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Physics · Textbook solutions

Kinetic Theory

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 22 questions

Worked Examples

9 q

Solved Examples

Worked · 9
  1. Eg 12.1
    The density of water is 1000 kg m−31000\,\text{kg m}^{-3}. The density of water vapour at 100∘C100^\circ\text{C} and 1 atm1\,\text{atm} pressure is 0.6 kg m−30.6\,\text{kg m}^{-3}. The volume of a molecule multiplied by the total number gives, what is called, molecular volume. Estimate the ratio (or fraction) of the molecular volume to the total volume occupied by the water vapour under the above conditions of temperature and pressure.
  2. Eg 12.2
    Estimate the volume of a water molecule using the data in Example 12.1.
  3. Eg 12.3
    What is the average distance between atoms (interatomic distance) in water? Use the data given in Examples 12.1 and 12.2.
  4. Eg 12.4
    A vessel contains two non-reactive gases : neon (monatomic) and oxygen (diatomic). The ratio of their partial pressures is 3:23:2. Estimate the ratio of (i) number of molecules and (ii) mass density of neon and oxygen in the vessel. Atomic mass of Ne =20.2 u= 20.2\,\text{u}, molecular mass of O2=32.0 u\text{O}_2 = 32.0\,\text{u}.
  5. Eg 12.5
    A flask contains argon and chlorine in the ratio of 2:12:1 by mass. The temperature of the mixture is 27∘C27^\circ\text{C}. Obtain the ratio of (i) average kinetic energy per molecule, and (ii) root mean square speed vrmsv_{\text{rms}} of the molecules of the two gases. Atomic mass of argon =39.9 u= 39.9\,\text{u}; Molecular mass of chlorine =70.9 u= 70.9\,\text{u}.
  6. Eg 12.6
    Uranium has two isotopes of masses 235 and 238 units. If both are present in Uranium hexafluoride gas which would have the larger average speed ? If atomic mass of fluorine is 19 units, estimate the percentage difference in speeds at any temperature.
  7. Eg 12.7
    (a) When a molecule (or an elastic ball) hits a (massive) wall, it rebounds with the same speed. When a ball hits a massive bat held firmly, the same thing happens. However, when the bat is moving towards the ball, the ball rebounds with a different speed. Does the ball move faster or slower? (Ch.5 will refresh your memory on elastic collisions.) (b) When gas in a cylinder is compressed by pushing in a piston, its temperature rises. Guess at an explanation of this in terms of kinetic theory using (a) above. (c) What happens when a compressed gas pushes a piston out and expands. What would you observe ? (d) Sachin Tendulkar used a heavy cricket bat while playing. Did it help him in anyway ?
  8. Eg 12.8
    A cylinder of fixed capacity 44.844.8 litres contains helium gas at standard temperature and pressure. What is the amount of heat needed to raise the temperature of the gas in the cylinder by 15.0∘C15.0^\circ\text{C} ? (R=8.31 J mol−1 K−1)(R = 8.31\,\text{J mol}^{-1}\,\text{K}^{-1}).
  9. Eg 12.9
    Estimate the mean free path for a water molecule in water vapour at 373 K373\,\text{K}. Use information from Exercises 12.1 and Eq. (12.41) above.

