Thermal Expansions, Physics tutorial

Thermal Expansion in Solids:

Matter is anything which has weight and occupies space. In that situation, solids, liquids and gases are forms of matter. When they are thus heated, they expand. Not only expansion is visible in matter when it is heated, but also the change in temperature is also visible as heat is absorbed by or removed from body. When solids are heated, effect of heat on them could be found in change of:

• The length
• The area and
• Volume of the solids as temperature changes.

The changes in length, area and volume of the solids depend on:

• The material making up solid;
• The range of the temperature change;
• The initial dimensions of solid.

From the three factors we could infer that

• The expansion of solids varies from one material to the other;
• The greater the range of temperature change, the greater the expansion;
• Expansion depends on the original length, area and volume of the solid.

Linear Expansion:

Consider linear that is straight-line expansion of material. This signifies we are considering expansion of the solid in one dimension only. Consider the metal rod with the original length lo. If such a length of material is heated from the initial temperature t1oC to t2oC, change in temperature ΔΘ is given as ΔΘ = (t2 - t1)oC

It will be seen that length of metal rod increased from lo to lt lt is new length at temperature t2oC and lo is the original length at temperature t1oC.

New length lt is thus given as

lt = lO + Δl Where Δl is the change in length of rod when heated from t1oC to t2oC

Δl = lt - lO .............................. Eq.1

It is found experimentally that for the given material, increase in length, Δl, is proportional directly to

(i) Original length lO and

(ii) Change in temperature ΔΘ

Δl = α lO ΔΘ............................................. E.q.2

Where, α is constant of proportionality that is called as coefficient of linear expansion of solid or in short, linear expansivity. Consequently α can be provided as, α = Δl lO ΔΘ.

Therefore by definition, α, linear expansivity is increase in length of material (Δl) per original lO length per degree Celsius change in temperature (ΔΘ).

The unit of α is oC-1or per degree Celsius. On comparing Equations 1 and 2, we get

lt - lO = αlO ΔΘ

lt - lO = αlO ΔΘ

lt - lO = αlO (t2 - t1).....................

Determination of Linear Expansion (α):

We can estimate value of α by different methods. We have the following methods utilized in determining linear expansivity (α) of the metal:

• Optical lever method
• Screw gauge method
• Comparator method
• Henning's tube method
• Fizeau's method

All the above methods are different in manner in which increase in length is estimated. Specimen to be estimated is in form of the bar or tube and this comprises:

• the measurement of the length of the bar,
• the rise in temperature during the experiment and

Increase in length of bar consequent on the rise in temperature.

The Screw Gauge Method:

Screw gauge is the instrument utilized for estimating correctly the diameter of the thin wire or thickness of the sheet of metal.  It comprises of the U-shaped frame fitted with screwed spindle that is joined to a thimble.

Parallel to axis of the thimble, the scale graduated in mm is engraved. This is known as pitch scale. A sleeve is joined to head of the screw.

The head of screw has the ratchet that avoids undue tightening of screw. On thimble there is circular scale called as head scale that is divided in 50 or 100 equal parts. When screw is worked, sleeve moves over pitch scale.

The stud with the plane end surface known as anvil is attached on U frame accurately opposite to tip of screw. When tip of the screw is in contact with anvil, generally, zero of the head scale coincides with zero of pitch scale.

Superficial Thermal Expansion:

It is increase in surface area of solid on heating. Assume S is original surface of the solid and consider xS be the small increase in area of solid, when the temperature is raised by small amount xT. It is found that:

xS = b S xT

Where b is constant of proportionality and is known as coefficient of superficial expansion of solids. We may state b as a small change in surface area per unit original area per degree Celsius.

Cubical Thermal Expansion:

It is a increase in volume of solid on heating. Assume V is original volume of the solid. Consider xV be the small increase in volume of solid, when the temperature is raised by small amount xT. It is found that

xV = g V xT

Where g is constant of proportionality and is known as coefficient of cubical expansion of solids. We may state g as small change in volume per unit original volume per degree Celsius change in temperature.

Applications of Expansivity:

The following are some practical applications of expansivity:

1) Some metals like platinum and tungsten have their linear expansivity very close to that of glass. Beside the fact that linear expansivity of platinum in approximately equal to that of glass, behavior of the solids are very much alike. This features therefore allows us to seal electrodes through glass without occurrence of breakage through cooling and heating processes.

2) Linear expansivity is also applied in formation of bimetallic element that are used as:

(i) Thermostatic control switches;

(ii) In construction of expansion loops for use in steam lines;

(iii) bimetallic thermometers.

3) Linear expansivity is also utilized in construction of bridges where gaps are left between girders to hold expansion. Such gaps are also between iron rails in construction of railway lines.

Thermal Expansion in Liquids:

Real and Apparent Expansion of Liquids Experience has shown that it is impossible to estimate real or absolute thermal expansion of the liquid by direct volume determinations. This is due to liquids are contained in vessels that also expand when heated. Therefore, expansion of content of the vessel is always relative or apparent. Apparent expansion of liquid is hence less that real expansion of liquid. Volume dilatometers are utilized in determination of thermal expansion of liquids. Mean coefficient of apparent expansion of the liquid (αapp ) between temperature t1 and t2 is given as:

αapp = [(V2 - V1)/(V1(t2 - t1))]

Where V2 is finial volume at t2oC, V1 is initial volume and (t2 - t1) change in temperature. This is general definition of coefficient of apparent expansion. It applies to such experiments as volume dilatometer, weight thermometer, relative density bottle and sinker methods of finding coefficient of apparent expansion.

Weights of volumes of liquid between t1 and t2 are compared which will be equal if vessel and sinker didn't expand.

αapp = mass of liquid expelled/mass remaining x temperature change

Thus αreal = αapp + γ

Where γ is coefficient of cubical expansion of material of vessel, αapp is the apparent coefficient of expansion of the liquid and αreal is real coefficient of liquid.

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