Physics · Textbook solutions

Kinetic Theory of Gases and Radiation

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 42 questions

3. Kinetic Theory of Gases and Radiation — worked examples

9 q

Solved Examples

Worked · 9
  1. Solved Ex.3.1
    Obtain the mean free path of nitrogen molecule at 0 ∘C0\ ^\circ \text{C} and 1.0 atm pressure. The molecular diameter of nitrogen is 324 pm (assume that the gas is ideal).
  2. Solved Ex.3.2
    At 300 K, what is the rms speed of Helium atom? [mass of He atom is 4u, 1u=1.66×10−271u = 1.66 \times 10^{-27} kg; kB=1.38×10−23k_{B} = 1.38 \times 10^{-23} J/K]
  3. Solved Ex.3.3
    Given the values of the two principal specific heats, SP=3400S_{P} = 3400 cal kg−1K−1\text{kg}^{-1} \text{K}^{-1} and SV=2400S_{V} = 2400 cal kg−1K−1\text{kg}^{-1} \text{K}^{-1} for the hydrogen gas, find the value of JJ if the universal gas constant R=8300R = 8300 J kg−1K−1\text{kg}^{-1} \text{K}^{-1}.
  4. Solved Ex.3.4
    The difference between the two molar specific heats of a gas is 8000 J kg−1K−1\text{kg}^{-1} \text{K}^{-1}. If the ratio of the two specific heats is 1.65, calculate the two molar specific heats.
  5. Solved Ex.3.5
    Calculate the value of λmax⁡\lambda_{\max} for solar radiation assuming that surface temperature of Sun is 5800 K (b=2.897×10−3 m K)\left( b = 2.897 \times 10^{-3} \text{ m K} \right). In which part of the electromagnetic spectrum, does this value lie?
  6. Solved Ex.3.6
    Calculate the energy radiated in one minute by a blackbody of surface area 200 cm2\text{cm}^{2} at 127 ∘C127\ ^\circ \text{C} (σ=5.67×10−8 J m−2s−1K−4)\left( \sigma = 5.67 \times 10^{-8} \text{ J m}^{-2} \text{s}^{-1} \text{K}^{-4} \right).
  7. Solved Ex.3.7
    A 60 watt filament lamp loses all its energy by radiation from its surface. The emissivity of the surface is 0.5. The area of the surface is 5×10−5 m25 \times 10^{-5} \text{ m}^{2}. Find the temperature of the filament (σ=5.67×10−8 J m−2s−1K−4)\left( \sigma = 5.67 \times 10^{-8} \text{ J m}^{-2} \text{s}^{-1} \text{K}^{-4} \right).
  8. Solved Ex.3.8
    Compare the rate of loss of heat from a metal sphere at 827 ∘C827\ ^\circ \text{C} with the rate of loss of heat from the same sphere at 427 ∘C427\ ^\circ \text{C}, if the temperature of the surrounding is 27 ∘C27\ ^\circ \text{C}.
  9. Solved Ex.3.9
    Assuming that the temperature at the surface of the Sun is 6000 K, find out the size of a virtual star (in terms of the size of Sun) whose surface temperature is 3000 K and the power radiated by the virtual star is 25 times the power radiated by the Sun. Treat both, the Sun and virtual star as a blackbody.

Exercises

33 q

Choose the correct option

Practice · 5
  1. Choose the correct option.
    Ex Q.1 (i)
    In an ideal gas, the molecules possess
    1. A.
      only kinetic energy
    2. B.
      both kinetic energy and potential energy
    3. C.
      only potential energy
    4. D.
      neither kinetic energy nor potential energy
  2. Ex Q.1 (ii)
    The mean free path λ\lambda of molecules is given by (where nn is the number of molecules per unit volume and dd is the diameter of the molecules)
    1. A.
      2πnd2\sqrt{\dfrac{2}{\pi n d^{2}}}
    2. B.
      1πnd2\dfrac{1}{\pi n d^{2}}
    3. C.
      12 πnd2\dfrac{1}{\sqrt{2}\, \pi n d^{2}}
    4. D.
      12πnd\dfrac{1}{\sqrt{2 \pi n d}}
  3. Ex Q.1 (iii)
    If pressure of an ideal gas is decreased by 10% isothermally, then its volume will
    1. A.
      decrease by 9%
    2. B.
      increase by 9%
    3. C.
      decrease by 10%
    4. D.
      increase by 11.11%
  4. Ex Q.1 (iv)
    If a=0.72a = 0.72 and r=0.24r = 0.24, then the value of trt_{\mathrm{r}} is
    1. A.
      0.02
    2. B.
      0.04
    3. C.
      0.4
    4. D.
      0.2
  5. Ex Q.1 (v)
    The ratio of emissive power of a perfect blackbody at 1327 ∘C1327\ ^\circ \text{C} and 527 ∘C527\ ^\circ \text{C} is
    1. A.
      4:14 : 1
    2. B.
      16:116 : 1
    3. C.
      2:12 : 1
    4. D.
      8:18 : 1

Answer in brief

Practice · 5
  1. Answer in brief.
    Ex Q.2 (i)
    What will happen to the mean square speed of the molecules of a gas if the temperature of the gas increases?
  2. Ex Q.2 (ii)
    On what factors do the degrees of freedom depend?
  3. Ex Q.2 (iii)
    Write ideal gas equation for a mass of 7 g of nitrogen gas.
  4. Ex Q.2 (iv)
    What is an ideal gas? Does an ideal gas exist in practice?
  5. Ex Q.2 (v)
    Define athermanous substances and diathermanous substances.

