JEE Important Chapter - Thermodynamics

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Name of the Concept

Key Points of the Concepts

1. Thermodynamics Process and Example of Thermodynamics

• When certain changes occur in the state of a thermodynamic system, i.e., the thermodynamic parameters of the system change over time, this is referred to as a thermodynamic process.

• Isothermal change is a change in the pressure ($P$) and volume ($V$) of gas without any change in its temperature. It is given as,

$PV= \text{constant}$

• When no heat is allowed to enter or depart from a gas, it undergoes an adiabatic change in pressure ($P$) and volume ($V$). It is given as

$PV^\gamma=\text{constant}$

• Melting ice and sitting in a crowded room are two real-life illustrations of thermodynamics.

2. Work done in Isothermal and Adiabatic Expansion

• Consider one gram mole of an ideal gas contained in a cylinder with flawlessly conducting walls and a completely frictionless and conducting piston. Let $P_1, V_1$ and $P_2, V_2$ be the initial and final pressure and volume of the gas and $T$ be its temperature then work done under isothermal expansion is given as,

$W=2.303 RT \log_{10}\dfrac{V_2}{V_1}$

• Consider a gram mole of an ideal gas confined in a cylinder with flawlessly nonconductive walls and a perfectly frictionless, nonconducting piston. Let $P_1, V_1, T_1$ and $P_2, V_2, T_2$ be the initial and final pressure, volume and temperature of the gas then work done under adiabatic expansion is given as,

$W=R\dfrac{(T_2-T_1)}{(1-\gamma)}$

3. First Law of thermodynamics and its Limitations

• According to this law, the energy ($dQ$) supplied to a system increases partly the internal energy of the system ($dU$) and the rest is spent on doing work ($dW$) on the environment. It is written as,

$dQ=dU+dW$

• There are three limitations of this law;

1. This law doesn’t indicate the direction in which the change can occur.

2. This law doesn’t give any idea about the extent of change.

3. This law gives no information about the source of heat i.e, whether it is hot or cold.

4. Cyclic Process and Non-Cyclic Process

• A cyclic process consists of a series of changes which return the system back to its initial state while a non-cyclic process consists of a series of changes which don’t return the system back to its initial state.

5. Heat Engine

• A heat engine is a mechanical device that transforms heat energy into mechanical energy. It consists of a source of heat at a higher temperature, a working substance and a sink of heat at a lower temperature.

• The thermal efficiency ($\eta$) of a heat engine is defined as the ratio of work done per cycle by the engine to the total amount of heat absorbed per cycle by the working substance from the source. It is given as,

$\eta=1-\dfrac{Q_1}{Q_2}$

Here, $Q_1$ = the amount of heat absorbed by a working substance from the source and $Q_2$=amount of heat rejected to the sink.

6. Refrigerator or Heat Pump 

• A heat pump is a device that is used for cooling things. It can be regarded as a heat engine working in the reverse direction.

• Thus in a refrigerator, the working substance would absorb a certain quantity of heat ($Q_2$) from the sink at a lower temperature ($T_2$) and reject a large amount of heat ($Q_1$) to the source at a higher temperature($T_1$).

• The coefficient of performance ($\beta$) of a refrigerator is defined as the ratio of the quantity of heat removed per cycle from the contents of the refrigerator ($Q_2$) to the energy spent per cycle ($W$) to remove this heat. It is given as,

$\beta=\dfrac{Q_2}{W}=\dfrac{Q_2}{Q_1-Q_2}$

7. Second Law of thermodynamics

• This law contains two important statements which are:

1. Kelvin Planck Statement: It is possible to construct a heat engine which would absorb heat from a reservoir and convert 100% of the heat absorbed into work.

2. Clausius Statement: It is impossible to design a self-acting machine unaided by any external agency, which would transfer heat from a body at a lower temperature to another body at a higher temperature.

8. Reversible and Irreversible processes

• A thermodynamic process taking a system from the initial state (i) to the final state (f) is reversible, if the process can be reversed, the system and its surroundings will revert to their previous conditions, with no additional changes occurring anywhere else in the universe.

• The following conditions have to be satisfied for a process to be reversible:

1. The process should proceed at an extremely slow rate i.e, the process is quasi-static so that the system is in equilibrium with its surroundings at every state.

2. The system should be free from dissipative forces like friction, inelasticity, viscosity etc.

• A process which doesn’t satisfy even one of the conditions for a reversible process is called an irreversible process.

9. Third Law of Thermodynamics

• According to this law, the entropy of a flawless crystal at zero Kelvin (absolute zero) is equal to zero.

10. Carnot Engine

• A Carnot engine is based upon the Carnot cycle which is explained below.

