Second Law Of Thermodynamics

First And Second Law Of Thermodynamics The Secondary Of Thermodynamics

The 2nd Law Of Thermodynamics

Let us consider a man holding a glass of hot tea. From our experience, we know that in such a case the hand gradually becomes hotter and the glass cooler. This is because some heat (say, 10 calories per second) is absorbed by the hand from the glass.

Second Law of Thermodynamics Explained

• Now, consider the opposite process. If 10 calories of heat is given by the cold hand to the hot glass per second, the energy would still be conserved, i.e., the first law of thermodynamics would still be obeyed. But, this opposite process never occurs in nature.
• In general, there is a natural direction in every real process. The first law of thermodynamics cannot determine this natural direction. So it is important to formulate a new law—the second law of thermodynamics.
• Scientists expressed the second law in different forms. However, all of the different forms are equivalent. They provide alternative statements of the same physical law.

Clausius and Kelvin Statements of the Second Law

Clausius’s statement: No self-acting machine can transfer heat from a lower to a higher temperature.

Kelvin-Flanck Statement: No self-acting machine can convert some amount of heat entirely into work.

Entropy: The glass and the hand together, is now regarded as a closed system. Energy can be transmitted from one part to another or can be transformed from one form to another in a closed system. But this type of process is irreversible. The direction of an irreversible process is determined by a change in a special property, called the entropy of the system.

The analysis of a reversible process, using the second law of thermodynamics, leads to the concept of entropy. Entropy, denoted by the letter S, is a property of all thermodynamic systems. It is defined by the relation

ds = \(\frac{dQ}{T}\)….(1)

where, dQ = heat exchange of a system in an infinitesimal reversible process at temperature T and dS = corresponding, change in entropy of the system.

From (1), dQ = TdS. Using this relation in the first law of thermodynamics, we get, dQ = dU+dW or, TdS = dU+pdV ….(2)

Now, we note that dQ and dW are quantities exchanged between the die system and the surroundings m a process, o thev depend on whether the process is reversible or irreversible.

But equation (2) does not contain any such exchange quarantine. So it is true for reversible as well as irreversible processes. Then, if we use equation (2) in thermodynamics, we need not worry about the nature of the process. This is the beauty of the ideal concept of reversibility.

The entropy principle: Thermodynamic analysis shows that in every real process in nature, the sum of the entropies of a system and its surroundings always increases. The opposite process, in which the sum of the entropies decreases, is not allowed in nature.

• We may compare the situation with the law of conservation of energy (first law of thermodynamics). This law states that the total energy of the universe is a constant—it can never increase or decrease. In analogy, the second law of.thermo¬dynamics states that the total entropy of the universe increases in every process it can never decrease.
• Every real process in nature occurs in such a direction that the total entropy of the universe increases. Alternatively, no process, in which the total entropy of the universe decreases, can occur in nature. This is known as the principle of increase of entropy.
• For a reversible process the total entropy of the universe remains constant while for an irreversible process. the entropy of the universe increases. As all natural processes in general are irreversible, every natural process results in an increase in entropy of the universe.

Entropy

This principle of increase of entropy is the most general statement of the second law of thermodynamics. The Clausius and the Kelvin-Planck statements can easily be derived from this principle.

In an adiabatic process, dQ = 0 . Then equation (1) gives that dS = 0 or, S = constant. So the entropy of a system remains constant in a reversible adiabatic process (just like temperature in an isothermal process). For this reason, a reversible adiabatic process is called an isentropic process.

In essence, each of the three laws of thermodynamics defines one important property of all thermodynamic systems:

1. Zeroth law: Temperature (T)
2. First law: Internal energy (U)
3. Second law: Entropy (S)

Unit 8 Thermodynamics Chapter 1 First And Second Law Of Thermodynamics Heat Engines And Refrigerators

Applications of the Second Law of Thermodynamics

Heat reservoir: A body whose temperature remains constant even when heat is gained or lost by it is called a heat reservoir. Every heat reservoir has its own characteristic temperature.

Generally, different heat reservoirs have different characteristic temperatures. They are drawn in the way as shown. The characteristic temperatures of the two reservoirs shown and T₁ and T₂ respectively.

• For example, the atmosphere or sea water may be taken as heat reservoir. If a burning oven is placed in the air, it rejects heat to the atmosphere. Or if a large block of ice is placed in the air, it receives heat from the atmosphere.
• From our daily experiences, it is known that we can ignore the change of temperature of the atmosphere in the above cases. Similarly, if a bucketful of boiling water is poured in the sea, sea water takes heat but its temperature does not change.
• It is obvious that the atmosphere and sea, being very large in size, behave as heat reservoirs. However, comparatively smaller bodies also may show similar properties.
• Suppose, the temperature of a coaloven when burning is 300°C. It continuously rejects heat to the surroundings. In spite of that, as long as coal burns, the temperature of the oven remains constant at 300°C. So in this case the oven acts as a heat reservoir.

