# Temperature and Its Measurement

The zeroth law of thermodynCimics is the basis of temperature measurement. In order to obtain a quantitative measure of temperature, a reference body of known physical characteristic which changes with temperature is choosen. the choosen characteristic is called thermometric property and the reference body is called thermometer. In common thermometer, expansion of mercury is used as thermometric property. Table 2.1 shows the types of thermometer with its thermometric property.

Table 2.1

S. No               Thermometer               Thermometric property                       Symbol

1.                     Mercury-in-glass                     Length                                      L

2.                     Constant volume gas               Pressure                                      p

3 .                    Electrical resistance                 Resistance                                R

4.            Constant pressure gas                         Volume                                     V

5.                     Thermocouple                        Thermal Electromotiveforce    E

2.2.1 Empirical Relations between Temperature and Thermometric Property

If xis the thermometric property and temperature is the linear fw1Ction (direct proportionality) of x. So, mathematically

f(x) = nx

where a is an arbitrary constant.

If x1 corresponds to f (x1), then x2 can be calculated as follows.

f(x1) = ax1

a = f (x) / x1 inserting this in equation  (2 .1 ) we get

f (x) = f (x) / x1 .x

x = x2 we get from (2.2)

f (x2) = f (x1) / x1 . x2

f (x2) / f (x1) = x2 / x1

Equation (2.3) shows that two temperatures on the linear x scale are to each other as the ratio of the corresponding readings i.e., x2 and x1.

2.2.2 Method Used Before 1954

Let                               f(x) = ax + b                                                   … (2.4)

The thermometer is first placed in contact with a standard system and measures the temperature of it. Let x1 is the reCiding, so,

f (x1) = ax1 + b

Then the thermometer is placed with an another standard system and measures the temperature of it and let x2 is the reading , so,

f (x2)  = ax2  +b

Solving (2.5) and (2.6) for a and b,

F(x1) – (x2) = a (x1 –x2)

A = f (x1) – f (x2) /  x1 –x2

(2.5)  => b = f (x1) – ax1

= f(x1) – f(x1) – f(x2) / x1 – x2 . x1

= x1 f (x1) – x2 f (x1) – x1 f( x1) + x1 f (x2) / x1 –x2

b = x1 f (x2) – x2 f(x1) / x1 –x2

Inserting the value of a and b in (2.4), we get

F(x) = f(x1) – f(x2) / x1 –x2 .x + x1 f(x2) – x2 f (x1) / x1 –x2

Note : If b = 0 then equation (2.7) reduces to

F(x) = f(x1) – f(x2) / x1 –x2 . x

and  we have

b = 0 = x1 f(x2) –x2 f (x1) / x1 –x2)

Two commonly used scales are Celsius scale and  Fahrenheit scale. Symbols C and Fare used to denote the readings on the two scales. Before 1954, the two fixed points used are steam point (boiling point of water at standard atmospheric pressure and the ice point (freezing point of water)

Temperature                Celsius scale               Fahrenheit scale

Steam point                             100                              212

Ice point                                  0                                  32

Interval                                    100 degrees                 180 degrees

100 -0 /C -0 = 212 – 32 /F -32

100 / C = 180 / F -32

5 /C = 9 / F -32

C /5 = F -32 / 9

2.2.3 Method in Use On and After 1954

On and after 1954, Kelvin suggested that only one fixed point is necessary to establish a temperature scale. This fixed point is the triple point of water. The value of the triple point of water is 0.01oC or 273.16 K. Correspondingly, the ice point of 0 o c on the Celsius scale becomes 273.15 K on Kelvin scale. Let us use to C for Celsius scale and T for Kelvin scale.

Then

Tin Kelvin= 273.15 + to C.

The triple point of water in the Fahrenheit scale is 32.02!!F and for Rankine scale is 491.69R

T  in Rankine scale = 459.67 toF

Table 2.2

Fixed point             Celsius             Kelvin                         Falzrenlieit                  Rankine

Steam point                 100                  373.15             212                              671.67

Triple point of water   0.01                 273.16            32.02                           491.69

Ice point                       0                     273.15             32                                491.67

Absolute zero              – 273.15          0                      – 459.67                          0