• Consider a cylinder containing one gram mole of an ideal gas. The initial volume, pressure, and temperature of the gas are $P_1,V_1,T_1$. The initial stage is represented by the point A on P-V diagram as shown in the diagram below:

• The Carnot cycle consists of the following four stages;

1. Isothermal expansion: Let the cylinder be placed on the source and the gas be allowed to expand by slow outward motion of the piston. Since it absorbs the required amount of heat from the source through the conducting base of the cylinder hence temperature remains constant. This operation is called isothermal expansion and is represented by isothermal curve AB in the diagram. Let the amount of heat absorbed be $Q_1$ and $W_1$ be the corresponding work done which is written as,

$Q_1=W_1=RT_1\log_{e}\dfrac{V_2}{V_1}=\text{area ABMKA}$

2. Adiabatic expansion: The cylinder is now removed from the source and placed on a perfectly insulating pad. The gas is allowed to expand further from B($V_2,P_2$) to C($V_3,P_3$). The temperature of the gas falls to $T_2$, the expansion is adiabatic and is represented by curve BC. Let $W_2$ be the work done by the gas in expanding adiabatically from B($V_2,P_2$) to C($V_3,P_3$) is,

$W_2=R\dfrac{(T_2-T_1)}{(1-\gamma)}=\text{area BCNMB}$

3. Isothermal compression: The cylinder is now removed from the insulating pad and is placed on the sink at a temperature $T_2$. The piston is moved inwards slowly so that gas is compressed until its pressure is $P_4$ and volume is $V_4$. This process is isothermal and represented by CD. Let $Q_2$ be the amount of heat energy required to sink and $W_3$ be the amount of work done on the gas in compressing it isothermally from a state C($V_3,P_3$) to D($V_4,P_4$) is given as,

$Q_2=W_3=-RT_2\log_{e}\dfrac{V_4}{V_3}=-\text{area CDLNC}$

4. Adiabatic Compression: The cylinder is put on the insulating pad once more. The piston is further moved downwards so that the gas is compressed to its initial volume $V_1$ and pressure $P_1$. This is an adiabatic change and represented by DA. Let $W_4$ be the work done on the gas in compressing it adiabatically from D($V_4, P_4$) to the state A($V_1,P_1$) is given as,

$W_4=-R\dfrac{(T_2-T_1)}{(1-\gamma)}=\text{area DAKLD}$

• Now the total work done($W$) is,

$W=W_1-W_3= \text{area ABCDA}$

11. Efficiency of Carnot Engine

• The efficiency ($\eta$) of a Carnot engine is defined as the ratio of net mechanical work done ($W$) per cycle by the gas to the amount of heat energy absorbed per cycle from the source($Q_1$). It is given as,

$\eta=\dfrac{W}{Q_1} =1-\dfrac{T_2}{T_1}$

S.No.

Name of the Concept

Formula

Thermodynamics equations for Isothermal and adiabatic change

• Isothermal change is given as,

$PV=\text{ constant}$

• Adiabatic change is given as,

$PV^\gamma=\text{constant}$

Work done in Isothermal and Adiabatic Expansion

• Work done under isothermal expansion is given as,

$W=2.303 RT \log_{10}\dfrac{V_2}{V_1}$

• Work done under adiabatic expansion is given as,

$W=R\dfrac{(T_2-T_1)}{(1-\gamma)}$

First Law of thermodynamics

• According to first law, we can write;

$dQ=dU+dW$

Heat Engine

• The efficiency of the heat engine is given as,

$\eta=1-\dfrac{Q_1}{Q_2}$

Refrigerator or Heat Pump

• The coefficient of performance of a refrigerator can be written as,

$\beta=\dfrac{Q_2}{W}=\dfrac{Q_2}{Q_1-Q_2}$

• Relation between coefficient of performance of a refrigerator and efficiency of heat engine is given as,

$\beta=\dfrac{1-\eta}{\eta}$

Carnot Engine

• Work done during isothermal expansion is,

$W_1=RT_1\log_{e}\dfrac{V_2}{V_1}$

• Work done during adiabatic expansion is,

$W_2=R\dfrac{(T_2-T_1)}{(1-\gamma)}$

• Work done during isothermal compression is,

$W_3=-RT_2\log_{e}\dfrac{V_4}{V_3}$

• Work done during adiabatic compression is,

$W_4=-R\dfrac{(T_2-T_1)}{(1-\gamma)}$

• Total work done by the engine per cycle is,

$W=W_1-W_3=RT_1\log_{e}\dfrac{V_2}{V_1}-RT_2\log_{e}\dfrac{V_4}{V_3}$

Efficiency of Carnot Engine

• The efficiency ($\eta$) of a Carnot engine is given as,

$\eta=\dfrac{W}{Q_1} =1-\dfrac{T_2}{T_1}$

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