## Heat

**Nature of heat:** Heat is a form of energy. When you give heat to a body, the molecules of the latter become more energetic and move about more rapidly thus increasing their kinetic energy. The body becomes hot.

**Temperature:** It is the degree of hotness or coldness of a body; the hotter the body, the higher its temperature. Thus it is different from heat; it is only one of its effects.

**Units of heat and temperature:** In the S. I. system, the unit of heat is the joule. In the C.G>S> system, it is the calorie. One calorie = 4.2 joules. The corresponding units of temperature are the degree Kelvin (k) in the S.I. system and degree Celsius ^{0}C (degree centigrade) in the C.G.S. system. However, the values of a degree is the same i.e. 1 ^{0}C = 1 K. In the Fahrenheit scale, the unit of temperature is the degree Fahrenheit (^{0}F). The relation between ^{0} C, K and ^{0}F is given in the next section.

**Thermometers:** These are instruments for measuring temperature. Two of the common forms used for elementary study are the Celsius and the Fahrenheit thermometers. Both use mercury as the thermometric substance and both work on the principle that mercury expands equally for equal rises of temperature. In the Celsius thermometer, the two fixed points are 0^{0} C and 100 ^{0} C. The space between the two fixed points is divided into 100 equal parts, each part being a degree Celsius. In the Fahrenheit thermometer, the two fixed points are 32 ^{0} F and 212 ^{0} F. The space between is divided into 180 equal parts, each part forming a degree Fahrenheit (^{0} F).

**Conversion from one scale to another:**

To convert the ^{0}C to ^{0}F and vice versa, we can use the formula

F = C × 9 / 5 + 32

This is easy to understand for 100 Celsius divisions occupy the whole space that is occupied by 180 Fahrenheit divisions. So 1^{0} C division is equal to 1 × 180 / 100 or 9 / 5 Fahrenheit divisions. Thus, 10 Celsius divisions will be = 10 × 9 / 5 Fahrenheit divisions.

So a reading of 10^{0} C in the Celsius thermometer will be shown as 10 × 9 / 5 + 32 in the Fahrenheit thermometer. Now instead of 10^{0} C in our example, we can generalize the relation and we get the formula

F = C × 9 / 5 +32

To convert the Celsius reading into the Kelvin scale, we add 273. Thus 20^{0} = (20 + 273) K or 293 K. Conversely, 423 K = (423 – 273 ) ^{0} or 150 ^{0} C. Here we subtract 273 from the Kelvin reading. Why we do so will be clear after we learn gas laws.

This is easy to understand for 100 Celsius divisions occupy the whole space that is occupied by 180 Fahrenheit divisions. So 1^{0} C division is equal to 1 × 180 / 100 or 9 / 5 Fahrenheit divisions. Thus, 10 Celsius divisions will be = 10 × 9 / 5 Fahrenheit divisions. So a reading of 10^{0} C in the Celsius thermometer will be shown as 10 × 9 / 5 + 32 in the Fahrenheit thermometer. Now instead of 10^{0} C in our example, we can generalize the relation and we get the formula

F = C × 9 / 5 +32

To convert the Celsius reading into the Kelvin scale, we add 273. Thus 20^{0} = (20 + 273) K or 293 K. Conversely, 423 K = (423 – 273 ) ^{0} or 150 ^{0} C. Here we subtract 273 from the Kelvin reading. Why we do so will be clear after we learn gas laws.

**Different types of thermometers: ** The thermometer used by doctors is called the clinical thermometer. It is graduated in Fahrenheit degrees, the range being only between 95^{0} F and 110 ^{0} F. This limited range is due to the fact that our body temperature cannot be lower than 95 ^{0} or higher than 110 ^{0} F. In fact our normal temperature is 98.4 ^{0} F. To help read the temperature after taking the instrument from the mouth of the patient, a constriction is provided which prevents the mercury level from going down.

We can however bring it down by giving the thermometer a gentle jerk when it will be ready for use with another patient. Another type of thermometer is the maximum and minimum thermometer which can help us to record the maximum and minimum temperature reached during a certain interval of time, say during the day or night. The description and working of this thermometer can be found in any Physics text book.

The above thermometers depend for their working on the expansion and contraction of mercury on heating. The expansion of metals on heating can also be used for the measurement of temperature Gas thermometers also work on similar principle. The increase in resistance of metals on heating can also provide a basis.

**Expansion of Solids: (i)** While mercury which is a liquid can expand in volume only, solids can expand in length, area and volume. Expansion in length is called linear expansion, area expansion is known as superficial expansion and volume expansion is cubic expansion.

(ii) While the fact of expansion due to heat is important, scientists and engineers are more interested in the quantitative aspect of it, i.e. how much it expands. This leads us to the concept of expansivity or coefficient of expansion.

**Coefficient of linear expansion of a solid i.e. the linear expansivity of the solid. **

This is the ratio of the increase in length of a rod to it original length per ^{0} C rise of temperature. It is denoted by the letter alpha ( α ).

Thus we have

α = ( l_{2} – l_{1} ) / l_{1} (t_{2} – t_{1} )………………(1)

Where l_{1} is the length at t_{1} ^{0} C, l _{2} is the length at t_{2} ^{0} C. α is thus a ratio. Its value for copper is 17 × 10^{-6} per ^{0} C. Since it is a ratio, the units of length chosen in the (C.G.S. or S.I units) will not alter its value.

By using the formula () 1 above, we can find any unknown quantity if the others are known. However, formula (2) below which can be deduced from (1) may sometimes be found more useful.

α = ( l_{2} – l_{1} ) / l_{1} (t_{2} – t_{1} )

Cross multiplying

l_{2} – l_{1} = α l_{1} (t_{2} – t_{1})

Therefore, l_{2} = l_{1} + α l_{1} (t_{2} – t_{1})

= l_{1} [ 1 + α (t_{2} - t_{1}) ]

i.e. l_{2} = l_{1} [ 1 + α (t_{2} - t_{1}) ]…………………………………(2)

(iv) Similarly, the coefficients of superficial ( area ) expansion (β) and cubical (volume) expansion (γ) can be written as

A _{2} A_{1} [ 1 + α (t_{2} - t_{1}) ]…………………………………….(3)

V _{2} V_{1} [ 1 + α (t_{2} - t_{1}) ]…………………………………….(4)

It can be shown that β = 2α and γ = 3α

Thus α for copper is 0.000017 per ^{0} C. Its β and γ values are

2 × 0.000017 per ^{0} C and 3 × 0.000017 per ^{0} C.

**To prove β = 2α**

We take a square sheet of metal of side unit length. Its area = 1 sq. unit. Suppose it is heated by 1 ^{0} C, each side will now expand and become (1 + α) where α is the coefficient of linear expansion of the metal. The new area is (1 + α) ^{2} i.e. 1 + 2α + α^{2}. [α being very small can be neglected]. So the new area becomes 1 + 2α.

Therefore, increase in area = 2α.

And this increase being on unit area for 1 ^{0} C rise of temperature, it is β by definition.

Therefore, we have β = 2 α

**To prove γ = 3 α**

Let us consider a cube of a metal of side 1 unit of length. Its volume = 1 unit of volume. If it is heated through 1 ^{0} C, each side will expand to (1 + α) and so the new volume will be (1 + α)^{3} = 1 + 3 α + 3 α^{2} + α^{3} = 1 + 3 α, as 3 α^{2} and α^{3} being very small can be neglected. So the increase in volume = 3 α. As this increase is on 1 unit of volume for 1 ^{0} C rise of temperature, it is γ. So we have γ = 3 α.

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