Choose the correct statement for processes $A$ and $B$ shown in the figure.

When a system is taken from a state $i$ to $f$ along the path $iaf$ (as shown in the figure),$Q = 50 \, cal$ and $W = 20 \, cal$. Along path $ibf$,$Q = 36 \, cal$.
$(i)$ What is $W$ along path $ibf$?
$(ii)$ If $W = -13 \, cal$ for path $fi$,what is $Q$ for the path $fi$?
$(iii)$ Take $E_{int,i} = 10 \, cal$,then what is $E_{int,f}$?

If heat energy $\Delta Q$ is supplied to an ideal diatomic gas,the increase in internal energy is $\Delta U$ and the amount of work done by the gas is $\Delta W$. The ratio $\Delta W: \Delta U: \Delta Q$ is

$A$ 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$)$

$A$ 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?

What are the limitations of the first law of thermodynamics?