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Ch 20: The Micro/Macro Connection
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Knight Calc 5th EditionCh 20: The Micro/Macro Connection문제 13
20장, 문제 13

A cylinder contains gas at a pressure of 2.0 atm and a number density of 4.2 x 1025 m-3. The rms speed of the atoms is 660 m/s. Identify the gas.

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1
Step 1: Use the ideal gas law in terms of number density to relate pressure, number density, and temperature. The formula is: P=nkT, where P is pressure, n is number density, k is Boltzmann's constant (1.38 × 10⁻²³ J/K), and T is temperature.
Step 2: Rearrange the formula to solve for temperature: T=Pnk. Substitute the given values for pressure (2.0 atm, converted to Pascals as 2.0 × 1.013 × 10⁵ Pa), number density (4.2 × 10²⁵ m⁻³), and Boltzmann's constant.
Step 3: Use the formula for the root mean square (rms) speed of gas molecules: vrms=3kTm, where m is the mass of one molecule of the gas. Rearrange the formula to solve for m: m=3kTvrms2. Substitute the values for T and vrms.
Step 4: Convert the molecular mass to molar mass using Avogadro's number (6.022 × 10²³ mol⁻¹). The molar mass is given by: M=mNA, where NA is Avogadro's number.
Step 5: Compare the calculated molar mass to known molar masses of common gases to identify the gas. For example, if the molar mass is approximately 4 g/mol, the gas is likely helium; if it is approximately 28 g/mol, the gas is likely nitrogen.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT. This law is fundamental in understanding the behavior of gases under various conditions and can help identify the type of gas based on its properties.
추천 영상:
가이드 코스
07:21
Ideal Gases and the Ideal Gas Law

Root Mean Square Speed (rms speed)

The root mean square speed is a measure of the average speed of particles in a gas and is given by the formula v_rms = √(3kT/m), where k is the Boltzmann constant, T is the temperature, and m is the mass of a gas particle. This concept is crucial for determining the molecular mass of the gas when combined with other properties.
추천 영상:
가이드 코스
05:21
Root-Mean-Square Speed of Ideal Gases

Number Density

Number density refers to the number of particles per unit volume, typically expressed in particles per cubic meter (m⁻³). It is an important parameter in gas physics, as it helps relate the macroscopic properties of the gas, such as pressure and temperature, to the microscopic behavior of its particles.
추천 영상:
가이드 코스
8:13
Intro to Density
관련 실천
교과서 질문

At 100℃ the rms speed of nitrogen molecules is 576 m/s. Nitrogen at 100℃ and a pressure of 2.0 atm is held in a container with a 10 cm x 10 cm square wall. Estimate the rate of molecular collisions (collisions/s) on this wall.

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교과서 질문

By what factor does the rms speed of a molecule change if the temperature is increased from 10℃ to 1000℃?

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교과서 질문

The molecules in a six-particle gas have velocities:

v⃗1=(20i^−30j^) m/sv⃗2=(40i^+70j^) m/sv⃗3=(−80i^+20j^) m/sv⃗4=30i^ m/sv⃗5=(40i^−40j^) m/sv⃗6=(−50i^−20j^) m/s \(\begin{aligned}\) \(\vec{v}\)_1 &= (20\(\hat{i}\) - 30\(\hat{j}\)) \(\text{ m/s}\) \\ \(\vec{v}\)_2 &= (40\(\hat{i}\) + 70\(\hat{j}\)) \(\text{ m/s}\) \\ \(\vec{v}\)_3 &= (-80\(\hat{i}\) + 20\(\hat{j}\)) \(\text{ m/s}\) \\ \(\vec{v}\)_4 &= 30\(\hat{i}\) \(\text{ m/s}\) \\ \(\vec{v}\)_5 &= (40\(\hat{i}\) - 40\(\hat{j}\)) \(\text{ m/s}\) \\ \(\vec{v}\)_6 &= (-50\(\hat{i}\) - 20\(\hat{j}\)) \(\text{ m/s}\) \(\end{aligned}\)

Calculate (a) v⃗avg\(\vec{v}\)_{\(\text{avg}\)}, (b) vavgv_{\(\text{avg}\)}, and (c) vrmsv_{\(\text{rms}\)}.

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1.0 mol of argon has 3100 J of thermal energy. What is the gas temperature in °C?

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Eleven molecules have speeds 15, 16, 17, …, 25 m/s. Calculate (a) vavg and (b) vrms.

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The rms speed of molecules in a gas is 600 m/s. What will be the rms speed if the gas pressure and volume are both halved?

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