$A$ monoatomic gas at pressure $P$ having volume $V$ expands isothermally to a volume $3V$ and then adiabatically to a volume $81V$. The final pressure of the gas is $(\gamma = 5/3)$.

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 $,J$)?

$A$ real gas within a closed chamber at $27^{\circ} C$ undergoes a cyclic process $A \to B \to C \to A$. For the path $A \to B$, the gas follows the equation $PV^3 = RT$. For the path $B \to C$, the process is isobaric at $P = 10 \, Pa$ from $V = 4 \, m^3$ to $V = 2 \, m^3$. For the path $C \to A$, the process is isochoric at $V = 2 \, m^3$. Given $R = 8 \, J/mol \cdot K$ and $T = 300 \, K$ (constant for path $A \to B$), the net work done in the complete cycle is (in $J$):

$A$ monoatomic ideal gas is heated at constant pressure. The percentage of total heat used in increasing the internal energy and that used for doing external work is $A$ and $B$ respectively. Then the ratio,$A: B$ is

$A$ small particle of mass $m$ moving inside a heavy, hollow and straight tube along the tube axis undergoes elastic collision at two ends. The tube has no friction and it is closed at one end by a flat surface while the other end is fitted with a heavy movable flat piston as shown in the figure. When the distance of the piston from the closed end is $L = L_0$, the particle speed is $v = v_0$. The piston is moved inward at a very low speed $V$ such that $V \ll \frac{dL}{L} v_0$, where $dL$ is the infinitely small displacement of the piston. Which of the following statement(s) is/are correct?
$(1)$ The rate at which the particle strikes the piston is $v / (2L)$
$(2)$ After each collision with the piston, the particle speed increases by $2V$
$(3)$ The particle's kinetic energy increases by a factor of $4$ when the piston is moved inward from $L_0$ to $L_0 / 2$
$(4)$ If the piston moves inward by $dL$, the particle speed increases by $v \frac{dL}{L}$

$A$ gas is compressed from a volume of $2\,m^3$ to a volume of $1\,m^3$ at a constant pressure of $100\,N/m^2$. Then it is heated at constant volume by supplying $150\,J$ of energy. As a result,the internal energy of the gas