# Thermodynamics

 Question 1
Consider a mixture of two ideal gases, X and Y, with molar masses $\bar{M}_X=10kg/kmol$ and $\bar{M}_Y=20kg/kmol$, respectively, in a container. The total pressure in the container is 100 kPa, the total volume of the container is $10m^3$ and the temperature of the contents of the container is 300 K. If the mass of gas-X in the container is 2 kg, then the mass of gas-Y in the container is ____ kg. (Rounded off to one decimal place)
Assume that the universal gas constant is 8314 $J \;kmol^{-1}K^{-1}$
 A 2 B 4 C 6 D 8
GATE ME 2023      Thermodynamic System and Processes
Question 1 Explanation:

By gas law
$\mathrm{Pv}=\mathrm{nRT}$
where $\mathrm{n}=$ total value of gas in container
\begin{aligned} 100 \times 10 & =n \times 8.314 \times 300 \\ n & =0.4 \end{aligned}
and $\mathrm{n}=\mathrm{n}_{\mathrm{x}}+\mathrm{n}_{\mathrm{y}}$
Or, $0.4=\frac{m_{x}}{M_{x}}+\frac{m_{y}}{M_{y}}=\frac{2}{10}+\frac{m_{y}}{20}$ or, $m_{y}=4 \mathrm{~kg}$
 Question 2
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      First Law, Heat, Work and Energy
Question 2 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 3
Which one of the following statements is FALSE?
 A For an ideal gas, the enthalpy is independent of pressure. B For a real gas going through an adiabatic reversible process, the process equation is given by $PV^\gamma =$ constant, where $P$ is the pressure, $V$ is the volume and $\gamma$ is the ratio of the specific heats of the gas at constant pressure and constant volume. C For an ideal gas undergoing a reversible polytropic process $PV^{1.5} =$ constant, the equation connecting the pressure, volume and temperature of the gas at any point along the process is $\frac{P}{R}=\frac{mT}{V}$ where $R$ is the gas constant and $m$ is the mass of the gas. D Any real gas behaves as an ideal gas at sufficiently low pressure or sufficiently high temperature.
GATE ME 2023      Thermodynamic System and Processes
Question 3 Explanation:
Ideal gas follow the adiabatic equation $\mathrm{PV}^{\gamma}=\mathrm{C}$ and also follow the gas law $P V=m R T$
Any real gas at low pressure and high temperature follow gas law, e.g. Air conditioning system.
Real gas follow Van der waal's gas equation for an adiabatic process as
$\left(P+\frac{n^{2} a}{V^{a}}\right)(V-n b)^{\top}=k$
where
\begin{aligned} & \mathrm{T}=\frac{\mathrm{R}}{\mathrm{C}_{\mathrm{v}}}+1 \\ & \mathrm{k}=\mathrm{nRZ} \end{aligned}
Enthalpy of an ideal gas is independent of pressure at constant temperature and the internal energy of an ideal gas is independent of volume at constant temperature.
 Question 4
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      First Law, Heat, Work and Energy
Question 4 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 5
Consider an isentropic flow of air (ratio of specific heats = 1.4) through a duct as shown in the figure.
The variations in the flow across the cross-section are negligible. The flow conditions at Location 1 are given as follows:

??$P_1=100kPa, \rho _1=1.2kg/m^3,u_1=400m/s$

The duct cross-sectional area at Location 2 is given by $A_2=2A_1$, where$A_1$ denotes the duct cross-sectional area at Location 1. Which one of the given statements about the velocity $u_2$ and pressure $P_2$ at Location 2 is TRUE?

 A $u_2 \lt u_1, P_2 \lt P_1$ B $u_2 \lt u_1, P_2 \gt P_1$ C $u_2 \gt u_1, P_2 \lt P_1$ D $u_2 \gt u_1, P_2 \gt P_1$
GATE ME 2023      Power System
Question 5 Explanation:
Step -1: First identify type of flow - Subsonic or Supersonic by finding out Mach number

Mach no at start of flow
$M_a=\frac{u_1}{C_1},\; where, \; C_1=\sqrt{\gamma RT_1}$
$u_1$ is velocity of gas
$C_1$is velocity of sound

\begin{aligned} P_1 &=\rho _1RT_1 \\ T_1&= \frac{P_1}{\rho _1R}=\frac{100}{1.2 \times 0.287}=290.36K\\ C_1 &=\sqrt{1.4 \times 287 \times 290.36} \\ &= 341.56 m/sec\\ Ma_1 &=\frac{400}{341.56}=1.017 \end{aligned}
Flow is supersonic flow. Hence, diverging duct is nozzle so $u_2 \gt u_1$ and $P_2 \lt P_1$.

There are 5 questions to complete.

### 2 thoughts on “Thermodynamics”

1. Hello,
It would have been good if options weren’t provided for numerical type questions.