Mathematics TEMPERATURE & HEAT, IDEAL-GAS AND THERMAL EXPANSION

### Topic Covered

INTRODUCTION
TEMPERATURE AND HEAT
IDEAL-GAS EQUATION AND ABSOLUTE TEMPERATURE
THERMAL EXPANSION

### INTRODUCTION

Temperature is a measure of "hotness" of a body. A kettle with boiling water is hotter than a box containing ice. In physics, we need to define the notion of heat, temperature, etc., more carefully.

In this chapter, we will learn what heat is and how it is measured, and study the various proceses by which heat flows from one body to another. Along the way, you will find out why blacksmiths heat the iron ring before fitting on the rim of a wooden wheel of a bullock cart and why the wind at the beach often reverses direction after the sun goes down.

### TEMPERATURE AND HEAT

We can begin studying thermal properties of matter with definitions of temperature and heat. Temperature is a relative measure, or indication of hotness or coldness.

A hot utensil is said to have a high temperature, and ice cube to have a low temperature. An object that has a higher temperature than another object is said to be hotter. Note that hot and cold are relative terms, like tall and short.

We can perceive temperature by touch. However, this temperature sense is somewhat unreliable and its range is too limited to be useful for scientific purposes.

We know from experience that a glass of ice-cold water left on a table on a hot summer day eventually warms up whereas a cup of hot tea on the same table cools down.

It means that when the temperature of body, ice-cold water or hot tea in this case, and its surrounding medium are different, heat transfer takes place between the system and the surrounding medium, until the body and the surrounding medium are at the same temperature.

We also know that in the case of glass tumbler of ice cold water, heat flows from the environment to the glass tumbler, whereas in the case of hot tea, it flows from the cup of hot tea to the environment.

So, we can say that heat is the form of energy transferred between two (or more) systems or a system and its surroundings by virtue of temperature difference.

The SI unit of heat energy transferred is expressed in joule (J) while SI unit of temperature is kelvin (K), and °C is a commonly used unit of temperature. When an object is heated, many changes may take place. Its temperature may rise, it may expand or change state. We will study the effect of heat on different bodies in later sections.

### MEASUREMENT OF TEMPERATURE

A measure of temperature is obtained using a thermometer. Many physical properties of materials change sufficiently with temperature to be used as the basis for constructing thermometers.

The commonly used property is variation of the volume of a liquid with temperature. For example, a common thermometer (the liquid-in-glass type) with which you are familiar. Mercury and alcohol are the liquids used in most liquid-in-glass thermometers.

Thermometers are calibrated so that a numerical value may be assigned to a given temperature. For the definition of any standard scale, two fixed reference points are needed.

Since all substances change dimensions with temperature, an absolute reference for expansion is not available. However, the necessary fixed points may be correlated to physical phenomena that always occur at the same temperature.

The ice point and the steam point of water are two convenient fixed points and are known as the freezing and boiling points. These two points are the temperatures at which pure water freezes and boils under standard pressure.

The two familiar temperature scales are the Fahrenheit temperature scale and the Celsius temperature scale. The ice and steam point have values 32 °F and 212 °F respectively, on the Fahrenheit scale and 0 °C and 100 °C on the Celsius scale. On the Fahrenheit scale, there are 180 equal intervals between two reference points, and on the celsius scale, there are 100.

A relationship for converting between the two scales may be obtained from a graph of Fahrenheit temperature (t_F) versus celsius temperature (t_C) in a straight line (Fig. 11.1), whose equation is

color{orange}(((t_F-32)/(180)=(t_c)/(100)).........(11.1)

### IDEAL-GAS EQUATION AND ABSOLUTE TEMPERATURE

Liquid-in-glass thermometers show different readings for temperatures other than the fixed points because of differing expansion properties. A thermometer that uses a gas, however, gives the same readings regardless of which gas is used.

Experiments show that all gases at low densities exhibit same expansion behaviour. The variables that describe the behaviour of a given quantity (mass) of gas are pressure, volume, and temperature (P, V, and T )(where T = t + 273.15; t is the temperature in °C).

When temperature is held constant, the pressure and volume of a quantity of gas are related as P V = constant. This relationship is known as Boyle’s law, after Robert Boyle the English Chemist who discovered it.

