
Consider that specific heat (0 to \(50~^\circ\mathrm{C}\)) of water, water vapour and air remains constant: \(4.48\), \(1.88\) and \(1.0~\mathrm{kJ/(kg\^\circ C)}\), respectively. Assuming the heat energy required to convert \(1~\mathrm{kg}\) of water to water vapour at \(0~^\circ\mathrm{C}\) is \(2000~\mathrm{kJ}\), the enthalpy (in kJ/kg dry air) of atmospheric air containing \(0.05~\mathrm{kg}\) water vapour per kg dry air at \(50~^\circ\mathrm{C}\) is ________. (rounded off to 1 decimal place)
In hot weather, a human body cools by evaporation of sweat. The amount of water that must evaporate to cool the body by \(1~^\circ\mathrm{C}\) is __________________________% of the body mass. (Round off to two decimal places)
[Given: latent heat of vaporization of water \(L_v=2.25\times10^6~\mathrm{J\,kg^{-1}}\); specific heat capacities of body and water \(c=4.2\times10^3~\mathrm{J\,kg^{-1}\,K^{-1}}\).]
The center of a disk of radius $ r $ and mass $ m $ is attached to a spring of spring constant $ k $, inside a ring of radius $ R>r $ as shown in the figure. The other end of the spring is attached on the periphery of the ring. Both the ring and the disk are in the same vertical plane. The disk can only roll along the inside periphery of the ring, without slipping. The spring can only be stretched or compressed along the periphery of the ring, following Hooke’s law. In equilibrium, the disk is at the bottom of the ring. Assuming small displacement of the disc, the time period of oscillation of center of mass of the disk is written as $ T = \frac{2\pi}{\omega} $. The correct expression for $ \omega $ is ( $ g $ is the acceleration due to gravity): 
Let $ a_0, a_1, ..., a_{23} $ be real numbers such that $$ \left(1 + \frac{2}{5}x \right)^{23} = \sum_{i=0}^{23} a_i x^i $$ for every real number $ x $. Let $ a_r $ be the largest among the numbers $ a_j $ for $ 0 \leq j \leq 23 $. Then the value of $ r $ is ________.
Thermodynamics in physics is a branch that deals with heat, work and temperature, and their relation to energy, radiation and physical properties of matter.
The first law of thermodynamics, also known as the Law of Conservation of Energy, states that energy can neither be created nor destroyed; energy can only be transferred or changed from one form to another.
The second law of thermodynamics says that the entropy of any isolated system always increases. Isolated systems spontaneously evolve towards thermal equilibrium—the state of maximum entropy of the system. More simply put: the entropy of the universe (the ultimate isolated system) only increases and never decreases.
The third law of thermodynamics states that the entropy of a system approaches a constant value as the temperature approaches absolute zero. The entropy of a system at absolute zero is typically zero, and in all cases is determined only by the number of different ground states it has. Specifically, the entropy of a pure crystalline substance (perfect order) at absolute zero temperature is zero