# First Law, Heat, Work and Energy

 Question 1
Consider a fully adiabatic piston-cylinder arrangement as shown in the figure. The piston is massless and cross-sectional area of the cylinder is $A$. The fluid inside the cylinder is air (considered as a perfect gas), with $\gamma$ being the ratio of the specific heat at constant pressure to the specific heat at constant volume for air. The piston is initially located at a position $L_1$. The initial pressure of the air inside the cylinder is $P_1 \gt \gt P_0$, where $P_0$ is the atmospheric pressure. The stop $S_1$ is instantaneously removed and the piston moves to the position $L_2$, where the equilibrium pressure of air inside the cylinder is $P_2 \gt \gt P_0$.
What is the work done by the piston on the atmosphere during this process?

 A $0$ B $P_0A(L_2-L_1)$ C $P_1AL_1 \ln \frac{L_1}{L_2}$ D $\frac{(P_2L_2-P_1L_1)A}{(1-\gamma )}$
GATE ME 2023   Thermodynamics
Question 1 Explanation:

Initial volume $\mathrm{V}_{1}=\mathrm{L}_{1} \times \mathrm{A}$
Final volume $V_{2}=L_{2} \times A$
Work done by atmospheric air $=\int_{V_{1}}^{V_{2}} P d v$
\begin{aligned} & =P_{0} \int_{V_{1}}^{V_{2}} d V \\ & =P_{0}\left(L_{2} A-L_{1} A\right) \\ & =P_{0} A\left(L_{2}-L_{1}\right) \end{aligned}
 Question 2
A heat engine extracts heat $(Q_H)$ from a thermal reservoir at a temperature of $1000 K$ and rejects heat $(Q_L)$ to a thermal reservoir at a temperature of $100 K$, while producing work $(W)$. Which one of the combinations of $(Q_H,Q_L,W)$ given is allowed?
 A $Q_H=2000J, Q_L=500J, W=1000J$ B $Q_H=2000J, Q_L=750J, W=1250J$ C $Q_H=6000J, Q_L=500J, W=5500J$ D $Q_H=6000J, Q_L=600J, W=5500J$
GATE ME 2023   Thermodynamics
Question 2 Explanation:
For a reversible engine, the rate of heat rejection is minimum.

For process to be feasible
$\oint \frac{dQ}{T}\leq 0$
for option (b) $\oint \frac{dQ}{T}=\frac{Q_H}{T_1}-\frac{Q_2}{T_2}=\frac{2000}{1000}-\frac{750}{100} \lt 0$
So cyclic process is possible.

 Question 3
Consider 1 kg of an ideal gas at 1 bar and 300 K contained in a rigid and perfectly insulated container. The specific heat of the gas at constant volume $c_v$ is equal to $750 \; Jkg^{-1}K^{-1}$. A stirrer performs 225 kJ of work on the gas. Assume that the container does not participate in the thermodynamic interaction. The final pressure of the gas will be ______ bar (in integer).
 A 1 B 2 C 3 D 4
GATE ME 2022 SET-2   Thermodynamics
Question 3 Explanation:
$m = 1 kg, P_1 = 1 \;bar, T_1 = 300 \;K$

\begin{aligned} V &= \text{Constant} \\ W_{expansion}&= 0\\ C_V &=750\frac{J}{kgK}=0.75\frac{kJ}{kgK} \\ W_{stirrer}&= 225kJ\;\;\;(-ve \;\; work)\\ P_2&=? \\ \therefore W&= W_{expansion}+W_{stirrer}\\ &=0-225=-225kJ \end{aligned}
Using Ist law of thermodynamics
\begin{aligned} Q-W&=dU=mc_v(T_2-T_1)\\ 0-(-225)&=1 \times 0.75(T_2-300)\\ T_2&=600K\\ \therefore \frac{P_2}{P_1}&=\frac{T_2}{T_1}\\ P_2&=\frac{600}{300} \times 1 =2\; bar \end{aligned}
 Question 4
A polytropic process is carried out from an initial pressure of 110 kPa and volume of 5 $m^3$ to a final volume of 2.5 $m^3$. The polytropic index is given by n = 1.2. The absolute value of the work done during the process is _______ kJ (round off to 2 decimal places).
 A 408.92 B 215.58 C 852.36 D 789.14
GATE ME 2022 SET-1   Thermodynamics
Question 4 Explanation:
Polytropic process]
\begin{aligned} P_1 &= 110 kPa,\\ V_1&= 5m^3,\\ V_2&=2.5 m^3,\\ n&=1.2\\ \Rightarrow \; P_1V_1^n&=P_2V_2^n\\ P_2&=P_1\left ( \frac{V_1}{V_2} \right )^n\\ &=110 \times \left ( \frac{5}{2.5} \right )^{1.2}\\ P_2&=252.71 kPa\\ W&=\frac{P_1V_1=P_2V_2}{n-1}\\ &=\frac{110 \times 5-252.71 \times 2.5}{1.2-1}\\ W&=-408.92kJ\\ |W|&=408.92kJ \end{aligned}
 Question 5
In a steam power plant, superheated steam at 10 MPa and 500$^{\circ}C$, is expanded isentropically in a turbine until it becomes a saturated vapour. It is then reheated at constant pressure to 500$^{\circ}C$. The steam is next expanded isentropically in another turbine until it reaches the condenser pressure of 20 kPa. Relevant properties of steam are given in the following two tables. The work done by both the turbines together is ______ kJ/kg (roundoff to the nearest integer).
 A 1513 B 1245 C 832 D 1825
GATE ME 2020 SET-2   Thermodynamics
Question 5 Explanation:
Given data: $h_{1}=3373.6 \mathrm{kJ} / \mathrm{kg}, h_{3}=3478.4 \mathrm{kJ} / \mathrm{kg}, h_{2}=2778.1 \mathrm{kJ} / \mathrm{kg}, s_{1}=s_{2}$ (as from table )

\begin{aligned} s_{3} &=s_{4} \\ s_{3} &=7.7621=0.8319+x+(7.9085-0.8319) \\ x_{4} &=0.9793 \\ h_{4} &=h_{f}+x_{4} \times\left(h_{g}-h_{f}\right)=2560.91 \mathrm{kJ} / \mathrm{kg} \\ W_{T} &=\left(h_{1}-h_{2}\right)+\left(h_{3}-h_{4}\right)=1512.95 \mathrm{kJ} / \mathrm{kg} \end{aligned}

There are 5 questions to complete.