When the pressure is held constant, the volume of a quantity of the gas is related to the temperature as V/T = constant. This relationship is known as Charles’ law, after the French scientist Jacques Charles (1747- 1823). Low density gases obey these laws, which may be combined into a single relationship.

"Note" that since PV = "constant" and V//T = "constant" for a given quantity of gas, then PV//T should also be a constant. This relationship is known as ideal gas law. It can be written in a more general form that applies not just to a given quantity of a single gas but to any quantity of any dilute gas and is known as ideal-gas equation:

color{orange}((PV)/T = muR)

or color{orange}(PV = μRT)......(11.2)

In Eq. 11.2, we have learnt that the pressure and volume are directly proportional to temperature : PV ∝ T.

This relationship allows a gas to be used to measure temperature in a constant volume gas thermometer. Holding the volume of a gas constant, it gives P ∝T.

Thus, with a constant-volume gas thermometer, temperature is read in terms of pressure. A plot of pressure versus temperature gives a straight line in this case, as shown in Fig. 11.2.

However, measurements on real gases deviate from the values predicted by the ideal gas law at low temperature. But the relationship is linear over a large temperature range, and it looks as though the pressure might reach zero with decreasing temperature if the gas continued to be a gas.

The absolute minimum temperature for an ideal gas, therefore, inferred by extrapolating the straight line to the axis, as in Fig. 11.3. This temperature is found to be – 273.15 °C and is designated as absolute zero. Absolute zero is the foundation of the Kelvin temperature scale or absolute scale temperature.

named after the British scientist Lord Kelvin. On this scale, – 273.15 °C is taken as the zero point, that is 0 K (Fig. 11.4).

The size of the unit for Kelvin temperature is the same celsius degree, so temperature on these scales are related by

color{green}(T = t_C + 273.15)...(11.3)

### THERMAL EXPANSION

You may have observed that sometimes sealed bottles with metallic lids are so tightly screwed that one has to put the lid in hot water for sometime to open the lid. This would allow the metallic cover to expand, thereby loosening it to unscrew easily.

In case of liquids, you may have observed that mercury in a thermometer rises, when the thermometer is put in a slightly warm water. If we take out the thermometer from the warm water the level of mercury falls again.

Similarly, in the case of gases, a balloon partially inflated in a cool room may expand to full size when placed in warm water. On the other hand, a fully inflated balloon when immersed in cold water would start shrinking due to contraction of the air inside.

It is our common experience that most substances expand on heating and contract on cooling. A change in the temperature of a body causes change in its dimensions. The increase in the dimensions of a body due to the increase in its temperature is called "thermal expansion."

The expansion in length is called linear expansion. The expansion in area is called area expansion. The expansion in volume is called volume expansion (Fig. 11.5).

If the substance is in the form of a long rod, then for small change in temperature, ΔT, the fractional change in length, Δl//l, is directly proportional to ΔT.

color{orange}((Deltal)/l = alpha_1DeltaT).......(11.4)

where α_1 is known as the coefficient of linear expansion and is characteristic of the material of the rod.

In Table 11.1 are given typical average values of the coefficient of linear expansion for some materials in the temperature range 0 °C to 100 °C. From this Table, compare the value of α_1 for glass and copper.
We find that copper expands about five times more than glass for the same rise in temperature. Normally, metals expand more and have relatively high values of α_1.

Similarly, we consider the fractional change in volume, (DeltaV)/V . of a substance for temperature change ΔT and define the
coefficient of volume expansion, alpha_V

color{blue}(alpha_V= (DeltaV/V)1/(DeltaT))......(11.5)

Here α_V is also a characteristic of the substance but is not strictly a constant. It depends in general on temperature (Fig 11.6). It is seen that α_V becomes constant only at a high temperature.

Table 11.2 gives the values of co-efficient of volume expansion of some common substances in the temperature range 0 –100 °C. You can see that thermal expansion of these substances (solids and liquids) is rather small, with materials like pyrex glass and invar (a special iron-nickel alloy) having particularly low values of α_V.

From this Table we find that the value of α_v for alcohol (ethyl) is more than mercury and expands more than mercury for the same rise in temperature.

