Kinetic Theory of Gases and Radiation
Kinetic Theory of Gases and Radiation
- The r.m.s. speed of the molecules of a gas at 27 °C is \(v\). The temperature at which the r.m.s. speed becomes \(2v\) is Answer: D. 927 °C v_rms∝√(T) : T₂=4×300=1200 K =927 °C. 1
- If the absolute temperature of a perfect black body is doubled, the energy radiated by it per second becomes Answer: B. 16 times Stefan's law: E∝ T⁴ , so 2⁴=16 times. 1
- The wavelength of maximum emission from the sun is 480 nm. Taking Wien's constant \(b=2.88\times10^{-3}\) m K, the surface temperature of the sun is Answer: C. 6000 K T=fracbλ_max=frac2.88×10⁻³480×10⁻⁹=6000 K. 1
- An ideal gas has density 1.2 kg/m3 at a pressure of \(1\times10^5\) Pa. The r.m.s. speed of its molecules is Answer: B. 500 m/s P=1/3ρ v_rms²⇒ v_rms=√(3P/ρ)=√(3×10⁵/1.2)=500 m/s. 1
- Find the temperature of a blackbody if its spectrum has a peak at (a) λmax = 700 nm = 700×10^(-9 )m, (b) λmax = 3 cm = 3×10^(-2 )m, (c) λmax = 3 m b = 2.897 × 10⁻³ m K (a) T= b/λmax = (2.897 × 10⁻³ )/ (700×... 1
- A small-blackened solid copper sphere of radius 2.5 cm is placed in an evacuated chamber. Given : r = 2.5 cm = 2.5 ×10⁻²m, T₀ = 273+100 = 373 K, T = 273+110 = 383 K, σ = 5.76 × 10⁻⁸ J s⁻¹ m⁻² K⁻⁴ The rate at which energy must be s... 2
- Earth’s mean temperature can be assumed to be 280 K. Given: T = 280 K, Wien's constant b = 2.897×10⁻³ m.K λ_maxT = b λ_max = b /T = 2.897×10⁻³ / 280 = 1.035 ×10⁻⁵ m This value lies in the infra... 1
- Compare the rates of emission of heat by a blackbody maintained at 727 ºC and at 227ºC Given: T₁ = 273 + 727 = 1000 K, T₂ = 273 + 227 = 500 K, T₀ = 273 + 27 = 300 K i) The rate of emission of heat, dQ/dt = σAT⁴ we assume that t... 1
- The emissive power of a sphere of area 0.02 m2 is 0.5 kcal s-1 m-2. What is the amount of heat radiated Given : R = 0.5 kcal-s⁻¹ 'm⁻² , A =0.02 m², t=20s Q = RAt = (0.5) (0.02) (20) = 0.2 kcal 0.2 kcal is the required quantity. 1
- Energy is emitted from a hole in an electric furnace at the rate of 20 W, when the temperature of the furnace is 727 ºC. Given: Q/t = 20W, T=273+727=1000 K, σ = 5.7×10⁻⁸ Js⁻¹/m².K⁻⁴ Q/t = σAT⁴ ∴ The area of the hole, A = (Q/t)/(σT⁴ ) = 20(5.7×10⁻⁸ ×1000⁴ ) = 20... 1
- Calculate the energy radiated in one minute by a blackbody of surface area 100 cm2 when it is maintained at 227 ºC. Given: t = one minute = 60 s, A = 100 cm² = 100×10⁻⁴ m² = 10⁻² m², T = 273 + 227 = 500 K, σ= 5.67×10⁻⁸ W/m².K⁴ The energy radiated, Q = σAT⁴... 1
- Calculate the average molecular kinetic energy (i) per kmol (ii) per kg (iii) per molecule of oxygen at 127 ºC Given : T = 273+127 = 400 K, molecular weight=32 \ molar mass = 32 kg/kmol, R = 8.31 Jmol⁻¹ K⁻¹, N_A = 6.02 x 10²³ molecules mol⁻¹ (i) The a... 1
- Find kinetic energy of 5 litre of a gas at S.T.P. given standard pressure is 1.013 × 105 N/m2 Given : P 1.013 x 10⁵ N/m², V = 5 litres = 5 x 10⁻³ m E = 3/2 PV = 3/2 (1.013 x 10⁵) (5 x 10⁻³) = 7.5 x 1.013 x 10² J = 7.597 x 10² J This i... 1
- Compare the rms speed of hydrogen molecules at 127 ºC with rms speed of oxygen molecules at 27 ºC M_(o1 )(hydrogen) = 2 g/mole M_(o2 )(oxygen) = 32 g/mole T_(1 )(hydrogen) = 273+127=400 K T_(2 )(oxygen) = 273+27=300 K The rms speed v_rms... 1
- At what temperature will oxygen molecules have same rms speed as helium molecules at S.T.P.? Given : T_(2 )= 273 k, M_o1 (oxygen)= 32 ×10^(-3 )kg/mole, M_o2 (hydrogen)= 4 ×10^(-3 )kg/mole v_rms = √(3RT/M_o) The rms speed of oxygen mo... 1
- Show that rms velocity of an oxygen molecule is 2 times that of a sulfur dioxide molecule at S.T.P. Given : m_o(SO₂)/m_o(O₂) = 64/32 =2 The rms speed v_rms = √(3RT/M_o) v_rms ∝ 1/√(M_o) at constant T fracv_rms(O₂)v_rms(SO₂) = √(M_o(SO₂)/M_o... 1
- A gas in a cylinder is at pressure P. If the masses of all the molecules are made one third of their original Given: m_(2 )= m₁/3 v_(2 )= 2v₁ Pressure P= 1/3 mN/vv² ∴P₁= 1/3 m₁N/vv₁² ∴P₂= 1/3 m₂N/vv₂² ∴ P₁/P₂ =(m₂/m₁)(v₂²/v₁²) = m₁/3/m₁(2²) = 4/3 P₂... 1
- Two vessels A and B are filled with same gas where volume Given : V_(A )= V_(B ), T_A=2T_(B )P_A=2P_B PV=Nk_BT ∴ No of molecules, N=PV/k_BT N_A=P_AV_A/k_BT_(A )and N_B=P_BV_B/k_BT_B ∴ N_A/N_(B )= (P... 1
- Calculate the ratio of mean square speeds of molecules of a gas at 30 K and 120 K. Given : T₁ = 30K, T₂= 120K The mean square speed v² = 3RT/M_o ∴ v₁²/v₂² =T₁/T₂ for given gas ∴ v₁²/v₂² =30K/120K = 1/4 1/4 is the ratio 1
- State and prove Kirchoff’s law of heat radiation. Kirchhoff’s law of heat radiation: At a given temperature, the ratio of the emissive power to the coefficient of absorption of a body is equ... 1
- Explain spectral distribution of blackbody radiation. Answer: the spectral distribution of blackbody radiation is the graph of energy emitted per unit wavelength ( E_λ ) against wavelength ( λ )... 1
- state Wein’s displacement law. Wien’s displacement law : The wavelength for which the emissive power of a blackbody is maximum, is inversely proportional to the absolute t... 1
- state Stefan-Boltmann law The Stefan-Boltzmann law : The rate of emission of radiant energy per unit area or the power radiated per unit area of a perfect blackbody i... 1
- State the law of equipartition of energy and hence calculate molar specific heat of mono- and di-atomic gases Law of equipartition of energy states that for a dynamical system in thermal equilibrium the total energy of the system is shared equally by... 1