As shown in the figure, an insulated container is fitted with a thermally conducting but immovable partition ($P_1$) and a freely movable but thermally insulated piston ($P_2$). The partition $P_1$ with thermal conductivity $K$, cross sectional area $A$ and width $x$ divides the container into two sections, $S_1$ and $S_2$, each containing one mole of a monoatomic gas. The piston $P_2$ moves freely such that the gas in $S_2$ is always at the atmospheric pressure. Initially, the difference between the temperatures of $S_1$ and $S_2$ is $\Delta T_0$. The time it takes for the temperature difference to become $\frac{\Delta T_0}{2}$ is $nxR/KA$, where $R$ is the universal gas constant. The value of $n$ is:

[ Given: $ln 2 \approx 0.7$ ]

An ideal monatomic gas of $n$ moles is taken through a cycle $W X Y Z W$ consisting of consecutive adiabatic and isobaric quasi-static processes, as shown in the schematic $V-T$ diagram. The volume of the gas at $W, X$ and $Y$ points are, $64 \mathrm{~cm}^3, 125 \mathrm{~cm}^3$ and $250 \mathrm{~cm}^3$, respectively. If the absolute temperature of the gas $T_W$ at the point $W$ is such that $n R T_W=1 \mathrm{~J}$ ( $R$ is the universal gas constant), then the amount of heat absorbed (in J ) by the gas along the path $X Y$ is ___________.

The left and right compartments of a thermally isolated container of length $L$ are separated by a thermally conducting, movable piston of area $A$. The left and right compartments are filled with $\frac{3}{2}$ and 1 moles of an ideal gas, respectively. In the left compartment the piston is attached by a spring with spring constant $k$ and natural length $\frac{2 L}{5}$. In thermodynamic equilibrium, the piston is a distance $\frac{L}{2}$ from the left and right edges of the container as shown in the figure. Under the above conditions, if the pressure in the right compartment is $P=\frac{k L}{A} \alpha$, then the value of $\alpha$ is __________.

Two identical plates P and Q , radiating as perfect black bodies, are kept in vacuum at constant absolute temperatures $\mathrm{T}_{\mathrm{P}}$ and $\mathrm{T}_{\mathrm{Q}}$, respectively, with $\mathrm{T}_{\mathrm{Q}}<\mathrm{T}_{\mathrm{P}}$, as shown in Fig. 1. The radiated power transferred per unit area from P to Q is $W_0$. Subsequently, two more plates, identical to P and Q , are introduced between P and Q, as shown in Fig. 2. Assume that heat transfer takes place only between adjacent plates. If the power transferred per unit area in the direction from $P$ to $Q$ (Fig. 2) in the steady state is $W_S$, then the ratio $\frac{W_0}{W_S}$ is ________.