Corollaries (conclusion) of 1st Law of Thermodynamics

Corollaries (conclusion) of 1st Law of Thermodynamics are as follows -

1. Internal energy is the property of the system.

2. Heat is a path function.

3. Internal energy of an isolated system remains constant.

4. Perpetual motion machine of 1st kind is impossible.

1. Internal energy is the property of the system.

Let us consider a process which goes from state 1 to state 2 by following path A.

Let the system again comes to its initial position either by following path B or path C. Therefore we have two cycles 1-ᴀ-2-ʙ-1 and 1-ᴀ-2-c-1.

Fig. P-V diagram

Considering cycle 1-ᴀ-2-ʙ-1

For process 1-ᴀ-2

Q1-ᴀ-2 = dU1-ᴀ-2 + W1-ᴀ-2

Q1-ᴀ-2 - W1-ᴀ-2 = dU1-ᴀ-2 ---(1)

Similarly for process 2-ʙ-1

Q2-ʙ-1 – W2-ʙ-1 = dU2-ʙ-1 --(2)

For cycle 1-ᴀ-2-ʙ-1 from 1st Law of Thermodynamics

ƩQ = ƩW

⇒ Q1-ᴀ-2 + Q2-ʙ-1 = W1-ᴀ-2 + W2-ʙ-1

⇒ Q1-ᴀ-2 - W1-ᴀ2 = W2-ʙ-1 - Q2-ʙ-1

⇒ dU1-ᴀ-2 = - dU2-ʙ-1 -----(3)

Now considering cycle 1-ᴀ-2-c-1

For process 1-ᴀ-2

Q1-ᴀ-2 = dU1-ᴀ-2 + W1-ᴀ-2

Q1-ᴀ-2 - W1-ᴀ-2 = dU1-ᴀ-2 --(4)

For process 2-c-1

Q2-c-1 – W2-c-1 = dU2-c-1 --(5)

For cycle 1-ᴀ-2-c-1 from 1st Law of Thermodynamics

ƩQ = ƩW

⇒ Q1-ᴀ-2 + Q2-c-1 = W1-ᴀ-2 + W2-c-1

⇒ Q1-ᴀ-2 - W1-ᴀ-2 = W2-c-1 - Q2-c-1

⇒ dU1-ᴀ-2 = - dU2-c-1 -----(6)

From equation (3) and (6)

- dU2-ʙ-1 = - dU2-c-1

dU2-ʙ-1 = dU2-c-1 -----(7)

From eq. (7) it is clear that change in internal energy when the system goes from point 2-1 remains same irrespective of the path followed during change of state, hence it is a point function and property of system.

2. Heat is a path function

Let us consider a process which goes from state 1-2 by following path A.

Let the system again comes to it initial position either by following path B or path C. Therefore we have two cycles 1-ᴀ-2-ʙ-1 and 1-ᴀ-2-c-1.

Fig. P-V diagram

For cycle 1-ᴀ-2-ʙ-1 from 1st Law of Thermodynamics

ƩQ = ƩW

⇒ Q1-ᴀ-2 + Q2--1 = W1-ᴀ-2 + W2-ʙ-1

⇒ Q1-ᴀ-2 - W1-ᴀ-2 = W2-ʙ-1 - Q2-ʙ-1 -----(1)

For cycle 1-ᴀ-2-c-1 from 1st Law of Thermodynamics

ƩQ = ƩW

⇒ Q1-ᴀ-2 + Q2-c-1 = W1-ᴀ-2 + W2-c-1

⇒ Q1-ᴀ-2 - W1-ᴀ-2 = W2-c-1 - Q2-c-1 -----(2)

From equation (1) and (2)

W2-ʙ-1 - Q2-ʙ-1 = W2-c-1 - Q2-c-1

Q2-c-1 - Q2-ʙ-1 = W2-c-1 - W2-ʙ-1 -----(3)

As we know that work is a path function therefore although all the terms given in eq. (3) are working under same initial and final condition but there will be some definite value of W2-c-1 - W2-ʙ-1 . Hence to deal with this eq. Q2-c-1 - Q2-ʙ-1 will also have the same definite value. Hence we can say that heat is not a property of system and therefore is a path function.

3. Internal energy of an isolate system remains constant

We know that an isolated system is a system in which there is not energy interaction in the form of heat and work.

From 1st Law of Thermodynamics for any system undergoing any process

Q1-2 = dU + W1-2 -----(1)

In case of isolated system

W­1-2 = 0

Q1-2 = 0

Therefore dU1-2 = 0

⇒ U2 – U1 = 0

⇒ U2 = U1 -----(2)

4. Perpetual Motion Machine of 1st kind ( PMM-1) is impossible

PMM-1 is a hypothetical machine which produces work in a cyclic process without consuming any heat.

As per 1st Law of Thermodynamics for any system undergoing cycle

ƩQ = ƩW

Q = 0 →PMM-1 →W = some value

For PMM-1

ƩW = some value

ƩQ = 0

⇒ in case of PMM-1

ƩQ ≠ ƩW

Which violates the 1st Law of Thermodynamics hence we can say that PMM-1 is impossible or in other words we cannot create any machine which continuously produces work without taking any energy.