Solve the following

Practice · 23
  1. Ex Q.3
    When a gas is heated its temperature increases. Explain this phenomenon based on kinetic theory of gases.
  2. Ex Q.4
    Explain, on the basis of kinetic theory, how the pressure of gas changes if its volume is reduced at constant temperature.
  3. Ex Q.5
    Mention the conditions under which a real gas obeys ideal gas equation.
  4. Ex Q.6
    State the law of equipartition of energy and hence calculate molar specific heat of mono- and di-atomic gases at constant volume and constant pressure.
  5. Ex Q.7
    What is a perfect blackbody? How can it be realized in practice?
  6. Ex Q.8
    State (i) Stefan-Boltzmann law and (ii) Wein's displacement law.
  7. Ex Q.9
    Explain spectral distribution of blackbody radiation.
  8. Ex Q.10
    State and prove Kirchoff's law of heat radiation.
  9. Ex Q.11
    Calculate the ratio of mean square speeds of molecules of a gas at 30 K and 120 K.
  10. Ex Q.12
    Two vessels A and B are filled with same gas where volume, temperature and pressure in vessel A is twice the volume, temperature and pressure in vessel B. Calculate the ratio of number of molecules of gas in vessel A to that in vessel B.
  11. Ex Q.13
    A gas in a cylinder is at pressure PP. If the masses of all the molecules are made one third of their original value and their speeds are doubled, then find the resultant pressure.
  12. Ex Q.14
    Show that rms velocity of an oxygen molecule is 2\sqrt{2} times that of a sulfur dioxide molecule at S.T.P.
  13. Ex Q.15
    At what temperature will oxygen molecules have same rms speed as helium molecules at S.T.P.? (Molecular masses of oxygen and helium are 32 and 4 respectively)
  14. Ex Q.16
    Compare the rms speed of hydrogen molecules at 127 ∘C127\ ^\circ \text{C} with rms speed of oxygen molecules at 27 ∘C27\ ^\circ \text{C} given that molecular masses of hydrogen and oxygen are 2 and 32 respectively.
  15. Ex Q.17
    Find kinetic energy of 5000 cc of a gas at S.T.P. given standard pressure is 1.013×105 N/m21.013 \times 10^{5} \text{ N/m}^{2}.
  16. Ex Q.18
    Calculate the average molecular kinetic energy (i) per kmol (ii) per kg (iii) per molecule of oxygen at 127 ∘C127\ ^\circ \text{C}, given that molecular weight of oxygen is 32, RR is 8.31 J mol−1K−1\text{mol}^{-1} \text{K}^{-1} and Avogadro's number NAN_{\mathrm{A}} is 6.02×10236.02 \times 10^{23} molecules mol−1\text{mol}^{-1}.
  17. Ex Q.19
    Calculate the energy radiated in one minute by a blackbody of surface area 100 cm2\text{cm}^{2} when it is maintained at 227 ∘C227\ ^\circ \text{C}. (Take Stefen's constant σ=5.67×10−8 J m−2s−1K−4\sigma = 5.67 \times 10^{-8} \text{ J m}^{-2} \text{s}^{-1} \text{K}^{-4})
  18. Ex Q.20
    Energy is emitted from a hole in an electric furnace at the rate of 20 W, when the temperature of the furnace is 727 ∘C727\ ^\circ \text{C}. What is the area of the hole? (Take Stefan's constant σ\sigma to be 5.7×10−8 J s−1m−2K−45.7 \times 10^{-8} \text{ J s}^{-1} \text{m}^{-2} \text{K}^{-4})
  19. Ex Q.21
    The emissive power of a sphere of area 0.02 m2\text{m}^{2} is 0.5 kcal s−1m−2\text{s}^{-1} \text{m}^{-2}. What is the amount of heat radiated by the spherical surface in 20 second?
  20. Ex Q.22
    Compare the rates of emission of heat by a blackbody maintained at 727 ∘C727\ ^\circ \text{C} and at 227 ∘C227\ ^\circ \text{C}, if the blackbodies are surrounded by an enclosure (black) at 27 ∘C27\ ^\circ \text{C}. What would be the ratio of their rates of loss of heat?
  21. Ex Q.23
    Earth's mean temperature can be assumed to be 280 K. How will the curve of blackbody radiation look like for this temperature? Find out λmax⁡\lambda_{\max}. In which part of the electromagnetic spectrum, does this value lie? (Take Wien's constant b=2.897×10−3 m Kb = 2.897 \times 10^{-3} \text{ m K})
  22. Ex Q.24
    A small-blackened solid copper sphere of radius 2.5 cm is placed in an evacuated chamber. The temperature of the chamber is maintained at 100 ∘C100\ ^\circ \text{C}. At what rate energy must be supplied to the copper sphere to maintain its temperature at 110 ∘C110\ ^\circ \text{C}? (Take Stefan's constant σ\sigma to be 5.670×10−8 J s−1m−2K−45.670 \times 10^{-8} \text{ J s}^{-1} \text{m}^{-2} \text{K}^{-4}, π=3.1416\pi = 3.1416 and treat the sphere as a blackbody.)
  23. Ex Q.25
    Find the temperature of a blackbody if its spectrum has a peak at (a) λmax=700\lambda_{max} = 700 nm (visible), (b) λmax=3\lambda_{max} = 3 cm (microwave region) and (c) λmax=3\lambda_{max} = 3 m (short radio waves) (Take Wien's constant b=2.897×10−3 m Kb = 2.897 \times 10^{-3} \text{ m K}).