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Chapter 1: Q.1.23 (page 17)

Calculate the total thermal energy in a liter of helium at room temperature and atmospheric pressure. Then repeat the calculation for a liter of air.

Short Answer

Expert verified

The total thermal energy in a liter is for heliumU=151.987J and for airU=253.312J.

Step by step solution

01

Step 1. Given data.

A liter of helium at ambient temperature at the pressure of one atmosphere1atm=101325Pa.

02

Step 2. Explanation.

Whereas a monatomic gas, like helium, has only three translational degrees of freedom, it is the simplest instance.

U=f2NkT=32NkT

Thermal energy,

U=32NkT=32PV=32×101325×1×10-3=151.987J

03

Step 3. Total thermal energy.

A rotation around the bond's axis isn't counted because it doesn't change the molecule's position.

f=5

U=f2NkT=52NkTU=52PV=52×101325×1×10-3=253.312.J

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Most popular questions from this chapter

The specific heat capacity of Albertson's Rotini Tricolore is approximately 1.8 J/g oC . Suppose you toss 340 g of this pasta (at 25oC ) into 1.5 liters of boiling water. What effect does this have on the temperature of the water (before there is time for the stove to provide more heat)?

Estimate how long it should take to bring a cup of water to boiling temperature in a typical 600-wattmicrowave oven, assuming that all the energy ends up in the water. (Assume any reasonable initial temperature for the water.) Explain why no heat is involved in this process.

Suppose you have a gas containing hydrogen molecules and oxygen molecules, in thermal equilibrium. Which molecule are moving faster, on average? By what factor?

Make a rough estimate of the total rate of conductive heat loss through the windows, walls, floor, and roof of a typical house in a cold climate. Then estimate the cost of replacing this lost energy over a month. If possible, compare your estimate to a real utility bill. (Utility companies measure electricity by the kilowatt-hour, a unit equal to MJ. In the United States, natural gas is billed in terms, where 1 therm = 105 Btu. Utility rates vary by region; I currently pay about 7 cents per kilowatt-hour for electricity and 50 cents per therm for natural gas.)

By applying Newton’s laws to the oscillations of a continuous medium, one can show that the speed of a sound wave is given by

cs=Bρ,

where ρis the density of the medium (mass per unit volume) and B is the bulk modulus, a measure of the medium’s stiffness? More precisely, if we imagine applying an increase in pressure ΔPto a chunk of the material, and this increase results in a (negative) change in volume ΔV, then B is defined as the change in pressure divided by the magnitude of the fractional change in volume:

B=ΔP–ΔV/V

This definition is still ambiguous, however, because I haven't said whether the compression is to take place isothermally or adiabatically (or in some other way).

  1. Compute the bulk modulus of an ideal gas, in terms of its pressure P, for both isothermal and adiabatic compressions.
  2. Argue that for purposes of computing the speed of a sound wave, the adiabatic B is the one we should use.
  3. Derive an expression for the speed of sound in an ideal gas, in terms of its temperature and average molecular mass. Compare your result to the formula for the RMS speed of the molecules in the gas. Evaluate the speed of sound numerically for air at room temperature.
  4. When Scotland’s Battlefield Band played in Utah, one musician remarked that the high altitude threw their bagpipes out of tune. Would you expect altitude to affect the speed of sound (and hence the frequencies of the standing waves in the pipes)? If so, in which direction? If not, why not?
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