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An Introduction to Thermal Physics

Daniel V. Schroeder

Physics

1st Edition · 356 pages

ISBN: 9780201380279

Chapter 1: Q. 1.18 (page 13)

Calculate the rms speed of a nitrogen molecule at room temperature.

Short Answer

Expert verified

Root mean square speed is 515.9m/s.

Step by step solution

01

Given information

Gas is Nitrogen.

Temp = room temp = 25^oC= 298 K

02

Explanation

The root mean square velocity is calculated as

$v = \sqrt{\frac{3 k T}{m}} . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)$

where

T = temperature

k = Boltzmann constant

m = mass

Mass of Nitrogen is calculated as

$m = 28 u = 28 \times 1.6 \times 10^{- 27} kg = 4.65 \times 10^{- 26} kg$

Boltzmann constant is $= 1.38 \times 10^{- 23} m^{2} {kgs}^{- 2} K^{- 1}$

Substitute the values in equation (1), we get

$v = \sqrt{\frac{3 \times (1.38 \times 10^{- 23} m^{2} k g s^{- 2} K^{- 1}) \times (298 K)}{4.65 \times 10^{- 26} k g}} v = 515.9 {ms}^{- 1}$

Root mean square speed is 515.9m/s.

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Most popular questions from this chapter

In Problem 1.16 you calculated the pressure of the earth’s atmosphere as a function of altitude, assuming constant temperature. Ordinarily, however, the temperature of the bottommost 10-15 km of the atmosphere (called the troposphere) decreases with increasing altitude, due to heating from the ground (which is warmed by sunlight). If the temperature gradient $| d T / d z |$ exceeds a certain critical value, convection will occur: Warm, low-density air will rise, while cool, high-density air sinks. The decrease of pressure with altitude causes a rising air mass to expand adiabatically and thus to cool. The condition for convection to occur is that the rising air mass must remain warmer than the surrounding air despite this adiabatic cooling.

a. Show that when an ideal gas expands adiabatically, the temperature and pressure are related by the differential equation

$\frac{d T}{d P} = \frac{2}{f + 2} \frac{T}{P}$

b. Assume that $d T / d z$ is just at the critical value for convection to begin so that the vertical forces on a convecting air mass are always approximately in balance. Use the result of Problem 1.16(b) to find a formula for $d T / d z$ in this case. The result should be a constant, independent of temperature and pressure, which evaluates to approximately $- 10 ° C / k m$ . This fundamental meteorological quantity is known as the dry adiabatic lapse rate.

If you poke a hole in a container full of gas, the gas will start leaking out. In this problem, you will make a rough estimate of the rate at which gas escapes through a hole. (This process is called effusion, at least when the hole is sufficiently small.)

Consider a small portion (area = $A$ ) of the inside wall of a container full of gas. Show that the number of molecules colliding with this surface in a time interval $Δ t$ is role="math" localid="1651729685802" $P A Δ t / (2 m \bar{v_{x}})$ , where width="12" height="19" role="math">Pis the pressure, is the average molecular mass, and $v_{x}$ is the average $x$ velocity of those molecules that collide with the wall.
It's not easy to calculate $v_{x}$ , but a good enough approximation is ${(\bar{v_{x}^{2}})}^{1 / 2}$ , where the bar now represents an average overall molecule in the gas. Show that ${(\bar{v_{x}^{2}})}^{1 / 2} = \sqrt{k T / m}$ .
If we now take away this small part of the wall of the container, the molecules that would have collided with it will instead escape through the hole. Assuming that nothing enters through the hole, show that the number $N$ of molecules inside the container as a function of time is governed by the differential equation
$\frac{d N}{d t} = - \frac{A}{2 V} \sqrt{\frac{k T}{m} N}$
Solve this equation (assuming constant temperature) to obtain a formula of the form $N (t) = N (0) e^{- t / r}$ , where $r$ is the “characteristic time” for $N$ (and $P$ ) to drop by a factor of $e$ .
Calculate the characteristic time for gas to escape from a 1-liter container punctured by a $1 - {mm}^{2}$ ? hole.
Your bicycle tire has a slow leak so that it goes flat within about an hour after being inflated. Roughly how big is the hole? (Use any reasonable estimate for the volume of the tire.)
In Jules Verne’s Around the Moon, the space travelers dispose of a dog's corpse by quickly opening a window, tossing it out, and closing the window. Do you think they can do this quickly enough to prevent a significant amount of air from escaping? Justify your answer with some rough estimates and calculations.

In analogy with the thermal conductivity, derive an approximate formula for the viscosity of an ideal gas in terms of its density, mean free path, and average thermal speed. Show explicitly that the viscosity is independent of pressure and proportional to the square root of the temperature. Evaluate your formula numerically for air at room temperature and compare to the experimental value quoted in the text.

In analogy with the thermal conductivity, derive an approximate formula for the diffusion coefficient of an ideal gas in terms of the mean free path and the average thermal speed. Evaluate your formula numerically for air at room temperature and atmospheric pressure, and compare to the experimental value quoted in the text. How does D depend on T, at fixed pressure?

During a hailstorm, hailstones with an average mass of 2g and a speed of 15 m/s strike a window pane at a 45^o angle. The area of the window is 0.5 m² and the hailstones hit it at a rate of 30 per second. What average pressure do they exert on the window? How does this compare to the pressure of the atmosphere?

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