Liquid nitrogen comes in large cylinders that required special tank carts and hold 120 L of liquid at 77 K. Given the density of liquid nitrogen of 0.840 g/cm 3 , using the van der Waals equation, the volume of the nitrogen gas after it evaporates at 77 K and atmospheric pressure is to be estimated. Concept introduction: The ideal gas law considered the molecules of a gas as point particles with perfectly elastic collisions among them in nature. This works importantly well for gases at dilution and at low pressure in many experimental calculations. But the gas molecules are not performing as point masses, and there are situations where the properties of the gas molecules have measurable effect by experiments. Thus, a modification of the ideal gas equation was coined by Johannes D. van der Waals in 1873 to consider size of molecules and the interaction forces among them. It is generally denoted as the van der Waals equation of state.

Question

In a commonly used model of the atmosphere, the atmospheric pressure varies with altitude, h, according to the barometric formula:p = p0e–h/Hwhere p0 is the pressure at sea level and H is a constant approximately equal to 8 km. More specifically, H = RT/Mg, where M is the average molar mass of air and T is the temperature at the altitude h. This formula represents the outcome of the competition between the potential energy of the molecules in the gravitational field of the Earth and the stirring effects of thermal motion. Derive this relation by showing that the change in pressure dp for an infinitesimal change in altitude dh where the mass density is ρ is dp = −ρgdh. Remember that ρ depends on the pressure. Evaluate (a) the pressure difference between the top and bottom of a laboratory vessel of height 15 cm, and (b) the external atmospheric pressure at a typical cruising altitude of an aircraft (11 km) when the pressure at ground level is 1.0 atm.

Answer 1

Question

Chapter 1, Problem 1.37E

Interpretation Introduction

Interpretation:

Liquid nitrogen comes in large cylinders that required special tank carts and hold 120 L of liquid at 77 K. Given the density of liquid nitrogen of 0.840 g/cm³, using the van der Waals equation, the volume of the nitrogen gas after it evaporates at 77 K and atmospheric pressure is to be estimated.

Concept introduction:

The ideal gas law considered the molecules of a gas as point particles with perfectly elastic collisions among them in nature. This works importantly well for gases at dilution and at low pressure in many experimental calculations. But the gas molecules are not performing as point masses, and there are situations where the properties of the gas molecules have measurable effect by experiments. Thus, a modification of the ideal gas equation was coined by Johannes D. van der Waals in 1873 to consider size of molecules and the interaction forces among them. It is generally denoted as the van der Waals equation of state.

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In a commonly used model of the atmosphere, the atmospheric pressure varies with altitude, h, according to the barometric formula:p = p0e–h/Hwhere p0 is the pressure at sea level and H is a constant approximately equal to 8 km. More specifically, H = RT/Mg, where M is the average molar mass of air and T is the temperature at the altitude h. This formula represents the outcome of the competition between the potential energy of the molecules in the gravitational field of the Earth and the stirring effects of thermal motion. Derive this relation by showing that the change in pressure dp for an infinitesimal change in altitude dh where the mass density is ρ is dp = −ρgdh. Remember that ρ depends on the pressure. Evaluate (a) the pressure difference between the top and bottom of a laboratory vessel of height 15 cm, and (b) the external atmospheric pressure at a typical cruising altitude of an aircraft (11 km) when the pressure at ground level is 1.0 atm.

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