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Learn/ 12th Class Physics/ Unit 13 /Numericals

Unit 13: Thermal Physics — Numericals

12th Class Physics · Unit 13: Thermal Physics

CRQ 13.2.What is the average kinetic energy of the oxygen and nitrogen molecules in a room at room temperature?

Data T = 300 K (room temperature), kB = 1.38 × 10⁻²³ J K⁻¹.
Formula: <K.E.> = (3/2)kBT.
Substitution: <K.E.> = (3/2)(1.38 × 10⁻²³ J K⁻¹)(300 K) = 6.21 × 10⁻²¹ J.
This value is the same for both O₂ and N₂ molecules since <K.E.> depends only on temperature, not on molecular mass. In electron-volts, using kBT ≈ 0.026 eV at 300 K, <K.E.> = (3/2)(0.026 eV) ≈ 0.039 eV.
Result: <K.E.> ≈ 6.21 × 10⁻²¹ J (≈ 0.039 eV) for each gas.

CRQ 13.3.Consider a gas enclosed in a sealed container. By what factor does the gas temperature change if: (a) the volume and pressure are both doubled? (b) the volume is halved and the pressure is tripled?

Formula PV = NkBT, so T ∝ PV (N fixed, sealed container).
(a) New pressure = 2P, new volume = 2V. Tnew/Told = (2P × 2V)/(PV) = 4. The temperature increases by a factor of 4.
(b) New pressure = 3P, new volume = V/2. Tnew/Told = (3P × V/2)/(PV) = 3/2. The temperature increases by a factor of 1.5 (increases by 50%).

NP 13.1.Calculate the rms velocity of hydrogen molecules at standard temperature and pressure (STP). Density of hydrogen at STP, ρ = 8.957 × 10⁻² kg m⁻³. Density of mercury = 13600 kg m⁻³, g = 9.8 m s⁻².
NP 13.2.Determine the pressure of oxygen gas at 0 °C if the density of oxygen is 1.44 kg m⁻³ at STP and rms speed at STP is 456.4 m s⁻¹.

Data ρ = 1.44 kg m⁻³, v(rms) = 456.4 m s⁻¹.
Formula: P = (1/3)ρ<v²> = (1/3)ρv(rms)².
Substitution: P = (1/3)(1.44 kg m⁻³)(456.4 m s⁻¹)² = (1/3)(1.44)(208301 m² s⁻²).
P = (0.48)(208301) ≈ 99985 N m⁻².
Result: P ≈ 1.0 × 10⁵ N m⁻².

NP 13.3.At what temperature will the rms speed of the molecules of a gas be three times its value at STP?

Data T(STP) = 273 K, new rms speed = 3 × v(rms, STP).
Formula: v(rms) ∝ √T, so (v2/v1)² = T2/T1.
Substitution: (3)² = T2/273, so T2 = 9 × 273 K = 2457 K.
Converting to Celsius: T2 = 2457 − 273 = 2184 °C.
Result: T ≈ 2184 °C.

NP 13.4.If a gas is at temperature 80 °C and pressure 5 × 10⁻¹⁰ N m⁻², having 1 m³ volume, what is the number of molecules per cubic metre, where Boltzmann's constant is 1.38 × 10⁻²³ J K⁻¹?
NP 13.5.The temperature and pressure in the Sun's atmosphere are 2.00 × 10⁶ K and 0.0300 Pa respectively. Calculate the rms speed of free electrons (mass of the electron = 9.11 × 10⁻³¹ kg).
NP 13.6.A star has a mass enclosed within radius r equal to 2.0 × 10³⁰ kg, density 1500 kg m⁻³ and radius 7.0 × 10⁸ m. Calculate the gravitational force inside the star.