Ch 20: The Micro/Macro Connection

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 20: The Micro/Macro Connection

Problem 55

Chapter 20, Problem 55

5.0 x 10²³ nitrogen molecules collide with a 10 cm² wall each second. Assume that the molecules all travel with a speed of 400 m/s and strike the wall head-on. What is the pressure on the wall?

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Convert the given area of the wall from cm² to m². Since 1 cm² = 1 × 10⁻⁴ m², multiply 10 cm² by 1 × 10⁻⁴ to get the area in m².

Calculate the momentum change for a single nitrogen molecule during a collision. The momentum of a molecule is given by p = mv, where m is the mass of the molecule and v is its velocity. Since the molecule rebounds with the same speed but in the opposite direction, the change in momentum is Δp = 2mv.

Determine the total momentum change per second. Multiply the momentum change for a single molecule (Δp) by the number of molecules colliding with the wall per second (5.0 × 10²³).

Use the relationship between force and momentum change: F = Δp/Δt. Here, Δt is 1 second, so the total force exerted on the wall is equal to the total momentum change per second.

Calculate the pressure on the wall using the formula P = F/A, where F is the total force exerted on the wall and A is the area of the wall in m². Substitute the values to find the pressure.

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Key Concepts

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

Pressure

Pressure is defined as the force exerted per unit area. In the context of gas molecules colliding with a surface, it can be calculated using the formula P = F/A, where P is pressure, F is the total force from the collisions, and A is the area of the surface. Understanding pressure is crucial for analyzing how gas molecules interact with surfaces.

Momentum Transfer

When gas molecules collide with a wall, they transfer momentum to the wall, which contributes to the force exerted on it. The change in momentum during a collision can be calculated as Δp = mv, where m is the mass of the molecule and v is its velocity. This concept is essential for determining the total force from multiple collisions over time.

Ideal Gas Behavior

Ideal gas behavior assumes that gas molecules are point particles that do not interact except during elastic collisions. This simplification allows for the use of kinetic theory to relate molecular motion to macroscopic properties like pressure and temperature. Understanding this behavior is important for applying the principles of gas dynamics in the given problem.

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Textbook Question

On earth, STP is based on the average atmospheric pressure at the surface and on a phase change of water that occurs at an easily produced temperature, being only slightly cooler than the average air temperature. The atmosphere of Venus is almost entirely carbon dioxide (CO₂), the pressure at the surface is a staggering 93 atm, and the average temperature is 470℃. Venusian scientists, if they existed, would certainly use the surface pressure as part of their definition of STP. To complete the definition, they would seek a phase change that occurs near the average temperature. Conveniently, the melting point of the element tellurium is 450℃. What are (a) the rms speed and (b) the mean free path of carbon dioxide molecules at Venusian STP based on this phase change in tellurium? The radius of a CO₂ molecule is 1.5 x 10^-10 m.

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Textbook Question

Uranium has two naturally occurring isotopes. $to the 238th power cap U$ has a natural abundance of $99.3 percent sign$ and $to the 235th power cap U$ has an abundance of $0.7 percent sign$ . It is the rarer $to the 235th power cap U$ that is needed for nuclear reactors. The isotopes are separated by forming uranium hexafluoride, $UF sub 6$ , which is a gas, then allowing it to diffuse through a series of porous membranes. $to the 235th power U F sub 6$ has a slightly larger rms speed than $to the 238th power U F sub 6$ and diffuses slightly faster. Many repetitions of this procedure gradually separate the two isotopes. What is the ratio of the rms speed of $to the 235th power U F sub 6$ to that of $to the 238th power U F sub 6$ ?

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Textbook Question

A 100 cm³ box contains helium at a pressure of 2.0 atm and a temperature of 100℃. It is placed in thermal contact with a 200 cm³ box containing argon at a pressure of 4.0 atm and a temperature of 400℃. What is the final thermal energy of each gas?

Equation 20.3 is the mean free path of a particle through a gas of identical particles of equal radius. An electron can be thought of as a point particle with zero radius. Electrons travel 3.0 km through the Stanford Linear Accelerator. In order for scattering losses to be negligible, the pressure inside the accelerator tube must be reduced to the point where the mean free path is at least 50 km. What is the maximum possible pressure inside the accelerator tube, assuming T = 20℃? Give your answer in both P_a and atm.

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Textbook Question

FIGURE P20.57 shows the thermal energy of 0.14 mol of gas as a function of temperature. What is Cᵥ for this gas?

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5.0 x 1023 nitrogen molecules collide with a 10 cm2 wall each second. Assume that the molecules all travel with a speed of 400 m/s and strike the wall head-on. What is the pressure on the wall?

Verified video answer for a similar problem:

Key Concepts

Pressure

Momentum Transfer

Ideal Gas Behavior

5.0 x 10²³ nitrogen molecules collide with a 10 cm² wall each second. Assume that the molecules all travel with a speed of 400 m/s and strike the wall head-on. What is the pressure on the wall?