Ch 19: Work, Heat, and the First Law of Thermodynamics

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 19: Work, Heat, and the First Law of Thermodynamics

Problem 49a

Chapter 19, Problem 49a

A 6.0-cm-diameter cylinder of nitrogen gas has a 4.0-cm-thick movable copper piston. The cylinder is oriented vertically, as shown in FIGURE P19.49, and the air above the piston is evacuated. When the gas temperature is 20°C, the piston floats 20 cm above the bottom of the cylinder. What is the gas pressure?

Verified step by step guidance

Determine the area of the piston. The piston is circular, so its area can be calculated using the formula for the area of a circle:

A = π r 2^{2}

, where

r

is the radius of the piston. The diameter of the piston is 6.0 cm, so the radius is half of that:

r = 3.0 cm

Calculate the weight of the copper piston. The weight is given by

W = m g

, where

m

is the mass of the piston and

g

is the acceleration due to gravity (approximately

9.8 m/s 2^{2}

). To find the mass, use the density of copper (

8.96 g/cm 3^{3}

) and the volume of the piston. The volume of the piston is the area of its base multiplied by its thickness:

V = A h

, where

h

is the thickness (4.0 cm).

Relate the gas pressure to the weight of the piston. Since the air above the piston is evacuated, the gas pressure inside the cylinder is the only force supporting the piston. The pressure is given by

P = F / A

, where

F

is the force exerted by the piston (equal to its weight) and

A

is the area of the piston.

Substitute the values for the weight of the piston and the area of the piston into the pressure formula. Ensure that all units are consistent (e.g., convert cm to m if necessary).

Simplify the expression to find the gas pressure. The result will be in Pascals (Pa), which is the SI unit for pressure.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Gas Pressure

Gas pressure is defined as the force exerted by gas molecules colliding with the walls of a container per unit area. It is influenced by the number of gas molecules, their temperature, and the volume of the container. In this scenario, the pressure of the nitrogen gas can be calculated using the ideal gas law or by considering the weight of the piston and the area it covers.

Ideal Gas Law

The ideal gas law is a fundamental equation in thermodynamics that relates the pressure, volume, temperature, and number of moles of a gas. It is expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature in Kelvin. This law helps in calculating the state of a gas under various conditions, such as the one described in the question.

Hydrostatic Pressure

Hydrostatic pressure refers to the pressure exerted by a fluid at equilibrium due to the force of gravity. In this case, the weight of the copper piston creates a downward force that must be balanced by the pressure of the nitrogen gas inside the cylinder. The relationship between the weight of the piston and the area of the cylinder's base is crucial for determining the gas pressure.

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Textbook Question

The beaker in FIGURE P19.45, with a thin metal bottom, is filled with 20 g of water at 20°C. It is brought into good thermal contact with a 4000 cm³ container holding 0.40 mol of a monatomic gas at 10 atm pressure. Both containers are well insulated from their surroundings. What is the gas pressure after a long time has elapsed? You can assume that the containers themselves are nearly massless and do not affect the outcome.

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Textbook Question

A 6.0-cm-diameter cylinder of nitrogen gas has a 4.0-cm-thick movable copper piston. The cylinder is oriented vertically, as shown in FIGURE P19.49, and the air above the piston is evacuated. When the gas temperature is 20°C, the piston floats 20 cm above the bottom of the cylinder. What is the new equilibrium temperature of the gas?

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Textbook Question

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