Exercises

13 q
  1. Ex 12.1
    Estimate the fraction of molecular volume to the actual volume occupied by oxygen gas at STP. Take the diameter of an oxygen molecule to be 3 A˚3\,\mathring{A}.
  2. Ex 12.2
    Molar volume is the volume occupied by 1 mol of any (ideal) gas at standard temperature and pressure (STP : 1 atmospheric pressure, 0∘C0^\circ\text{C}). Show that it is 22.422.4 litres.
  3. Ex 12.3(a)
    Figure 12.8 shows plot of PV/TPV/T versus PP for 1.00×10−3 kg1.00 \times 10^{-3}\,\text{kg} of oxygen gas at two different temperatures. (a) What does the dotted plot signify?
  4. Ex 12.3(b)
    Figure 12.8 shows plot of PV/TPV/T versus PP for 1.00×10−3 kg1.00 \times 10^{-3}\,\text{kg} of oxygen gas at two different temperatures. (b) Which is true: T1>T2T_1 > T_2 or T1<T2T_1 < T_2?
  5. Ex 12.3(c)
    Figure 12.8 shows plot of PV/TPV/T versus PP for 1.00×10−3 kg1.00 \times 10^{-3}\,\text{kg} of oxygen gas at two different temperatures. (c) What is the value of PV/TPV/T where the curves meet on the yy-axis?
  6. Ex 12.3(d)
    Figure 12.8 shows plot of PV/TPV/T versus PP for 1.00×10−3 kg1.00 \times 10^{-3}\,\text{kg} of oxygen gas at two different temperatures. (d) If we obtained similar plots for 1.00×10−3 kg1.00 \times 10^{-3}\,\text{kg} of hydrogen, would we get the same value of PV/TPV/T at the point where the curves meet on the yy-axis? If not, what mass of hydrogen yields the same value of PV/TPV/T (for low pressure high temperature region of the plot) ? (Molecular mass of H2=2.02 u\text{H}_2 = 2.02\,\text{u}, of O2=32.0 u\text{O}_2 = 32.0\,\text{u}, R=8.31 J mol−1 K−1R = 8.31\,\text{J mol}^{-1}\,\text{K}^{-1}.)
  7. Ex 12.4
    An oxygen cylinder of volume 30 litre has an initial gauge pressure of 15 atm15\,\text{atm} and a temperature of 27∘C27^\circ\text{C}. After some oxygen is withdrawn from the cylinder, the gauge pressure drops to 11 atm11\,\text{atm} and its temperature drops to 17∘C17^\circ\text{C}. Estimate the mass of oxygen taken out of the cylinder (R=8.31 J mol−1 K−1(R = 8.31\,\text{J mol}^{-1}\,\text{K}^{-1}, molecular mass of O2=32 u)\text{O}_2 = 32\,\text{u}).
  8. Ex 12.5
    An air bubble of volume 1.0 cm31.0\,\text{cm}^3 rises from the bottom of a lake 40 m40\,\text{m} deep at a temperature of 12∘C12^\circ\text{C}. To what volume does it grow when it reaches the surface, which is at a temperature of 35∘C35^\circ\text{C} ?
  9. Ex 12.6
    Estimate the total number of air molecules (inclusive of oxygen, nitrogen, water vapour and other constituents) in a room of capacity 25.0 m325.0\,\text{m}^3 at a temperature of 27∘C27^\circ\text{C} and 1 atm1\,\text{atm} pressure.
  10. Ex 12.7
    Estimate the average thermal energy of a helium atom at (i) room temperature (27∘C)(27^\circ\text{C}), (ii) the temperature on the surface of the Sun (6000 K)(6000\,\text{K}), (iii) the temperature of 10 million kelvin (the typical core temperature in the case of a star).
  11. Ex 12.8
    Three vessels of equal capacity have gases at the same temperature and pressure. The first vessel contains neon (monatomic), the second contains chlorine (diatomic), and the third contains uranium hexafluoride (polyatomic). Do the vessels contain equal number of respective molecules ? Is the root mean square speed of molecules the same in the three cases? If not, in which case is vrmsv_{\text{rms}} the largest ?
  12. Ex 12.9
    At what temperature is the root mean square speed of an atom in an argon gas cylinder equal to the rms speed of a helium gas atom at −20∘C-20^\circ\text{C} ? (atomic mass of Ar =39.9 u= 39.9\,\text{u}, of He =4.0 u= 4.0\,\text{u}).
  13. Ex 12.10
    Estimate the mean free path and collision frequency of a nitrogen molecule in a cylinder containing nitrogen at 2.0 atm2.0\,\text{atm} and temperature 17∘C17^\circ\text{C}. Take the radius of a nitrogen molecule to be roughly 1.0 A˚1.0\,\mathring{A}. Compare the collision time with the time the molecule moves freely between two successive collisions (Molecular mass of N2=28.0 u\text{N}_2 = 28.0\,\text{u}).