The internal energy of an ideal diatomic gas corresponding to volume $V$ and pressure $P$ is $2.5 PV$. The gas expands from $1 \text{ litre}$ to $2 \text{ litre}$ at a constant pressure of $10^5 \text{ N/m}^2$. The heat supplied to the gas is: (in $\text{ J}$)

When $100 \ J$ of heat is supplied to a gas,the increase in the internal energy of the gas is $60 \ J$. Then the gas is/can

Fill in the blanks:
$1.$ The heat required to raise the temperature of a body by $1\,K$ is called ...... .
$2.$ $A$ temperature difference of $10\,^{\circ}C$ is equivalent to a temperature difference of ...... on the Fahrenheit temperature scale.
$3.$ For an isothermal process of an ideal gas,$\frac{dP}{P} = .........$

Heat is supplied to a diatomic gas at constant pressure. The ratio of $\Delta Q : \Delta U : \Delta W$ is

Answer the following by appropriately matching the lists based on the information given in the paragraph.
In a thermodynamics process on an ideal monatomic gas,the infinitesimal heat absorbed by the gas is given by $T \Delta X$,where $T$ is the temperature of the system and $\Delta X$ is the infinitesimal change in a thermodynamic quantity $X$ of the system. For a mole of monatomic ideal gas,$X = \frac{3}{2} R \ln \left(\frac{T}{T_A}\right) + R \ln \left(\frac{V}{V_A}\right)$. Here,$R$ is the gas constant,$V$ is the volume of the gas,$T_A$ and $V_A$ are constants.
The $List-I$ below gives some quantities involved in a process and $List-II$ gives some possible values of these quantities.

List-$I$	List-$II$
$(I)$ Work done by the system in process $1 \rightarrow 2 \rightarrow 3$	$(P)$ $\frac{1}{3} R T_0 \ln 2$
$(II)$ Change in internal energy in process $1 \rightarrow 2 \rightarrow 3$	$(Q)$ $\frac{1}{3} R T_0$
$(III)$ Heat absorbed by the system in process $1 \rightarrow 2 \rightarrow 3$	$(R)$ $R T_0$
$(IV)$ Heat absorbed by the system in process $1 \rightarrow 2$	$(S)$ $\frac{4}{3} R T_0$
	$(T)$ $\frac{1}{3} R T_0 (3 + \ln 2)$
	$(U)$ $\frac{5}{6} R T_0$

If the process carried out on one mole of monatomic ideal gas is as shown in the figure in the $PV$-diagram with $P_0 V_0 = \frac{1}{3} R T_0$,the correct match is:
$(1)$ $I \rightarrow Q, II \rightarrow R, III \rightarrow P, IV \rightarrow U$
$(2)$ $I \rightarrow S, II \rightarrow R, III \rightarrow Q, IV \rightarrow T$
$(3)$ $I \rightarrow Q, II \rightarrow R, III \rightarrow S, IV \rightarrow U$
$(4)$ $I \rightarrow Q, II \rightarrow S, III \rightarrow R, IV \rightarrow U$
If the process on one mole of monatomic ideal gas is as shown in the $TV$-diagram with $P_0 V_0 = \frac{1}{3} R T_0$,the correct match is:
$(1)$ $I \rightarrow S, II \rightarrow T, III \rightarrow Q, IV \rightarrow U$
$(2)$ $I \rightarrow P, II \rightarrow R, III \rightarrow T, IV \rightarrow S$
$(3)$ $I \rightarrow P, II \rightarrow R, III \rightarrow Q, IV \rightarrow T$
$(4)$ $I \rightarrow P, II \rightarrow R, III \rightarrow T, IV \rightarrow P$
Give the answer for question $(1)$ and $(2)$.

Three processes form a thermodynamic cycle as shown on the $P-V$ diagram for an ideal gas. Process $1 \rightarrow 2$ takes place at a constant temperature $(300 \ K)$. Process $2 \rightarrow 3$ takes place at a constant volume. During this process,$40 \ J$ of heat leaves the system. Process $3 \rightarrow 1$ is adiabatic and the temperature $T_3$ is $275 \ K$. The work done by the gas during the process $3 \rightarrow 1$ is ..... $J$.