Thermometry -

It is a branch of physics to study the measurement of temperature.

Heat

It is a form of energy due to which we feel the sensation of hotness and coldness. The energy that is transferred from one body to the other is due to only temperature difference. i.e. without doing any mechanical work involved is called heat.

Once heat is transferred, it becomes the internal energy of the receiving body.

The energy because of the random motion of molecules and configuration is called internal energy. This internal energy when transferred from one body to the other due to temperature difference is called heat.

Zeroth law of thermodynamics

Two bodies are said to be in thermal equilibrium if no heat is transferred from one body to another.

” According to Zeroth law of thermodynamics, if two bodies A and B are in thermal equilibrium and A and C are also in thermal equilibrium, then B and C are also in thermal equilibrium.

Temperature

It is the degree of hotness or coldness. It is a numerical value of hotness or coldness. Two or more bodies in contact are said to be in thermal equilibrium if the temperature of all bodies are same.

According to the kinetic theory of gases, temperature ( macroscopic physics quantity) is a measure of the average translational K.E of a molecule ( microscopic physics quantity)

Temperature ∝ Kinetic Energy $\fn_cm \left [ As\; E=\frac{3}{2}RT \right ]$

The instrument used to measure the temperature of a body is called a Thermometer.

Thermometers are based on the principle, that some physical properties of a substance change with temperature (linearly). Ex- the volume of a liquid, length of a solid, pressure of a gas at constant volume, the volume of a gas at constant pressure, the electric resistance of a conductor. All vary with temperature.

Before 1950/1954…….. There are two fixed points of temperature, ice point and steam point. But due to various technical reasons, after 1954 there is only one fixed point ( triple point of water, All three states of water co-exist at this temperature). The temperature of the triple point of water is arbitrarily assigned a value of 273.16K.

Scale of thermometer

For any scale

$\fn_cm \frac{C-0}{100-0}=\frac{F-32}{212-32}=\frac{K-273.15}{373.15-273.15}=\frac{R-0}{80-0}=\frac{R_n-492}{672-492}$

$\fn_cm \frac{C}{100}=\frac{F-32}{180}=\frac{K-273.15}{100}=\frac{R}{80}=\frac{R_n-492}{180}$

$\fn_cm \left [ K=C+273.15 \right ]$

Exercise

The steam point and the ice point of a Hg thermometer are marked as 80° and 10°. At what temperature on centigrade scale, the reading of this thermometer will be 59°? ( 70°C)
At what temperature do the Celsius and Fahrenheit readings have the same numerical value? ( -40°)
Convert following temperature into Kelvin (a) 37°C, (b) 80°F, (c) -196°F ( 310K, 299.66K, 77K)
Convert 27°C and 40°C into Kelvin and Fahrenheit.
If ΔT=40°C in Celsius, then find ΔT in Fahrenheit. ( 72°F)
A faulty thermometer reads the freezing point and the °boiling point of water is 5°C and 95°C resp. What is the correct value of temperature as it reads 60°C on a faulty thermometer? ( 61.11°C)
The normal temperature of the human body is 98.4°F. Find the temperature on Celsius and Absolute scale.
A thermometer has the wrong calibration. It reads the melting point of ice -10°C. It reads 60° C in place of 50° C. Calculate the temperature of the boiling point of water in this scale. (130° C)
Why is the bulb of the thermometer cylindrical?
Generally, mercury is used as thermometer liquid in all thermometers. why?
A difference of temperature of 45° F is equivalent to a difference of Celsius. (25° C)
A Fahrenheit thermometer reads 113° F while a faulty Celsius thermometer reads 44° C. Find the correction required to be applied to the Celsius thermometer. (+1° C)

Methods of Calibration

For two fixed points thermometer

Consider a Hg thermometer for a centigrade scale. The relation between temperature and length of the mercury column may be written as

