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Chapter 20: Q. 43 (page 568)

You are watching a science fiction movie in which the hero shrinks down to the size of an atom and fights villains while jumping from air molecule to air molecule. In one scene, the hero's molecule is about to crash head-on into the molecule on which a villain is riding. The villain's molecule is initially $50$ molecular radii away and, in the movie, it takes $3.5 s$ for the molecules to collide. Estimate the air temperature required for this to be possible. Assume the molecules are nitrogen molecules, each traveling at the rms speed. Is this a plausible temperature for air?

Short Answer

Expert verified

Temperature for the air is $5.8 脳 10^{- 22} K$ ,unrealistically low.

Step by step solution

01

Formula for root mean square

Root mean square with distance be $d$ covered in time $t$ ,

$v_{rms} = 2 \frac{d}{t} . . . . . . . 1$

We may calculate the rms speed using the formula,

$v_{蟿尘蝉} = \sqrt{\frac{3 k_{B} T}{m}} . . . . . . . . . . . . . . . . 2$

02

Calculation for temperature

where $m$ is the mass of one particle.

Equating $1$ and $2$ equation,

localid="1648641658334" $2 \frac{d}{t} = \sqrt{\frac{3 k_{B} T}{m}}$

localid="1648641663890" $\frac{3 k_{B} T}{m} = \frac{4 d^{2}}{t^{2}} 鈬� T = \frac{4 {md}^{2}}{3 k_{B} t^{2}}$

Substituting localid="1648642043882" $m = \frac{M}{N_{A}}$ andlocalid="1648641689763" $k_{B} N_{A} = R$ we get,

localid="1648641709109" $T = \frac{4 {Md}^{2}}{3 k_{B} N_{A} t^{2}} = \frac{4 {Md}^{2}}{3 {Rt}^{2}}$

localid="1648641723376" $T = \frac{4 (50 r)^{2} M}{3 {Rt}^{2}}$

localid="1648641744912" $T = \frac{4 路 {(5 路 10^{- 9})}^{2} 路 0.028}{3 路 8.314 路 3 . 5^{2}} = 5.8 路 10^{- 22} K$ ,unrealistically low

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Most popular questions from this chapter

A monatomic gas and a diatomic gas have equal numbers of moles and equal temperatures. Both are heated at constant pressure until their volume doubles. What is the ratio $Q_{d i a t o m i c} / Q_{m o n a t o m i c}$ ?

A $10 c m 脳 10 c m 脳 10 c m$ box contains $0.010 m o l$ of nitrogen at $20 掳 C$ . What is the rate of collisions (collisions/s) on one wall of the box?

An experiment you're designing needs a gas with $纬 = 1.50$ . You recall from your physics class that no individual gas has this value, but it occurs to you that you could produce a gas with $纬 = 1.50$ by mixing together a monatomic gas and a diatomic gas. What fraction of the molecules need to be monatomic?

Consider a container like that shown in Figure $20.12$ , with $n_{1}$ moles of a monatomic gas on one side and $n_{2}$ moles of a diatomic gas on the other. The monatomic gas has initial temperature $T_{1 i}$ . The diatomic gas has initial temperature $T_{2 i}$ .
a. Show that the equilibrium thermal energies are

$\begin{aligned} E_{1 f} = \frac{3 n_{1}}{3 n_{1} + 5 n_{2}} (E_{1 i} + E_{2 i}) \\ E_{2 f} = \frac{5 n_{2}}{3 n_{1} + 5 n_{2}} (E_{1 i} + E_{2 i}) \end{aligned}$

b. Show that the equilibrium temperature is

$T_{f} = \frac{3 n_{1} T_{1 i} + 5 n_{2} T_{2 i}}{3 n_{1} + 5 n_{2}}$

c. $2.0 g$ of helium at an initial temperature of role="math" localid="1648474536876" $300 K$ interacts thermally with $8.0 g$ of oxygen at an initial temperature of $600 K$ . What is the final temperature? How much heat energy is transferred, and in which direction?

The rms speed of the molecules in $1.0 g$ of hydrogen gas is $1800 \frac{m}{s}$ .
a. What is the total translational kinetic energy of the gas molecules?
b. What is the thermal energy of the gas?
c. $500 J$ of work are done to compress the gas while, in the same process, $1200 J$ of heat energy are transferred from the gas to the environment. Afterward, what in the rms speed of the molecules?

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