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Chapter 1: Q. 1.19 (page 13)

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

Short Answer

Expert verified

Hydrogen molecule will be moving faster

Step by step solution

01

Given information

At room temp (298 K ) Hydrogen and Oxygen molecule is given

02

Explanation

Root mean square velocity of molecule is given by

v=3kTm.....................(1)

Use this to find the rms of Oxygen and and Hydrogen, And then get their ratio

As both molecules hydrogen and oxygen are in equilibrium so they both have the same temperature 298 K. and Boltzmann constant is same .

vHvo=3kTHmH3kTOmOvHvo=mOmH.......................................(2)

Substitute mO=32 and mH=2 in equation (2) and calculate ratio

vHvO=322vHvo=4vH=4vO

So velocity of Hydrogen is four times more than that of oxygen molecules.

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Most popular questions from this chapter

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?

Measured heat capacities of solids and liquids are almost always at constant pressure, not constant volume. To see why, estimate the pressure needed to keep Vfixed as Tincreases, as follows.

(a) First imagine slightly increasing the temperature of a material at constant pressure. Write the change in volume,dV1, in terms of dTand the thermal expansion coefficient βintroduced in Problem 1.7.

(b) Now imagine slightly compressing the material, holding its temperature fixed. Write the change in volume for this process, dV2, in terms of dPand the isothermal compressibility κT, defined as

κT≡−1V∂V∂PT

(c) Finally, imagine that you compress the material just enough in part (b) to offset the expansion in part (a). Then the ratio of dPtodTis equal to (∂P/∂T)V, since there is no net change in volume. Express this partial derivative in terms of βandκT. Then express it more abstractly in terms of the partial derivatives used to define βandκT. For the second expression you should obtain

∂P∂TV=−(∂V/∂T)P(∂V/∂P)T

This result is actually a purely mathematical relation, true for any three quantities that are related in such a way that any two determine the third.

(d) Compute β,κT,and(∂P/∂T)Vfor an ideal gas, and check that the three expressions satisfy the identity you found in part (c).

(e) For water at 25∘C,β=2.57×10−4K−1andκT=4.52×10−10Pa−1. Suppose you increase the temperature of some water from 20∘Cto30∘C. How much pressure must you apply to prevent it from expanding? Repeat the calculation for mercury, for which (at25∘C)β=1.81×10−4K−1andκT=4.04×10−11Pa−1

Given the choice, would you rather measure the heat capacities of these substances at constant vor at constant p?

Does it ever make sense to say that one object is "twice as hot" as another? Does it matter whether one is referring to Celsius or Kelvin temperatures? Explain.

Suppose you open a bottle of perfume at one end of a room. Very roughly, how much time would pass before a person at the other end of the room could smell the perfume, if diffusion were the only transport mechanism? Do you think diffusion is the dominant transport mechanism in this situation?

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.
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