Explanation Given: The diameter of tube is D = 10 μm . The temperature of gases is T = 23 o C . Formula used: The expression for transition density is given as, ρ c = 100 k B M 2 π d 2 R D Here, k B is the Boltzmann constant, d is the molecular diameter, ρ c is the density, R is the ideal gas constant, and M is the molecular mass. Calculation: The expression for mean free path is given as, λ m f p = k B T 2 π d 2 p ...... ( 1 ) The expression for pressure from ideal gas equation is given as, p = ρ R T M ...... ( 2 ) The effect of micro scale comes to action when, λ m f p D ≈ 0.01...... ( 3 ) On combining the above three equation, transition density can be obtained. λ m f p D = 0.01 0.01 = k B M 2 π d 2 ρ c R D ρ c = 100 k B M 2 π d 2 R D Refer table A.4 “Thermo physical properties of gases at atmospheric pressure”. Molecular density of air ρ = 1.161 kg / m 3 . Molecular density of hydrogen ρ = 0.0808 kg / m 3 . Molecular density of air ρ = 1.773 kg / m 3 . Refer figure 2.8 for the molecular weight of substances. Molecular mass of air is M = 28.97 kmol / kg . Molecular mass of hydrogen is M = 2.016 kmol / kg . Molecular weight of carbon dioxide is M = 44.01 kmol / kg . Refer Figure 2.8 “The temperature dependence of the thermal conductivity of selected gases at normal pressures” The molecular diameter forair is d = 0.327 nm . The molecular diameter for hydrogen is d = 0.274 nm . The molecular diameter for carbon dioxide is d = 0.464 nm . The transition density of air can be calculated as, ρ c , a i r = 100 k B M 2 π d 2 R D = 100 ( 1.381 × 10 − 23 J / K ) ( 28.97 kmol / kg ) 2 π ( 0.327 nm ( 1 m 10 9 nm ) ) 2 ( 8315 J / kmol ⋅ K ) ( [ 10 μm ( 1 m 10 6 μm ) ] ) = 1.0127 kg / m 3 The transition density of hydrogen can be calculated as, ρ c , a i r = 100 k B M 2 π d 2 R D = 100 ( 1.381 × 10 − 23 J / K ) ( 2

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ISBN: 9780470917855

Author: Bergman, Theodore L./

Publisher: John Wiley & Sons Inc

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Chapter 8, Problem 8.108P

To determine

The transition density for hydrogen

The transition density for air.

The transition density carbon dioxide.

The comparison of calculated transition density to the gas density at T=23oC .

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Chapter 8, Problem 8.107P

Chapter 8, Problem 8.109P

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The transition density for hydrogen The transition density for air. The transition density carbon dioxide. The comparison of calculated transition density to the gas density at T = 23 o C .

Solution Summary: The author compares the transition density for hydrogen and carbon dioxide to the gas density at T=23oC.

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The transition density for hydrogen

The transition density for air.

The transition density carbon dioxide.

The comparison of calculated transition density to the gas density at T=23oC .

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Chapter 8 Solutions

The transition density for hydrogen The transition density for air. The transition density carbon dioxide. The comparison of calculated transition density to the gas density at T = 23 o C .

Solution Summary: The author compares the transition density for hydrogen and carbon dioxide to the gas density at T=23oC.

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The transition density for hydrogen The transition density for air. The transition density carbon dioxide. The comparison of calculated transition density to the gas density at T=23oC .

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Chapter 8 Solutions

The transition density for hydrogen

The transition density for air.

The transition density carbon dioxide.

The comparison of calculated transition density to the gas density at T=23oC .