physics-class-11-unit-12-chapter-04-carnot-engine-and-carnot-theorem-l-8-9-xn-ytv87w3k

### <Success style="font-size:25px"> Carnot Engine And Carnot Theorem L-8 </Success>
### <primary style="font-size:24px"> Heat Engine </primary>
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- Works in a complete cycle (Reversible)

- $Q_1=Q_2+W $

- $\eta=\frac{W}{Q_1}=\frac{Q_1-Q_2}{Q_1} =1-\frac{Q_2}{Q_1}$

- $Q_2 = 0 \Rightarrow \eta = 1 ? \longrightarrow$ 2nd law

- Can I construct a heat engine with efficiency 1 ? No.

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<footer style = "font-size: 18px"> $\rightarrow$ Carnot Engine and Carnot Theorem $\rightarrow$ <span style = "color:blue"> Heat Engine </span> $\rightarrow$ Refrigerator $\rightarrow$ Refrigerator </footer>

### <Success style="font-size:25px"> Carnot Engine And Carnot Theorem L-8 </Success>
### <primary style="font-size:24px"> Refrigerator </primary>
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- $T_1>T_2 $

- $Q_1=Q_2+W $

- $\phi=\frac{Q_2}{Q_1-Q_2}$

- $Q_1-Q_2=W=0 ? \longrightarrow$  2nd law

- Energy conservation is satisfied.

- Can the COP be infinity ? $NO$

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<footer style = "font-size: 18px">Carnot Engine and Carnot Theorem $\rightarrow$ Heat Engine $\rightarrow$ <span style = "color:blue"> Refrigerator </span> $\rightarrow$ Refrigerator $\rightarrow$ Second Law of Thermodynamics </footer>

### <Success style="font-size:25px"> Carnot Engine And Carnot Theorem L-8 </Success>
### <primary style="font-size:24px"> Refrigerator </primary>
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- Perpetual Motion of the first kind produces work without the input of energy (heat).

- No: First law (conservation of energy) forbids that! (Isolated system).

- Perpetual Motion of the second kind.

- Can we convert the entire heat energy extracted from the hot reservoir to work? $\eta=1$

- Assume cylic process: Change in $U$ is zero

- Then conservation of energy is satisfied.

- No: the second law forbids that.

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<footer style = "font-size: 18px">Heat Engine $\rightarrow$ Refrigerator $\rightarrow$ <span style = "color:blue"> Refrigerator </span> $\rightarrow$ Second Law of Thermodynamics $\rightarrow$ Carnot Engine </footer>

### <Success style="font-size:25px"> Carnot Engine And Carnot Theorem L-8 </Success>
### <primary style="font-size:24px"> Second Law of Thermodynamics </primary>
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- <span style="color:blue">Kelvin-Planck Statement:</span> No (cyclia) process is possible whose sole result is the absorption of heat from a reservoir and the complete conversion of the heat to work.

- $\eta \neq 1$, $\eta<1$

- No heat engine with efficiency one!

- <span style="color:blue">Clausius Statement:</span> No (cyclic) process is possible whose sole result is the transfer of heat from a colder object to a hotter object.

- No refrigerator with $\quad \varphi \rightarrow \infty \Rightarrow W=0$

- These two are equivalent statements!

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<footer style = "font-size: 18px">Refrigerator $\rightarrow$ Refrigerator $\rightarrow$ <span style = "color:blue"> Second Law of Thermodynamics </span> $\rightarrow$ Carnot Engine $\rightarrow$ Nicolas Léonard Sadi Carnot </footer>

### <Success style="font-size:25px"> Carnot Engine And Carnot Theorem L-8 </Success>
### <primary style="font-size:24px"> Nicolas Léonard Sadi Carnot </primary>
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"Nicolas Léonard Sadi Carnot (French: [kasno]; 1 June 1796 - 24 August 1832) was a French military engineer and physicist, often described as the "father of thermodynamics". In his only publication, the 1824 monograph Reflections on the Motive Power of Fire, Carnot gave the first successful theory of the maximum efficiency of heat engines. Carnot's work attracted little attention during his lifetime, but it was later used by Rudolf Clausius and Lord Kelvin to formalize the second law of thermodynamics and define the concept of entropy."

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<footer style = "font-size: 18px">Second Law of Thermodynamics $\rightarrow$ Carnot Engine $\rightarrow$ <span style = "color:blue"> Nicolas Léonard Sadi Carnot </span> $\rightarrow$ The Carnot Engine $\rightarrow$ Carnot Engine </footer>

### <Success style="font-size:25px"> Carnot Engine And Carnot Theorem L-8 </Success>
### <primary style="font-size:24px"> The Carnot Engine </primary>
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- The Carnot Engine: Reversible

- Ideal Gas one mole

- <span style="color:blue">Four processes:</span>

- 1. Isothermal expansion $P_1, V_1, T_1 \rightarrow P_2, V_2, T_1$

- 2. Adiabatic Expansion $P_2, V_2, T_1 \rightarrow P_3, V_3, T_2$

- 3. Isothermal compression $P_3, V_3, T_2 \rightarrow P_4, V_4, T_2$

- 4. Adiabatic Compression $P_4, V_4, T_2 \rightarrow P_1, V_1, T_1$

- closed loop

- These processes can be executed in any order ensuring the initial and final states are the same. We shall calculate the work done and the heat absorbed. The change over a cycle in internal energy is zero.
	
