Thermodynamics and Heat Engine4 (3+1)

LESSON - 4 TEMPERATURE, EQUALITY OF TEMPERATURE, THE ZEROTH LAW OF THERMODYNAMICS AND TEMPERATURE SCALE

4.2. EQUALITY OF TEMPERATURE OR THERMAL EQUILIBRIUM

Any two bodies are said to have ‘equality of temperature’ when no change in any observable property occurs when they are brought in for thermal communication. The equality of temperature is also termed 'thermal equilibrium'.

The following example is illustrated in order to understand the equality of temperature:

Consider two blocks of copper, one hot and the other cold. Each of blocks has a mercury thermometer and an arrangement to measure electric resistance and length of block so that any change in length of mercury column, resistance and length of block could be measured for each block due to temperature change. Let these two blocks of copper are brought in contact for thermal communication.We would then observe that:

• Height of mercury column of thermometer in the hot block would drop and in cold body it would rise at first and after sometime, no further change in height would be observed.

• Electrical resistance of hot block decreases with time and for the cold block it increases with time and after a period of time, no further changes in resistance would be observed.

• Length of a hot block decreases and for the cold block it increases with time, and after a period of time, no further change in length of either of the blocks would be observed.

So we may say that the two blocks of copper have equality of temperature after a period of time when no change in observable properties like height of mercury column, electrical resistance and block length would be observed.

4.3. THE ZEROTH LAW OF THERMODYNAMICS

The Zeroth law of thermodynamics states that when two bodies satisfy thermal equilibrium with a third body individually, they in turn will be in thermal equilibrium with each other.

This law permits us to test the equality of temperature of two bodies without actually bringing them in thermal communication.

The following example is illustrated in order to under stand this:

Consider two copper blocks at the same temperature. Touch one block with a thermometer and keep it there till the height of mercury column becomes stand still. Then touch the second block with the thermometer and if there is no change in height of mercury, we can say the two copper blocks have equality of temperature.

The Zeroth law of thermodynamics is the basis of concept of temperature and it is also the basis of all temperature measurement, for numbers can be placed on the mercury thermometer.

WHY WE NEED STANDARD SCALE FOR TEMPERATURE MEASUREMENTS?

There is a problem of relating the temperatures that we might read on different mercury thermometers, or that we obtain when using different temperature-measuring devices, such as thermocouples and resistance thermometers. This problem suggests the need for a standard scale for temperature measurements.

4.5. TEMPERATURE SCALE

In order to measure the temperature of the system by thermometer, a temperature scale must be provided on the thermometer. The Fahrenheit and the Centigrade are two common temperature scales for measuring temperature.These temperature scales were based on some fixed points.

4.5.1.  Temperature scale based on two fixed points:

Until 1954, each of these scales was based on two fixed points, the ice point and the steam point.

Ice point:  The temperature of a mixture of ice and water which is in equilibrium with saturated air at a pressure of 1 atmosphere. On the Fahrenheit scale and Centigrade scale this point is assigned the numbers, 32 and 0, respectively.

Steam point: The temperature of water and steam which are in equilibrium at a pressure of 1 atmosphere. On the Fahrenheit scale and Centigrade scale this point is assigned the numbers, 212 and 100, respectively.

For simplicity, let us consider that the temperature ‘t’ be some linear function of asd thermometric property, say height L of the mercury column.

Therefore,      t = A .L + B                                                     …………………………………….(4.1)

For Celsius scale

       100 = A .L_s + B

        0     = AL_i + B     

               where,  L_i and L_s are the height  of the mercury column at ice point and steam point respectively. 

From these two equations we can obtain

A =   ;        B = − AL _i =  

By substituting the values of A and B in equation (4.1), we get

 

                                                  ……………………………………..(4.2)

Similarly for Fahrenheit scale we can obtain

                                                      …………………………….(4.3)

 From equations (4.2) and (4.3) we can establish a relation between Fahrenheit and Centigrade scales.

t^oF = (9/5)  t^oC + 32

So any two scales with different numerical values of temperature at two fixed points can be related with each other.   

