A small object is placed at the center of a large evacuated hollow spherical container. Assume that the container is maintained at 0 K. At time t = 0, the temperature of the object is 200 K. The temperature of the object becomes 100 K at t = t₁ and 50 K at t = t₂. Assume the object and the container to be ideal black bodies. The heat capacity of the object does not depend on temperature. The ratio (t₂/t₁) is ____________.

A thermally isolated cylindrical closed vessel of height 8 m is kept vertically. It is divided into two equal parts by a diathermic (perfect thermal conductor) frictionless partition of mass 8.3 kg. Thus the partition is held initially at a distance of 4 m from the top, as shown in the schematic figure below. Each of the two parts of the vessel contains 0.1 mole of an ideal gas at temperature 300 K. The partition is now released and moves without any gas leaking from one part of the vessel to the other. When equilibrium is reached, the distance of the partition from the top (in m) will be _______.
(take the acceleration due to gravity = 10 ms⁻² and the universal gas constant = 8.3 J mol⁻¹K⁻¹).

A spherical bubble inside water has radius R. Take the pressure inside the bubble and the water pressure to be p₀. The bubble now gets compressed radially in an adiabatic manner so that its radius becomes (R $$-$$ a). For a << R the magnitude of the work done in the process in given by (4$$\pi$$p₀Ra²)X, where X is a constant and $$\gamma$$ = C_p/C_v = 41/30. The value of X is ________.

A container with 1 kg of water in it is kept in sunlight, which causes the water to get warmer than the surroundings. The average energy per unit time per unit area received due to the sunlight is 700 Wm^$$-$$2 and it is absorbed by the water over an effective area of 0.05m². Assuming that the heat loss from the water to the surroundings is governed by Newton's law of cooling, the difference (in $$^\circ$$C) in the temperature of water and the surroundings after a long time will be ___________. (Ignore effect of the container, and take constant for Newton's law of cooling = 0.001 s^$$-$$1, Heat capacity of water = 4200 J kg^$$-$$1 K^$$-$$1)