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: 9780137344796Non è quello che usi tu?Cambia libro di testo

Knight Calc 5th Edition

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

Problema 49c

Capitolo 19, Problema 49c

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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Determine the initial state of the gas using the ideal gas law. The ideal gas law is given by:

P V = n R T

, where

P

is the pressure,

V

is the volume,

n

is the number of moles,

R

is the gas constant, and

T

is the temperature. Use the given dimensions of the cylinder and piston to calculate the initial volume of the gas.

Calculate the force acting on the piston due to the weight of the copper piston. The force is given by

F = m g

, where

m

is the mass of the piston and

g

is the acceleration due to gravity. Use the density of copper and the volume of the piston to find its mass.

Relate the pressure of the gas to the force exerted by the piston. The pressure is given by

P = \frac{F}{A}

, where

A

is the cross-sectional area of the piston. Use the diameter of the cylinder to calculate the area.

Apply the ideal gas law again to find the new equilibrium temperature. Since the piston is movable, the pressure remains constant. The relationship between the initial and final states of the gas can be expressed as:

\frac{V_{f}}{V_{i}} = \frac{T_{f}}{T_{i}}

, where

V_{i}

and

V_{f}

are the initial and final volumes, and

T_{i}

and

T_{f}

are the initial and final temperatures.

Solve for the final temperature

T_{f}

using the relationship derived in the previous step. Substitute the known values for the initial temperature, initial volume, and final volume to find the new equilibrium temperature.

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Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of an ideal gas through the equation PV = nRT. This law is essential for understanding how changes in temperature and volume affect the behavior of gases, particularly in scenarios involving pistons and varying pressures.

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Percorso guidato

07:21

Ideal Gases and the Ideal Gas Law

Thermal Equilibrium

Thermal equilibrium occurs when two systems reach the same temperature and no heat flows between them. In this context, it is crucial to determine the new equilibrium temperature of the gas after the piston moves, as it will influence the pressure and volume of the gas within the cylinder.

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Percorso guidato

05:21

Volume Thermal Expansion

Charles's Law

Charles's Law states that the volume of a gas is directly proportional to its temperature when pressure is held constant. This principle is vital for solving the problem, as it allows us to relate the initial and final states of the gas in the cylinder as the piston moves and the temperature changes.

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Percorso guidato

10:20

Gauss' Law

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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?

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