Water exhibits an anomalous behavour; it contracts on heating between 0 °C and 4 °C. The volume of a given amount of water decreases as it is cooled from room temperature, until its temperature reaches 4 °C, [Fig. 11.7(a)]. Below 4 °C, the volume increases, and therefore the density decreases [Fig. 11.7(b)].

This means that water has a maximum density at 4 °C. This property has an important environmental effect: Bodies of water, such as much as order(s) of magnitude larger than the coefficient of volume expansion of typical liquids.

There is a simple relation between the coefficient of volume expansion (α_v) and coefficient of linear expansion (α_l). Imagine a cube of length, l, that expands equally in all directions, when its temperature increases by ΔT. We have

color{orange}(Δl = α_1 ΔT)

so, color{red}(ΔV = (l+Δl)^3 – l^3 = 3l^2 Δl)..(11.7)

In equation (11.7), terms in (Δl)^2 and (Δl)^3 have been neglected since Δl is small compared to l. So

color{blue}(DeltaV= (3VDeltal)/l = 3Valpha_1DeltaT)...(11.8)

which gives color{blue}(alpha_V=3alpha_1)..........(11.9)

What happens by preventing the thermal expansion of a rod by fixing its ends rigidly? Clearly, the rod acquires a compressive strain due to the external forces provided by the rigid support at the ends.

The corresponding stress set up in the rod is called thermal stress. For example, consider a steel rail of length 5 m and area of cross section 40 cm^2 that is prevented from expanding while the temperature rises by 10 °C.

The coefficient of linear expansion of steel
is color{orange}(α_(l("steel")) = 1.2 xx10^(–5) K^(–1)).

Thus, the compressive strain is color{orange}((Deltal)/(l) = alpha_(l("steel")) DeltaT= 1.2 xx10^(-5) 1= 1.2\ \ 10^(-4))

Youngs modulus of steel is Y_(steel) = 2xx10^11 Nm^(-2) Therefore, the thermal stress developed is

color{orange}(F/a = Y _(steel) ((Deltal)/l) = 2.4xx 10^7 \ \N m^(-2)) , which corresponds to an external force of

color{navy}(DeltaF = AY _(steel) ((Deltal)/l) = 2.4 xx 10^7 \ \40 \ \ 10^(-4) = 10^5N)

If two such steel rails, fixed at their outer ends, are in contact at their inner ends, a force of this magnitude can easily bend the rails.

Q 3280312217

Show that the coefficient of area expansions, (ΔA//A)//ΔT, of a rectangular sheet of the solid is twice its linear expansivity, α_l .
Class 11 Chapter 11 Example 1
Solution:

Consider a rectangular sheet of the solid material of length a and breadth b (Fig. 11.8 ). When the temperature increases by ΔT, alpha
increases by Δa = α_l aΔT and b increases by Δb = α_1b ΔT. From Fig. 11.8, the increase in area

ΔA = ΔA_1 +ΔA_2 + ΔA_3
ΔA = a Δb + b Δa + (Δa) (Δb)
= a α_1b ΔT + b α_1 a ΔT + (α_1)^2 ab (ΔT)^2
= α_1 ab ΔT (2 + α_1 ΔT) = α_1 A ΔT (2 + α_1 ΔT)
Since α_1 approx 10^(–5) K^(–1), from Table 11.1, the product α_1 ΔT for fractional temperature is small in comparision with 2 and may be neglected. Hence,

((DeltaA)/A) 1/(DeltaT)=2 alpha_1
Q 3210312219

A blacksmith fixes iron ring on the rim of the wooden wheel of a bullock cart. The diameter of the rim and the iron ring are 5.243 m and 5.231 m respectively at 27 °C. To what temperature should the ring be heated so as to fit the rim of the wheel?
Class 11 Chapter 11 Example 2
Solution:

Given,
T_1 = 27^oC
L_(r1) = 5.231 m
L_(r2) = 5.243 m

So,
L_(r2) =L_(r1) [1+α_1 (T_2–T_1)]
5.243 m = 5.231 m [1 + 1.2010^(–5) K^(–1) (T_2–27 °C)]
or T_2 = 218 °C.