We know, if dT is the change of temperature of a body due to a heat exchange of dQ, then the thermal capacity of the body, C = \(\frac{dQ}{dT}\).

In case of a heat reservoir, dT =0 so whatever the value of heat gain or heat loss (dQ) may be, C → ∞, i.e., the thermal capacity of any heat reservoir is infinite.

Conversely, it can be said that if the thermal capacity of a body is infinite, the body will behave as a heat reservoir.

Heat engine: It is a mechanical device that converts heat into work.

• In general, a heat engine H takes heat from a source at a higher temperature converts a part of it into work, and gives out the rest to a body (sink) at lower temperature.
• In most cases, the source at a higher temperature and the sink at a lower temperature are two heat reservoirs, i.e., in a complete cycle their temperatures are fixed at T₁ and T₂. Obviously, T₁>T₂.
• Generally, a heat engine works in a cyclic process. It means that, after the completion of a cycle by converting heat into work, the working substance of the engine returns to its initial condition and becomes ready for the next cycle.
• The action of each cycle is equivalent, i.e., in each cycle the same amount of heat is converted into the same amount of work.

Efficiency of a heat engine: The object of a heat engine is to convert heat taken from the source into useful work. So the heat taken from the source is called the input, and the transformed work is called the output.

The efficiency of a heat engine is defined as the fraction of total heat taken from the source which is converted into work. Suppose in each cycle,

the heat is taken by the engine from the source at temperature T₁ = Q_1;

transformed work = W;

heat rejected by the engine to its surroundings (sink) at temperature T₂ = Q₂.

From the principle of conservation of energy, we can write, Q₁ = W+Q₂ or, W = Q₁-Q₂

So, efficiency, \(\eta=\frac{\text { output }}{\text { input }}=\frac{W}{Q_1}=\frac{Q_1-Q_2}{Q_1}\)

i.e., \(\eta=1-\frac{Q_2}{Q_1}\)…(1)

From equation (1) we get η ≤ 1, i.e., the efficiency of a heat engine can never be greater than 1 or 100%.

Class 11 Physics	Class 12 Maths	Class 11 Chemistry
NEET Foundation	Class 12 Physics	NEET Physics

Second Law of Thermodynamics in Heat Engines

Ideal heat engine: it is easily understood that the more a heat engine converts heat into work the more its efficiency will be. If any engine converts the whole amount of heat taken from the source into work, then the engine is called an ideal heat engine.

It is evident that in case of this type of engine, the heat is rejected to its surroundings (Q₂)zero. So the work obtained will be equal to the heat taken from the source (Q₁). Therefore, the efficiency of an ideal heat engine,

⇒ \(\eta=\frac{W}{Q_1}=\frac{Q_1}{Q_1}=1=100 \%\)

Alternatively, \(\eta=1-\frac{Q_2}{Q_1}=1-\frac{0}{Q_1}=1=100 \%\)

Kelvin-Planck’s statement of the second law of thermodynamics: No self-acting machine can convert some amount of heat entirely into work. On the basis of the discussion about heat engines, this statement can be expressed in the form of an easy alternative: An ideal heat engine does not exist in nature.

Refrigerator: A mechanical device that transfers heat from a colder to a hotter place is called a refrigerator and the working substance of a refrigerator is called a refrigerant.

• Suppose, a cold body, say an icebox is at a temperature of T₂. The surrounding temperature T₁ in most cases is greater than T₂ So the temperature of the cold body begins to increase due to receiving heat from the surroundings.
• Now if any mechanical device removes heat at the same rate at which the cold body receives heat, the temperature of the body will remain constant i.e., the body will remain in the same cold state. This mechanical arrangement is called a refrigerator, denoted by R in.
• Generally, with the help of some external work, the refrigerator transfers heat from the lower temperature T₂ to the higher temperature T₁ of the surroundings. Our household refrigerator is a familial example of this machine.

Like a heat engine, a refrigerator also works in a cyclic process. At the end of each cycle, the working substance returns to its initial state, and then the next cycle starts.

Coefficient of performance of a refrigerator: The aim of a refrigerator is to extract heat from a cold body at the expense of some external mechanical work. The external supplied work is known as the input and the heat extracted from the cold body is called the output.

Suppose, in each complete cycle, heat received by the refrigerator from the colder body at temperature T₂ = Q₂;

heat delivered by the refrigerator to the surroundings at higher temperatures T₁ = Q₁;

external work done = W

From the principle of conservation of energy, we can write,

W + Q₂ = Q₁

or, W = Q₁-Q₂

From the definition of the coefficient of performance (e) of a refrigerator, we have,

∴ e = \(\frac{\text { output }}{\text { input }}=\frac{Q_2}{W}=\frac{Q_2}{Q_1-Q_2}\)….(2)

Ideal refrigerator: Obviously, the less the amount of external work supplied to a refrigerator to run it, the better is its performance. If any refrigerator can transfer heat from low temperature to a higher temperature without any help of external work, then it is called an ideal refrigerator. For example, if any household refrigerator could run without any assistance of electricity, then that would be an ideal refrigerator.

