An engineer claims to have built an engine that delivers $10 \, kW$ of power with a fuel consumption of $1 \, g/s$. If the calorific value of the fuel is $2 \, kcal/g$, is the engineer's claim valid?

$A$ cycle followed by an engine (made of one mole of perfect gas in a cylinder with a piston) is shown in the figure.
$A$ to $B$: volume constant
$B$ to $C$: adiabatic
$C$ to $D$: volume constant
$D$ to $A$: adiabatic
$V_C = V_D = 2V_A = 2V_B$
$(a)$ In which part of the cycle is heat supplied to the engine from outside?
$(b)$ In which part of the cycle is heat given to the surrounding by the engine?
$(c)$ What is the work done by the engine in one cycle? Write your answer in terms of $P_A, P_B, V_A$.
$(d)$ What is the efficiency of the engine?
$(\gamma = 5/3, C_v = 3/2 R$ for one mole of the gas$)$

The figure shows a cylindrical adiabatic container of total volume $2V_0$ divided into two equal parts by a conducting piston (which is free to move). Each part contains an identical gas at pressure $P_0$. Initially,the temperature of the left and right parts is $4T_0$ and $T_0$ respectively. An external force is applied on the piston to keep it at rest. Find the value of the external force required when thermal equilibrium is reached. ($A =$ Area of the piston)

Three processes compose a thermodynamic cycle shown in the $PV$ diagram. Process $1\rightarrow 2$ takes place at constant temperature. Process $2\rightarrow 3$ takes place at constant volume,and process $3\rightarrow 1$ is adiabatic. During the complete cycle,the total amount of work done is $10\,J$. During process $2\rightarrow 3$,the internal energy decreases by $20\,J$ and during process $3\rightarrow 1$,$20\,J$ of work is done on the system. How much heat is added to the system during process $1\rightarrow 2$ (in $,J$)?

Three moles of an ideal monatomic gas perform a cycle $ABCDA$ as shown in the figure. The temperatures of the gas at the states $A, B, C$ and $D$ are $400 \, K, 800 \, K, 2400 \, K$ and $1200 \, K$, respectively. The work done by the gas during this cycle is ($R$ is the universal gas constant). (in $R$)

$A$ given amount of gas has an initial state $(P_i, V_i, T_i)$. It expands until its volume becomes $V_f$. Consider the following two cases:
$(a)$ The expansion occurs at constant temperature (isothermal).
$(b)$ The expansion occurs at constant pressure (isobaric).
Draw the $P-V$ diagram for each case. In which of the two cases is the work done by the gas greater?