Zeroth law of thermodynamics - KUET

Md. Kamrul Hasan Reza

Department of Physics

Khulna University of Engineering & Technology Khulna-9203, Bangladesh

Tel.: +880-41-769468~75 Ext. 587(O), 588 (R)

e-mail: mkhreza@phy.kuet.ac.bd, mkhreza1@gmail.com Website : www.kuet.ac.bd/phy/reza/

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Welcome to my Class (trial) Physics Ph 1101

11:40 PM July 15, 2020

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Thermal equilibrium and concept of temperature

Zeroth law of thermodynamics

This law states that if , of three systems, A, B and, C, A and B are separately in thermal, equilibrium with C, then A and B are also in thermal equilibrium with one another.

Conversely the law can be stated as follows :

If three or more systems are in thermal contact, each to each, By means of diathermal walls and are all in thermal "equilibrium together, then any two systems taken separately are in thermal equilibrium with one another.

Consider three fluids A, B and C. Let P_A, V_A represent the pressure and volume of A, P_B, V_B, the pressure and volume of B, and P_C, V_C are the pressure and volume of C.

If A and B are in thermal equilibrium, then

φ₁(P_A, V_A) =φ₂ (P_B, V_B)

or F₁ [P_A, V_A , P_B, V_B ]= 0 ……….(1) Expression (1) can be solved, and P_B= f₁ [P_A, V_A , V_B ] ……….(2)

If B and C are in thermal equilibrium

φ₂ (P_B, V_B) =φ₃ (P_C, V_C) or F₂ [P_B, V_B , P_C, V_C ]= 0

Aiso, P_B = f₂ [V_B , P_C, V_C ] ……….(3)

From equations (2) and (3) for A and C to be equilibrium separately, f₁ [P_A, V_A , V_B ] = f₂ [V_B , P_C, V_C ] ……..(4)

If A and C are in thermal equilibrium with B separately. Then according to the zeroth law, A and C are also in thermal equilibrium with one another.

∴ F₃ [P_A, V_A , P_C, V_C ] = 0 ……..(5)

Equation (4) contains a variable V_B whereas, equation (5) does not contain the variable V_B. It means

φ₁(P_A, V_A) = φ₃ (P_C, V_C) ……….(6)

References:

Heat and Thermodynamics – Brij Lal & N. Subrahmanyam and materials from internet resources

These three functions have the same numerical value through the parameters (P, V) of each are different. This numerical value is termed as temperature (T) of the body.

φ (P, V) = T ...(8)

This is called the equation of state of the fluid.

In general,

φ₁(P_A, V_A) = φ₂ (P_B, V_B) = φ₃ (P_C, V_C) ………(7)

First law of thermodynamics

Joule’s law W = J H ……(1)

The amount of heat given to a system is equal to the sum of internal energy and the external work done

δH = dU + δW …..(2)

For a cyclic process, the change in the internal energy is zero.

Therefore for a cyclic process

and ……….(3)

Joule’s Law

For a system undergoing any number of complete cycles U₂ – U₁ = 0 ……(4)

Then, …..(5)

∴ H = W ……(6) Both are in heat units

First law of thermodynamics for a change in state of a closed system

P

Fig.1: P-V diagram for a cyclic process

According to the first law of thermodynamics

………(1)

Now, consider a cyclic process in which the system changes from state 1 to state 2 along the path A and returns from state 2 to state 1 along the path B. For this cyclic process

Now, consider a cyclic process in which the system changes from state 1 to state 2 along the path A and returns from state 2 to state 1 along the path B. For this cyclic process

Now, consider the second cycle in which the system changes from state 1 to state 2 along the path A and returns from state 2 to state 1 along the path C. For this cyclic process

Now, consider the second cycle in which the system changes from state 1 to state 2 along the path A and returns from state 2 to state 1 along the path C. For this cyclic process

……..(2)

Subtracting equation (2) from (1)

……..(3)

The quantity (δH – δW) depends only on the initial and the final states of the system and is independent of the path followed between the two states

Let dE = (δH – δW)

From the above logic, it can be seen that

This naturally suggests that E is a point function and dE is an exact differential.

The point function E is a property of the system

∴ δH – δW = dE ……(4) or δH = dE + δW ……(5)

Integrating equation (5), from the initial state 1 to the final state 2

1H₂ = (E₂ - E₁) + ₁W₂

Here, ₁H₂ represents the heat transferred

1W₂ represents the work done

E₂ represents the total energy of the system in state 2 E₁ represents the total energy of the system in state 1

E = U + KE + PE + Others which depends upon chemical nature etc.

For a closed system (non chemical) the changes in all others except U are insignificant and

dE = dU From equation (5)

δH = dE + δW …..(6)

Here all the quantities are in consistent units

I Thank you