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 49c

Chapter 19, Problem 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?

Verified step by step guidance

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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Key Concepts

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

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.

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.

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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Gauss' Law

Related Practice

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 gas pressure?

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Textbook Question

2.0 mol of gas are at 30 °C and a pressure of 1.5 atm. How much work must be done on the gas to compress it to one third of its initial volume at constant pressure?

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Textbook Question

An ideal-gas process is described by p=cV^1/2, where c is a constant. Find an expression for the work done on the gas in this process as the volume changes from V₁ to V₂.

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Textbook Question

0.25 mol of a gas are compressed at a constant pressure of 250 kPa from 6000 cm³ to 2000 cm³, then expanded at a constant temperature back to 6000 cm³. What is the net work done on the gas?

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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 typical nuclear reactor generates 1000 MW (1000 MJ/s) of electric energy. In doing so, it produces 2000 MW of 'waste heat' that must be removed from the reactor to keep it from melting down. Many reactors are sited next to large bodies of water so that they can use the water for cooling. Consider a reactor where the intake water is at 18°C. State regulations limit the temperature of the output water to 30°C so as not to harm aquatic organisms. How many liters of cooling water have to be pumped through the reactor each minute?

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