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A cylinder of nitrogen gas has a volume of 15,000cm3and a pressure of100 atm.
a. What is the thermal energy of this gas at room temperature(20C)?
b. What is the mean free path in the gas?
c. The valve is opened and the gas is allowed to expand slowly and isothermally until it reaches a pressure of 1.0atm . What is the change in the thermal energy of the gas?

Short Answer

Expert verified

a)The Thermal energy of gas isE=379KJ.
b)The Mean free path in tha gas isλ=2.25nm.
c)The change in thermal energy of the gas is Zero.

Step by step solution

01

Step: 1   Finding the Thermal Energy of gas at  room temperature(20∘C): (part a)

Because nitrogen is a diatomic gas, the thermal energy is calculated as follows:

E=52NKBT

Now we know that , pV=NKBT

From equation localid="1648381889785" 1and2we get that

E=52pV

Given thatp=100atm=1.01×107N/m2and V=15000cm3=0.015m3,, then substituting these to the formula:

localid="1648390595486" E=52×1.01×107×0.015=379kJ.

02

Step: 2  Finding the Mean free path: (part b)

The mean free path is given by

λ=142π(N/V)r2

Now we have

NV=pKBT=1.01×1071.38×1023×293=2.5×1027

and rfor nitrogen is r=1×1010,then the mean free path will be:

λ=142π×1027×1×10102=2.25×109m=2.25nm.

03

Step: 3  Change in Thermal Energy of gas: (part c)

But even though the formula shows that thermal energy is proportional to the number of gas molecules, degrees of freedom, and gas temperature, none of these factors change in this isothermal expansion.

As a result, the total amount of thermal energy will remain unchanged. Thermal energy will remain unchanged.

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Most popular questions from this chapter

A monatomic gas is adiabatically compressed to 18of its initial volume. Does each of the following quantities change? If so, does it increase or decrease, and by what factor? If not, why not?

a. The rmsspeed.

b. The mean free path.

c. The thermal energy of the gas.

d. The molar specific heat at constant volume.

At what temperature does thermsspeed of (a)a nitrogen molecule and (b)a hydrogen molecule equal the escape speed from the earth's surface? (c)You'll find that these temperatures are very high, so you might think that the earth's gravity could easily contain both gases. But not all molecules move withVrms. There is a distribution of speeds, and a small percentage of molecules have speeds several times Vrms . Bit by bit, a gas can slowly leak out of the atmosphere as its fastest molecules escape. A reasonable rule of thumb is that the earth's gravity can contain a gas only if the average translational kinetic energy per molecule is less than 1%of the kinetic energy needed to escape. Use this rule to show why the earth's atmosphere contains nitrogen but not hydrogen, even though hydrogen is the most abundant element in the universe.

6. Suppose you could suddenly increase the speed of every molecule in a gas by a factor of 2.

a. Would the RMS speed of the molecules increase by a factor of 21/2,2,or22? Explain.

b. Would the gas pressure increase by a factor of 21/2,2or 22? Explain.

A 6.0m×8.0m×3.0m room contains air at 20OC. What is the room's thermal energy?

2.0mol of monatomic gas Ainitially has role="math" localid="1648440865095" 5000Jof thermal energy. It interacts with 3.0mol of monatomic gas B, which initially has 8000J of thermal energy.
a. Which gas has the higher initial temperature?
b. What is the final thermal energy of each gas?

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