The specific heat capacity of a substance is temperature dependent and is given by the formula $C=k T$, where $k$ is a constant of suitable dimensions in SI units, and $T$ is the absolute temperature. If the heat required to raise the temperature of $1 \mathrm{~kg}$ of the substance from $-73^{\circ} \mathrm{C}$ to $27^{\circ} \mathrm{C}$ is $n k$, the value of $n$ is ________.

[Given: $0 \mathrm{~K}=-273{ }^{\circ} \mathrm{C}$.]

One mole of an ideal gas undergoes two different cyclic processes I and II, as shown in the $P-V$ diagrams below. In cycle I, processes $a, b, c$ and $d$ are isobaric, isothermal, isobaric and isochoric, respectively. In cycle II, processes $a^{\prime}, b^{\prime}, c^{\prime}$ and $d^{\prime}$ are isothermal, isochoric, isobaric and isochoric, respectively. The total work done during cycle $\mathrm{I}$ is $W_I$ and that during cycle II is $W_{I I}$. The ratio $W_I / W_{I I}$ is ________.

A cylindrical furnace has height $(H)$ and diameter $(D)$ both $1 \mathrm{~m}$. It is maintained at temperature $360 \mathrm{~K}$. The air gets heated inside the furnace at constant pressure $P_a$ and its temperature becomes $T=360 \mathrm{~K}$. The hot air with density $\rho$ rises up a vertical chimney of diameter $d=0.1 \mathrm{~m}$ and height $h=9 \mathrm{~m}$ above the furnace and exits the chimney (see the figure). As a result, atmospheric air of density $\rho_a=$ $1.2 \mathrm{~kg} \mathrm{~m}^{-3}$, pressure $P_a$ and temperature $T_a=300 \mathrm{~K}$ enters the furnace. Assume air as an ideal gas, neglect the variations in $\rho$ and $T$ inside the chimney and the furnace. Also ignore the viscous effects.

[Given: The acceleration due to gravity $g=10 \mathrm{~m} \mathrm{~s}^{-2}$ and $\pi=3.14$ ]

Considering the air flow to be streamline, the steady mass flow rate of air exiting the chimney is _________ $\mathrm{gm} \mathrm{s}^{-1}$.

A cylindrical furnace has height $(H)$ and diameter $(D)$ both $1 \mathrm{~m}$. It is maintained at temperature $360 \mathrm{~K}$. The air gets heated inside the furnace at constant pressure $P_a$ and its temperature becomes $T=360 \mathrm{~K}$. The hot air with density $\rho$ rises up a vertical chimney of diameter $d=0.1 \mathrm{~m}$ and height $h=9 \mathrm{~m}$ above the furnace and exits the chimney (see the figure). As a result, atmospheric air of density $\rho_a=$ $1.2 \mathrm{~kg} \mathrm{~m}^{-3}$, pressure $P_a$ and temperature $T_a=300 \mathrm{~K}$ enters the furnace. Assume air as an ideal gas, neglect the variations in $\rho$ and $T$ inside the chimney and the furnace. Also ignore the viscous effects.

[Given: The acceleration due to gravity $g=10 \mathrm{~m} \mathrm{~s}^{-2}$ and $\pi=3.14$ ]

When the chimney is closed using a cap at the top, a pressure difference $\Delta P$ develops between the top and the bottom surfaces of the cap. If the changes in the temperature and density of the hot air, due to the stoppage of air flow, are negligible then the value of $\Delta P$ is __________ $\mathrm{N} \mathrm{m}^{-2}$.