4.5.2.  Temperature scale based on one fixed point:

At the Tenth Conference on Weights and Measures in 1954, the Centigrade (or Celsius) scale was redefined in terms of a single fixed point and the ideal-gas temperature scale.

The single fixed point is the triple point of water (The state in which solid, liquid, and vapor of a pure substance can co-exist in equilibrium).The triple point of water is assigned the value of 0.01 °C.

The Ideal-gas temperature scale: The magnitude of the degree is defined in terms of the ideal-gas temperature scale which is discussed below:

Both constant-pressure and constant-volume gas thermometers have been used for ideal-gas temperature scale. But pressure measurement at constant volume is easier than the measurement of volume at constant pressure. Hence, the constant volume gas thermometer is more commonly used.

4.5.2.1.  Constant-volume gas thermometer:

Fig. 4.1 shows a constant volume gas thermometer. It consists of a glass bulb with capillary tube having a definite mass of gas. A mercury filled U-shaped transparent flexible rubber pipe is connected to the capillary tube. The pressure on the gas (P_gas) is due to the ‘L’ difference in height of mercury column in U-tube together with atmospheric pressure acting on the open end of the flexible pipe i.e. P_gas = ρ.g.L + p_atm. The open end of the flexible pipe can be raised or lowered to keep the volume of the gas constant.

Let the gas bulb be placed in the system where the temperature ‘T’ is to be measured as shown in Fig. 4.1 (b). Depending on the system temperature, the gas in the gas bulb will expand and this results in increase in gas volume and change in mercury level in the right limb of U-tube from reference mark ‘A’ to ‘A₁’. Now, let the mercury column be so adjusted by raising the open end of the flexible pipe that the level of mercury stands at the reference mark ‘A’ as shown in Fig 4.1 (c). Thus the volume of gas in the gas bulb is maintained constant. At this point, let the pressure of the gas be P which is calculated based on new difference in height of mercury column ‘L₁’ of U-tube.

Let a similar measurement of pressure be made when the gas bulb is placed in the location where the triple point of water (T_t.p. = 273.16 K) is maintained. For triple point, let the pressure of the gas be P_t.p..

                         Fig. 4.1. Schematic diagram of a constant-volume gas thermometer.

Since for an ideal gas T varies as p, we have

                 

or     

By using the above relation, the unknown temperature T of a system could be determined from pressure measurement, P. The temperature so measured is referred to as the ideal-gas temperature.

On this scale, when the gas bulb is placed in the system having steam point, the steam point is experimentally found to be 100.0 °C. Thus, there is an essential agreement between the old and new temperature scales.

4.5.2.2.  Constant-pressure gas thermometer

A constant pressure thermometer also can be used to measure the temperature. In this case, while measuring the unknown temperature, the pressure on gas in the gas bulb is maintained constant and change in the volume of the system at unknown temperature ‘T’ and triple point are measured. By using the measured volumes in the following relation the unknown temperature of the system is thus obtained.

              T = 273.16

Here, V_t.p. is the volume of the gas at the triple point of water and V is the volume of the gas at unknown temperature T of a system.

Problem 4.1: Define a new temperature scale, say ^oN in which the boiling and freezing points of water (steam and ice points) are 400 and 100^oN respectively, and correlate this with centigrade scale.

Solution:

Given: Boiling point temperature = 400°N ; 

             Ice point temperature = 100°N

(i)   To determine ^oN temperature scale:

        Let the temperature t be some linear function of some observable property L. Ice point and steam point be represented by i and s, then

t = A.L + B                                                                                          ………………………………….(4.4)

For steam point of °N scale,     400 = A.L_s + B

For ice point of °N scale,          100 = AL_i + B

where  L_i and L_s are the height  of the mercury column at ice point and steam point be respectively. 

From these two equations, we have

       

Putting values of A and B in equation (4.4)

Therefore t ^oN = 

Answer:    t^oN =                                                                           ………………….. (4.5)

 

(ii)  To determine correlation between °N and °C:

For centigrade scale (see equation 4.2, already derived under section “Temperature scale based on two fixed points” of this Lesson), we have

                   t^oC =                                                                              ………………….. (4.6)

From equations (4.5) and (4.6), we have

Answer:    t ^oN = 3t ^oC + 100