For this ideal refrigerator, W = 0, i.e., Q₁ – Q₂ = 0 or, Q₁ = Q₂

So, coeffieicient of performance of an ideal refrigerator, e = \(\frac{Q_2}{W}=\frac{Q_2}{0} \rightarrow \infty\)

i.e., the coefficient of performance of an ideal refrigerator is infinite.

Clausius’s statement of the second law of thermodynamics: No self-acting machine can transfer heat from a lower to a higher temperature. On the basis of the discussions about refrigerators, this statement can be expressed in the form of an easy alternative: An ideal refrigerator does not exist in nature.

Refrigerator Discussions:

1. Comparing It is apparent that the working principles of a heat engine and a rein aerator are exactly opposite to each other. But it does not mean that these two machines are used for opposite purposes.
   • A heat engine is a machine for conversion of heat into work, but a refrigerator is not a machine for conversion of work into heat.
   • On the omer nana. I refrigerator is a machine that transfers heat from a low temperature to a higher temperature, but heat engines are never used to transfer heat from a higher to a lower temperature.
2. Again, on comparing it is found that if it is possible to conduct the different operations of a heat engine (ie.. heat intake from higher temperature.  transformation of heat into work, and rejection or heat at low temperature) in the reverse direction, then it will act as a refrigerator.
   • The necessary condition for this is that each operation should have to be a reversible process. Le.. the engine should have to be a reversible heat engine. In that case, the refrigerator running in the opposite direction will be reversible.
3. A reversible process is nothing but an ideal process. In nature this process does not exist—all real processes are irreversible. So, there is no opportunity to use a heat engine as a refrigerator, and vice versa.

First And Second Law Of Thermodynamics Useful Relations For Using Solving Problems

If W = work done and H = corresponding heat produced, then from Joule’s law, W ∝ H or, W = JH

where J = constant = mechanical equivalent of heat = 4.2 J · cal^-1 = 4.2 x 10⁷ erg  · cal^-1.

1 cal = 4.2 J = 4.2 x 10⁷ erg.

According to the first law of thermodynamics, heat taken by the system from the surroundings = change in internal energy + external work done

Q = (U_f – U_i) + W (integral form)

dQ = dU+ dW (differential form)

Infinitesimal work done by a system : dW = pdV

Work done in a finite process, W = ∫dW = ∫pdf

• c_v = specific heat of a substance at constant volume. The heat taken in a process is Q = \(m c_v t\), where m = mass of the substance and t = increase in temperature when the volume remains constant.
• Molar specific heat at constant volume is Cv = \(m c_v\), where M = molecular weight of the substance.
• c_p = specific heat of a substance at constant pressure. The heat taken in a process is Q = mc_pt, where m = mass of the substance and t = increase in temperature when the pressure remains constant.
• Molar specific heat at constant pressure is \(C_p=M c_p\), where M molecular weight of the substance.
• For an ideal gas, the difference between the molar specific heat is C_p – C_v = R
• If 1g is taken instead of 1 mol, then \(c_p-c_v=\frac{R}{M}\) m = molecular weight.

The ratio between the two specific heats is \(\gamma=\frac{C_p}{C_y}\) Also, C_p > C_v, γ> 1

For an isothermal process of n mol of an ideal gas, Q = \(W=n R T \ln \frac{V_f}{V_i}=n R T \ln \frac{p_i}{p_f}\)

In an adiabatic process of an ideal gas, p, V, and T are related as \(p V^\gamma\)=constant, \(T V^{\gamma-1}\)=constant, \(T^\gamma p^{1-\gamma}=\text { constant } \text {. }\) = constant.

Adiabatic work done for n mol of an ideal gas is W = \(n C_v\left(T_i-T_f\right)=\frac{n R}{\gamma-1}\left(T_i-T_f\right)=\frac{p_i V_i-p_f V_f}{\gamma-1}\)

Efficiency of a heat engine, \(\eta=1-\frac{Q_2}{Q_1}\)

where Q₁ =heat taken by the engine from the source at temperature T₁

and Q₂ =heat rejected by the engine to its surroundings (sink) at temperature T₂

Coefficient of performance of a refrigerator, e = \(\frac{Q_2}{Q_1-Q_2}\)

where Q₂ = heat received by the refrigerator from the colder body at temperature T₂,

and Q₁ = heat delivered by the refrigerator to the surroundings at higher temperatures T₁

The efficiency of a Carnot engine using an ideal gas. \(\eta=1-\frac{T_2}{T_1}\)

where T₁ = temperature of the source and T₂ = temperature of the sink

In the case of a Carnot refrigerator, work done and heat exchange are equal and opposite to the corresponding quantities of a Carnot engine. So, in this case, the coefficient of performance of a refrigerator,

∴ e = \(\frac{T_2}{T_1-T_2}\)