$\dpi{120} \fn_cm t=al+b$

Let $\dpi{120} \fn_cm l_0\rightarrow$ length of Hg at 0° C

$\dpi{120} \fn_cm l_{100}\rightarrow$ length of Hg at 100° C

$\dpi{120} \fn_cm l\rightarrow$ length of Hg at t° C

Now,

$\dpi{120} \fn_cm 0=al_0+b$

$\dpi{120} \fn_cm \left [ b=-al_0 \right ]$

Also,

$\dpi{120} \fn_cm 100=al_{100}+b$

$\dpi{120} \fn_cm 100=al_{100}-al_0$

$\dpi{120} \fn_cm \therefore a=\frac{100}{l_{100}-l_0}$

∴ From relation

$\dpi{120} \fn_cm t=\frac{100}{l_{100}-l_0}l-\frac{100}{l_{100}-l_0}l_0$

$\dpi{120} \fn_cm \left [ \mathbf{t=\frac{l-l_0}{l_{100}-l_0}\times 100^0} \right ]$

For Gas thermometer

$\dpi{120} \fn_cm \left [ \mathbf{t=\frac{P-P_0}{P_{100}-P_0}\times 100^0} \right ]$

For resistance thermometer

$\dpi{120} \fn_cm \left [ \mathbf{t=\frac{R-R_0}{R_{100}-R_0}\times 100^0} \right ]$

For one fixed thermometer

The above scale depends on the properties of the substance used. For ex- different resistance wires in the resistance thermometer used show different results. Also, the Hg thermometer, resistance thermometer, gas thermometer etc do not agree with each other.

So it is possible to define an absolute scale of temperature which doesn’t depend on the substance used. We now define the ideal gas thermometer.

In this scale, only one fixed point (triple point of water) is taken and temperature is just proportional to pressure.

$\dpi{120} \fn_cm i.e.\;\;T\propto P$

$\dpi{120} \fn_cm \left [ T=cP \right ]$

At the triple point of water

$\dpi{120} \fn_cm 273.16K=cP_{tr}$

$\dpi{120} \fn_cm \therefore {\color{Red} \mathbf{\left [ T=\frac{P}{P_{tr}}\times 273.16\; K \right ]}}\;\;\;\;\;(for\; any\; gas)$

we still have a problem with this thermometer. If we use it to measure, say the boiling point of water, we find that different gases give slightly different results. However as we use smaller and smaller amounts of gas, the reading nicely to a single temperature, no matter what gas we use.

i.e. equation becomes

$\dpi{120} \fn_cm {\color{Red} \mathbf{\left [T=(273.16\;K)\left [ \lim_{gas\rightarrow 0}\frac{P}{P_{tr}} \right ] \right ]}}$

NOTE:-

(1) Another type of thermometer is a total radiation pyrometer, which is used to measure very high temperatures.

When a body is at a high temperature, it emits radiation per second from the unit area of the surface of the body is proportional to the 4th power to the absolute temperature.

This radiation is measured by this thermometer and temperature is calculated. Its range is 800° C to no limit.

(2) Hg Thermometer Range (-30° C to 300° C)

Gas Thermometer Range (-268° C to 1500° C)

Resistance Thermometer (200°C to 1200° C)

(3) The triple point of water is independent of the external factor. It is the temperature at which the three phases of water, namely, ice, liquid water and water vapour are equally stable and coexistent. The triple point is suitable because it is unique, i.e., it occurs at one single temperature = 273.16 K and one single pressure of about 0.46 cm of the Hg column. This point is unique because any temperature or pressure higher or lower will change the phase of one or more states of water.

(4) The temperature which was first defined as the degree of hotness and later on the condition determining the flow of heat, is now regarded as one of the seven fundamental quantities like mass, length, time etc.

(5) The Zeroth law of thermodynamics was formulated by R.H. Fowler in 1931 long after the 1st and second laws of thermodynamics were stated. But as this law leads to the concept of the fundamental quantity temperature, so this law was called the zeroth law.

Exercise

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