# Carnot Cycle

## Carnot Cycle

Carnot used a reversible cycle to show the maximum convertibility of heat into work. It is a cyclic process carried out in four reversible steps alternatively isothermal and adiabatic. The first two steps being of expansion and the last two of compression. The engine that works by the above process is known as Carnot engine or ideal gas engine.The working substance of the engine is one mole of an ideal gas.

The engine works between two temperature(T

_{2}) and (T

_{1}). It takes heat from a source at higher temperature (T

_{2}) and does work and gives out the unused heat into a sink at lower temperature (T

_{1}).

**Stroke-1**

*Isothermal Expansion*

The gas is allowed to expand reversibly and isothermally at the temperature T

_{2}so that the volume increases from V

_{1}to V

_{2}.

We know that in isothermal expansion of an ideal gas ΔE = 0. So, from first law of thermodynamics-

ΔE = q - w

or, q = w

Let q

_{2}be the heat absorbed by the system at temperature T

_{2}and work w be the work done by the system on the surroundings.

then- q

_{2}= w

_{1}

= RT

_{2}ln V

_{2}/V

_{1}-----(equation-1)

**Stroke-2**

*Adiabatic Expansion*

The gas is now allowed to expand reversibly and adiabatically from volume V

_{2}to V

_{3}.

Since work has been done by the system adiabatically, where heat is not absorbed. So, the temperature of the system falls from T

_{2}to T

_{1}. Therefore, q = 0. Thus, the 1st law of thermodynamics becomes-

ΔE = - w

or, -ΔE = w

we know that-

C

_{v}= (δE/δT)

_{v}

or, C

_{v}. dT = dE

or, C

_{v}. (T

_{1}- T

_{2}) = -w

or, C

_{v}. (T

_{2}- T

_{1}) = w

Let work done by the system in this step is denoted by w

_{2}then-

w

_{2}= C

_{v}. (T

_{2}- T

_{1}) -----(equation-2)

**Stroke-3**

*Isothermal Compression*

Here gas is subjected to a reversible and isothermal compression at the lower temperature T

_{1}so the volume decreases from V

_{3}to V

_{4}.

In this case work is done on the system and hence heat will be produced and given up to the surroundings. Since compression takes place isothermally and reversibly,

so, ΔE = 0

and if q

_{1}be the heat given out to the surrounding at temperature T

_{1}and work w

_{3}be the work done on the system.

so in this process-

q

_{1}= -w

_{3}

or, -w

_{3}= RT

_{1}ln V

_{4}/V

_{3}-----(equation-3)

**Stroke-4**

*Adiabatic Compression*

Finally by adiabatic and reversible compression, the gas is restored to the original volume V

_{1}and temperature T

_{1}. In this case, work is done on the system. Hence, w is negative. Then, first law of thermodynamics becomes-

ΔE = q - (-w) = q + w

In adiabatic process, q = 0

Hence- ΔE = w = C

_{v}(T

_{2}- T

_{1})

Let w

_{4}be the work done in this stage-

then- w

_{4}= C

_{v}(T

_{2}- T

_{1})

or, -w

_{4}= -C

_{v}(T

_{2}- T

_{1}) -----(equation-4)

where, T

_{2}- T

_{1}is increase in temperature produced by the adiabatic compression.

The net heat absorbed q by an ideal gas in the whole cyclic process is-

q = q

_{2}+ (-q

_{1})

q = RT

_{2}ln V

_{2}/V

_{1}+ RT

_{1}ln V

_{4}/V

_{3}

or, q = RT

_{2}ln V

_{2}/V

_{1}- RT

_{1}ln V

_{3}/V

_{4}-----(equation-5)

According to the expression governing adiabatic changes-

(T

_{2}/T

_{1}) = (V

_{3}/V

_{2})

^{γ-1}---for adiabatic expansion

(T

_{1}/T

_{2}) = (V

_{1}/V

_{4})

^{γ-1}---for adiabatic compression

or, V

_{3}/V

_{2}= V

_{4}/V

_{1}Therefore, substituting the value of V

_{3}/V

_{4}in (equation-5), the Net Heat may be give as-

q = RT

_{2}ln V

_{2}/V

_{1}- RT

_{1}ln V

_{2}/V

_{1}

or, q = R(T

_{2}- T

_{1})ln V

_{2}/V

_{1}-----(equation-6)

Net work done in one cycle-

W = w

_{1}+ w

_{2}+ (-w

_{3}) + (-w

_{4})

W = RT

_{2}ln V

_{2}/V

_{1}+ C

_{v}(T

_{2}- T

_{1}) + (-RT

_{1}ln V

_{4}/V

_{3}) + (-C

_{v}(T

_{2}- T

_{1})

or, W = RT

_{2}ln V

_{2}/V

_{1}+ RT

_{1}ln V

_{3}/V

_{4}-----(equation-7)

Equation-7 is work done in one cycle.

or, q = RT

_{2}ln V

_{2}/V

_{1}- RT

_{1}ln V

_{2}/V

_{1}

or, q = R(T

_{2}- T

_{1})ln V

_{2}/V

_{1}(as V

_{2}/V

_{1}= V

_{3}/V

_{4}) -----(equation-8)

Equation-8 is Net Heat absorbed in one cycle.

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