rightarrow → Momentum Transferred → \rightarrow → → \rightarrow →
Kinetic Theory Of Gases Equation Of Ideal Gas L-01 Dulong-Petit Law
P V = 1 3 m N v rms 2 PV=\frac{1}{3} m N v_{\text {rms }}^2 P V = 3 1 m N v rms 2
1 2 m v r m s 2 = k B T = 1 2 I ω 2 \frac{1}{2} m v_{r m s}^2=k_B T = \frac{1}{2} I \omega^2 2 1 m v r m s 2 = k B T = 2 1 I ω 2
Total Translational → \rightarrow →
ε p = p 2 2 m = p c \varepsilon_p=\frac{p^2}{2 m} = p c ε p = 2 m p 2 = p c
Translational: ε t o t a l = 1 2 m N V 2 r m s \varepsilon_{total} = \frac{1}{2} m N{V^2}_{rms} ε t o t a l = 2 1 m N V 2 r m s
P V = 2 3 E PV=\frac{2}{3} E P V = 3 2 E
1 2 m v rms 2 = 3 2 k B T \frac{1}{2} m v_{\text {rms }}^2=\frac{3}{2} k_B T 2 1 m v rms 2 = 2 3 k B T
ε total = 3 N 2 k B T ⇒ ∂ ε total ∂ T = 3 N 2 k B \varepsilon_{\text {total }} = \quad \frac{3 N}{2} k_B T \Rightarrow \frac{\partial \varepsilon_{\text {total }}}{\partial T}=\frac{3 N}{2} k_B ε total = 2 3 N k B T ⇒ ∂ T ∂ ε total = 2 3 N k B
Pressure → \rightarrow → → \rightarrow → Dulong-Petit Law → \rightarrow → → \rightarrow →
Kinetic Theory Of Gases Equation Of Ideal Gas L-01 Equipartition of Energy
1 2 m v 2 = 3 2 k T = 1 2 k T + 1 2 \frac{1}{2} m v^2=\frac{3}{2} k T = \frac{1}{2} kT +\frac{1}{2} 2 1 m v 2 = 2 3 k T = 2 1 k T + 2 1 k T + 1 2 k T kT+\frac{1}{2} kT k T + 2 1 k T
1 2 m v ⃗ x 2 = 1 2 m v ⃗ y 2 = 1 2 m v ⃗ z 2 \frac{1}{2} m \vec v_x^2=\frac{1}{2} m \vec{v}_y^2=\frac{1}{2} m \vec v_z^2 2 1 m v x 2 = 2 1 m v y 2 = 2 1 m v z 2
1 2 m v ⃗ x 2 = 1 2 k T → \frac{1}{2} m \vec{v}_x^2=\frac{1}{2} k T \rightarrow 2 1 m v x 2 = 2 1 k T →
1 2 k T → \frac{1}{2} k T \rightarrow 2 1 k T →
Momentum Transferred → \rightarrow → → \rightarrow → Equipartition of Energy → \rightarrow → → \rightarrow →
Kinetic Theory Of Gases Equation Of Ideal Gas L-01 Thank You
Dulong-Petit Law → \rightarrow → → \rightarrow → Thank You → \rightarrow → → \rightarrow →
Resume presentation
Kinetic Theory Of Gases Equation Of Ideal Gas L-01 Kinetic Theory of Gases Equation of Ideal Gas $\rightarrow$ $\rightarrow$ Kinetic Theory of Gases Equation of Ideal Gas $\rightarrow$ Probability distribution $\rightarrow$ What did We Learn?
