Simple Nature - Light and Matter

ProblemsThe symbols √ , , etc. are explained on page 337.1 (a) Show that under conditions of standard pressure and temperature,the volume of a sample of an ideal gas depends only onthe number of molecules in it.(b) One mole is defined as 6.0 × 10 23 atoms. Find the volume of onemole of an ideal gas, in units of liters, at standard temperature andpressure (0 ◦ √C and 101 kPa).2 A gas in a cylinder expands its volume by an amount dV ,pushing out a piston. Show that the work done by the gas on thepiston is given by dW = P dV .3 (a) A helium atom contains 2 protons, 2 electrons, and 2√neutrons. Find the mass of a helium atom.√(b) Find the number of atoms in 1.0 kg of helium.(c) Helium gas is monoatomic. Find the amount of heat needed toraise the temperature of 1.0 kg of helium by 1.0 degree C. (This is√known as helium’s heat capacity at constant volume.)4 A sample of gas is enclosed in a sealed chamber. The gasconsists of molecules, which are then split in half through someprocess such as exposure to ultraviolet light, or passing an electricspark through the gas. The gas returns to thermal equilibrium withthe surrounding room. How does its pressure now compare with itspressure before the molecules were split?5 Most of the atoms in the universe are in the form of gas thatis not part of any star or galaxy: the intergalactic medium (IGM).The IGM consists of about 10 −5 atoms per cubic centimeter, witha typical temperature of about 10 3 K. These are, in some sense, thedensity and temperature of the universe (not counting light, or theexotic particles known as “dark matter”). Calculate the pressure ofthe universe (or, speaking more carefully, the typical pressure due√to the IGM).6 Estimate the pressure at the center of the Earth, assuming itis of constant density throughout. Note that g is not constant withrespect to depth — as shown in example 19 on page 105, g equalsGmr/b 3 for r, the distance from the center, less than b, the earth’sradius.√(a) State your result in terms of G, m, and b.(b) Show that your answer from part a has the right units for pressure.√(c) Evaluate the result numerically.(d) Given that the earth’s atmosphere is on the order of one thousandththe earth’s radius, and that the density of the earth is severalthousand times greater than the density of the lower atmosphere,check that your result is of a reasonable order of magnitude.Problems 335