Ch 18: A Macroscopic Description of Matter

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 18: A Macroscopic Description of Matter

Problem 53

Chapter 18, Problem 53

A 10-cm-diameter, 40-cm-tall gas cylinder, sealed at the top by a frictionless 50 kg piston, is surrounded by a bath of 20°C water. Then 50 kg of sand is slowly poured onto the top of the piston, where it stays. Afterward, what is the height of the piston?

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Determine the initial pressure inside the gas cylinder. The pressure is due to the weight of the piston and the atmospheric pressure. Use the formula for pressure: \( P = P_{\text{atm}} + \frac{F}{A} \), where \( F \) is the force due to the weight of the piston (\( F = m g \)) and \( A \) is the cross-sectional area of the cylinder (\( A = \pi r^2 \)).

Calculate the initial volume of the gas. The volume of a cylinder is given by \( V = A h \), where \( A \) is the cross-sectional area and \( h \) is the initial height of the gas column (40 cm).

Apply the ideal gas law to relate the initial and final states of the gas. The law is \( P_1 V_1 = P_2 V_2 \), assuming the temperature remains constant (isothermal process). Here, \( P_1 \) and \( V_1 \) are the initial pressure and volume, and \( P_2 \) and \( V_2 \) are the final pressure and volume.

Determine the final pressure \( P_2 \). The final pressure includes the weight of the piston, the added sand, and the atmospheric pressure. Use \( P_2 = P_{\text{atm}} + \frac{(m_{\text{piston}} + m_{\text{sand}}) g}{A} \).

Solve for the final height \( h_2 \) of the gas column. Rearrange the ideal gas law to \( h_2 = \frac{P_1 h_1}{P_2} \), substituting the known values for \( P_1 \), \( h_1 \), and \( P_2 \). This gives the final height of the piston.

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Key Concepts

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

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the force of gravity. It increases with depth in a fluid and is calculated using the formula P = ρgh, where P is the pressure, ρ is the fluid density, g is the acceleration due to gravity, and h is the height of the fluid column. In this scenario, the pressure exerted by the sand on the piston must be balanced by the pressure from the water and the gas in the cylinder.

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. In this case, the gas in the cylinder will respond to changes in pressure and volume as the piston moves. Understanding this relationship is crucial for determining how the height of the piston changes when additional weight is added.

Equilibrium of Forces

In this scenario, the system reaches equilibrium when the forces acting on the piston are balanced. The weight of the sand and the piston must equal the upward force exerted by the gas pressure in the cylinder. Analyzing the equilibrium of forces allows us to calculate the new height of the piston after the sand is added, ensuring that all forces are accounted for.

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The 50 kg circular piston shown in FIGURE P18.57 floats on 0.12 mol of compressed air. How far does the piston move if the temperature is increased by 100°C?

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The 50 kg circular piston shown in FIGURE P18.57 floats on 0.12 mol of compressed air. What is the piston height h if the temperature is 30°C?

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The interior of a Boeing 737-800 can be modeled as a 32-m-long, 3.7-m-diameter cylinder. The air inside, at cruising altitude, is 20°C at a pressure of 82 kPa. What volume of outside air, at −40°C and a pressure of 23 kPa, must be drawn in, heated, and compressed to fill the plane?

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