Question

Draw P-V diagram for Carnot cycle. Write the name of thermodynamic process carried out by each part of the cycle​

Answer 1

$\mathcal{CARNOT\:\:CYCLE}$

Carnot Engine is an ideal heat engine which is based on Carnot's reversible cycle.

The Carnot cycle consists of the following four stages:

1. Isothermal expansion
2. Adiabatic expansion
3. Isothermal expansion
4. Adiabatic compression

$\implies$Consider one gram mole of an ideal gas enclosed in the cylinder. Let $\sf{V_1,\:P_1,\:T_1}$ be the initial volume, pressure and temperature of the gas. The initial state of the gas is represented by the point A on P-V diagram. Now, the four processes are:

$\mathbb{1.\:ISOTHERMAL\:EXPANSION}$

Since the expansion is happening isothermally, therefore, temperature of the gas remains constant. This operation is represented by the isothermal curve AB. Let the amount of heat energy absorbed in the process be $\sf{Q_1\:and\:W_1}$ be the corresponding amount of work done by the gas in expanding isothermally from $\sf{A\left(V_1,\:P_1\right)}$ to $\sf{B\left(V_2,\:P_2\right)}$

$\sf{\therefore\:Q_1=W_1=\displaystyle\int\limits_{V_1}^{V_2}P\,dV=RT_1\log_e\dfrac{V_2}{V_1}}$

=area ABMKA

$\mathbb{2.\:ADIABATIC\:EXPANSION}$

The gas is allowed to expand further from $\sf{B\left(V_2,\:P_2\right)\:to\:C\left(V_3,\:P_3\right)}$ Temperature of gas falls to $\sf{T_2}$, the expansion is adiabatic and is represented by the adiabatic curve BC. Let $\sf{W_2}$ be the work done by the gas in expanding adiabatically.

$\sf{\therefore\:W_2=\displaystyle\int\limits_{V_2}^{V_3}P\,dV=\dfrac{R\left(T_2-T_1\right)}{1-\gamma}}$

=area BCNMB

$\mathbb{3\:ISOTHERMAL\:COMPRESSION}$

The gas is compressed until its pressure is $\sf{P_4}$ and volume is $\sf{V_4}$. This process is isothermal and is represented by the isothermal curve CD. Let $\sf{Q_2}$ be the amount of heat energy rejected to the sink and $\sf{W_3}$ be the amount of work done on the gas in compressing it isothermally.

$\sf{\therefore\:Q_2=W_3=\displaystyle\:\int\limits_{V_3}^{V_4}-P\,dV=-RT_2\log_e\dfrac{V_3}{V_4}}\\\sf{=RT_2\log_e\dfrac{V_3}{V_4}}$

=-area CDLNC

$\mathbb{4.\:ADIABATIC\:COMPRESSION}$

The gas is compressed to its initial volume $\sf{V_1}$ and pressure $\sf{P_1}$. Let $\sf{W_4}$ be the work done on the gas in compressing it adiabatically.

$\sf{\therefore\:W_4=\displaystyle\int\limits_{V_4}^{V_1}-P\,dV=\dfrac{-R\left(T_2-T_1\right)}{\left(1-\gamma\right)}}$

=-areaDAKLD

Work done by the engine per cycle,

Total work done by the gas=$\sf{W_1+W_2}$

Total work done on the gas=$\sf{W_3+W_4}$

Net work done by the gas in a complete cycle, $\sf{W=W_1+W_2-\left(W_3+W_4\right)}$

$\sf{W_2=W_4}$, in magnitude,

$\sf{\therefore\:W=W_1-W_3}$

$\boxed{\boxed{\sf{W=Q_1-Q_2}}}$

In terms of area,

W= area ABMKA + area BCNMB - area CDLNC - area DAKLD

$\boxed{\boxed{\sf{W=area\:ABCDA}}}$

Hence, in Carnot heat engine, net work done by the gas per cycle is numerically equal to the area of the loop representing the cycle.

Answer 2

Answer:

Explanation:

Carnot Engine is an ideal heat engine which is based on Carnot's reversible cycle.

The Carnot cycle consists of the following four stages:

Isothermal expansion

Adiabatic expansion

Isothermal expansion

Adiabatic compression

Consider one gram mole of an ideal gas enclosed in the cylinder. Let  be the initial volume, pressure and temperature of the gas. The initial state of the gas is represented by the point A on P-V diagram. Now, the four processes are:

Since the expansion is happening isothermally, therefore, temperature of the gas remains constant. This operation is represented by the isothermal curve AB. Let the amount of heat energy absorbed in the process be  be the corresponding amount of work done by the gas in expanding isothermally from  to

=area ABMKA

The gas is allowed to expand further from  Temperature of gas falls to , the expansion is adiabatic and is represented by the adiabatic curve BC. Let  be the work done by the gas in expanding adiabatically.

=area BCNMB

The gas is compressed until its pressure is  and volume is . This process is isothermal and is represented by the isothermal curve CD. Let  be the amount of heat energy rejected to the sink and  be the amount of work done on the gas in compressing it isothermally.

=-area CDLNC

The gas is compressed to its initial volume  and pressure . Let  be the work done on the gas in compressing it adiabatically.

=-areaDAKLD

Work done by the engine per cycle,

Total work done by the gas=

Total work done on the gas=

Net work done by the gas in a complete cycle,

, in magnitude,

In terms of area,

W= area ABMKA + area BCNMB - area CDLNC - area DAKLD

Hence, in Carnot heat engine, net work done by the gas per cycle is numerically equal to the area of the loop representing the cycle.