Three Carnot engines operate in series between a heat source at a temperature T₁ and a heat sink at temperature T₄ (see figure). There are two other reservoirs at temperature T₂ and T₃, as shown, with T₁ > T₂ > T₃ > T₄. The three engines are equally efficient if -

Two Carnot engines A and B are operated in series. The first one, A, receives heat at T₁ (= 600 K) and rejects to a reservoir at temperature T₂. The second engine B receives heat rejected by the first engine and, in tum, rejects to a heat reservoir at T₃ (=400 K). Calculate the temperature T₂ if the work outputs of the two engines are equal :

A 15 g mass of nitrogen gas is enclosed in a vessel at a temperature 27^oC. Amount of heat transferred to the gas, so that rms velocity of molecules is doubled, is about : [Take R = 8.3 J/K mole]

A mixture of 2 moles of helium gas (atomic mass = 4 u), and 1 mole of argon gas (atomic mass = 40 u) is kept at 300 K in a container. The ratio of their rms speeds $$\left[ {{{{V_{rms}}\,(helium)} \over {{V_{rms}}\,(\arg on)}}} \right],$$ is close to :