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<footer style = "font-size: 18px">Carnot Engine $\rightarrow$ Nicolas Léonard Sadi Carnot $\rightarrow$ <span style = "color:blue"> The Carnot Engine </span> $\rightarrow$ Carnot Engine $\rightarrow$ The Carnot Engine </footer>

### <Success style="font-size:25px"> Carnot Engine And Carnot Theorem L-8 </Success>
### <primary style="font-size:24px"> The Carnot Engine </primary>
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- Step I: $W_1=R T_1 \ln \frac{V_2}{V_1} ;\quad Q_1=R T_1 \ln \left(\frac{V_2}{V_1}\right)$ (heat absorbed)

- Step II:$W_2=\frac{R\left(T_1-T_2\right)}{\gamma-1} ;\quad Q_2=0$

- $\delta Q=0 $

- $\delta W=-d U$

- $W_3=R T_2 \ln \left(\frac{V_4}{V_3}\right)=-\left(R T_2 \ln \frac{V_3}{V_4}\right) ;\quad Q_2=-R T_2 \ln \left(\frac{V_3}{V_4}\right)$ (releases heat)

- $W_4=\frac{R\left(T_2-T_1\right)}{\gamma-1} ;\quad Q_4=0$

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<footer style = "font-size: 18px">The Carnot Engine $\rightarrow$ Carnot Engine $\rightarrow$ <span style = "color:blue"> The Carnot Engine </span> $\rightarrow$ The Carnot Engine $\rightarrow$ Carnot theorem </footer>

### <Success style="font-size:25px"> Carnot Engine And Carnot Theorem L-8 </Success>
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- $\left(W=Q_1-Q_2\right)$

- $\eta=\frac{Q_1-Q_2}{Q_1}=1-\left(\frac{T_2}{T_1}\right)\left(\frac{\ln \frac{V_3}{V_4}}{\ln \frac{V_2}{V_1}}\right)$

- <span style="color:blue">Adiabatic Path:</span> $ P_A V_A^\gamma=C_A \Rightarrow T_A V_A^{\gamma-1}=C $

- $T_1 V_2^{\gamma-1}=T_2 V_3^{\gamma-1} ; T_1 V_1^{\gamma-1}=T_2 V_4^{\gamma-1}$

- $\frac{V_2}{V_3}=\left(\frac{T_2}{T_1}\right)^{1 /(\gamma-1)} \Longrightarrow \frac{V_1}{V_4}=\left(\frac{T_2}{T_1}\right)^{1 /(\gamma-1)}$

- $\frac{V_3}{V_4}=\frac{V_2}{V_1} \Rightarrow n=1-\frac{T_2}{T_1}$
	
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<footer style = "font-size: 18px">Carnot Engine $\rightarrow$ The Carnot Engine $\rightarrow$ <span style = "color:blue"> The Carnot Engine </span> $\rightarrow$ Carnot theorem $\rightarrow$ Proof of Carnot theorem </footer>

### <Success style="font-size:25px"> Carnot Engine And Carnot Theorem L-8 </Success>
### <primary style="font-size:24px"> Carnot theorem </primary>
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- Given two heat reservoirs a Carnot engine has the maximum efficiency.

- Efficiency of all reversible engines working between two given reservoirs is the same: regardless of the working Substance employed or the Operation details.

- $\eta=1- \frac{T_2}{T_1}$

- Why ideal gas? Easier and Specific heat is independent of temperature.
	
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<footer style = "font-size: 18px">The Carnot Engine $\rightarrow$ The Carnot Engine $\rightarrow$ <span style = "color:blue"> Carnot theorem </span> $\rightarrow$ Proof of Carnot theorem $\rightarrow$ Composite System </footer>

### <Success style="font-size:25px"> Carnot Engine And Carnot Theorem L-8 </Success>
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- Consider a Carnot Engine C which is operated as refrigerator and and an irreversible engine I: both are being operated between a hot reservoir $T_1$ and a cold reservoir $T_2$.

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<footer style = "font-size: 18px">The Carnot Engine $\rightarrow$ Carnot theorem $\rightarrow$ <span style = "color:blue"> Proof of Carnot theorem </span> $\rightarrow$ Composite System $\rightarrow$ Thank You </footer>

### <Success style="font-size:25px"> Carnot Engine And Carnot Theorem L-8 </Success>
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- Assume $W^{\prime}>W$ and look at the composite system

- $W^{\prime}>W$ is it possible?

- In a closed loop:

- No change in the hot reservoir

- Heat absorbed from the cold reservoir:

- $\left(Q_1-W\right)-\left(Q_1-W^{\prime}\right)=W^{\prime}-W>0$

- Net Work $W^{\prime}-W$ an Engine that violates $2^{\text {nd }}$ Law.

- $\eta_c>\eta_I$

- $W>W^{\prime} \Rightarrow \frac{W}{Q_1} >\frac{W^{\prime}}{Q_1} \Rightarrow \eta_c>\eta_I$

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<footer style = "font-size: 18px">Carnot theorem $\rightarrow$ Proof of Carnot theorem $\rightarrow$ <span style = "color:blue"> Composite System </span> $\rightarrow$ Thank You $\rightarrow$  </footer>