Notes for Heat Transfer chapter of class 11 physics. Dronstudy provides free comprehensive chapterwise class 11 physics notes with proper images & diagram.
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TEMPERATURE
It is a relative measure or indication of hotness or coldness.
Scales of Temperature
There are three different scales of temperature.
(a)Â The Celsius Scale:
This scale was devised by Anders Celsius in the year 1710. The interval between the lower fixed point and the upper fixed points is divided into 100 equal parts. Each division of the sale is called one degree centigrade or one degree Celsius (1^{o}C). At normal pressure, the melting point of ice is 0^{o}C. This is the lower fixed point of the Celsius scale. At normal pressure, the boiling point of water is 100^{o}C. This is the upper fixed point of the Celsius scale.
(b) The Fahrenheit Scale.
This scale was devised by Gabriel Fahrenheit in the year 1717. The interval between the lower and the upper fixed points is divided into 180 equal parts. Each division of this scale is called one degree Fahrenheit (1^{o}F). On this scale, the melting point of ice at normal pressure is 32^{o}F. This is the lower fixed point. The boiling point of water at normal pressure is taken as 212^{o}F. This is the upper fixed point.
(c)Â The Reaumer Scale
This scale was devised by R. A. Reaumer in the year 1730.
The interval between the lower and the upper fixed point is divided into 80 equal parts. Each division is called one degree Reaumer (1^{o}R). on this scale, the melting point of ice at normal pressure 0^{o}R. This is lower fixed point. The boiling point of water at normal pressure is 80^{o}R. This is the upper fixed point.
In order to convert temperature from one scale to another, following relation is used.
â¸« Â Â
or Â
Equilibrium state of energy thermodynamic system is completely described by specific values of some macroscopic variables or state variables. The relation between the state variables is called the equation of state. The thermodynamic state variables are of two types (i) extensive and (ii) intensive.
Extensive variables depend on the quantities of system e.g. volume, mass etc.
Intensive variables are independent of quantity of system e.g. pressure, density, etc.
HEAT
It is the form of energy transferred between two or more systems or a system and its surrounding by virtue of temperature difference.
IDEAL GAS EQUATION
Combining first four laws (i.e. Boyleâ€™s law, Charleâ€™s Law, Gay â€“ Lussacâ€™s Law and Avagadroâ€™s law) we get one single equation for an ideal gas, i.e.
PV = nRT Â Â Â Â Â Â for n moles of gas
orÂ Â = R = constant
Here,Â R = universal gas constant = 8.31 J/mol-KÂ = 2 calorie/mol-K
For 1 mole of a gasÂ n = 1
i.e.Â Â PV = RT
orÂ = constant
THERMAL EXPANSION
Experiments show that most of bodies increase their volume upon heating. The extent of expansion of various bodies is characterized by the temperature coefficient of expansion, or simply the coefficient of expansion. While considering solid which retain their shape during temperature variations, the distinction is made between (a) a change in their linear dimensions (viz. the dimensions in a certain direction), i.e. linear expansion, and (b) a change in the volume of a body, i.e. cubic expansion.
The coefficient of linear expansion is the quantity equal to the fraction of the initial length by which a body taken at 0^{o}C has elongated as a result of heating it by 1^{o}C (or by 1 K):
where l_{Âo} is the initial length at 0^{o}C and l_{t} is the length at a temperature t. From this expansion, we can find
The dimensions of are K^{-1}Â (orÂ Â ^{o}C^{-1}).
The coefficient of cubic expansion is the quantity equal to the fraction of the initial volume by which the volume of a body taken at 0^{o}C has increased upon heating it by 1^{o}C (or by 1 K):
,
where V_{o} is the volume of a body at 0^{o}C and V_{t} is its volume at a temperature t. From this equation, we obtain
The quantity has also the dimensions of K^{-1} (orÂ ^{o}C^{-1}).
The coefficient of cubic expansion is about three times larger than the coefficient of linear expansion:
The coefficient of cubic expansion for liquids are somewhat higher than for solid bodies, ranging between 10^{-3} and 10^{-4} K^{-1}.
What obeys the general laws of thermal expansion only at a temperature above 4 ^{o}C. FromÂ 0 ^{o}C to 4 ^{o}C, water contracts rather than expands. At 4 ^{o}C, water occupies the smallest volume, i.e. it has the highest density. At the bottom of deep lakes, there is denser water in winter, which remains the temperature of 4 ^{o}C even after the upper layer has been frozen.
Application 1
The lengths l_{1i} = 100 m of iron wire and l_{1c} = 100 m of copper wire are marked off at t_{1} = 20 ^{o}C. What is the difference in lengths of the wires at t_{2} = 60 ^{o}C? The coefficients of linear expansion for iron and copper are Â = 1.2 xÂ 10^{-5} K^{-1} and Â = 1.7 xÂ 10^{-5} K^{-1}.
Solution:
Â andÂ
The elongation of the iron wires is
.
Substituting Â , we find the elongation of the iron and copper wires
Â Â Â Â Â Â Â (1)
Â Â Â Â Â Â (2)
Subtracting (1) from (2) and considering that , we obtain
For low values of temperature t, when , it is not necessary to reduce l_{1} and l_{2} to l_{o1} and l_{Âo2} at t = 0 ^{o}C. To a sufficiently high degree of accuracy, we an assume that . Under this assumption, the problem can be solved in a simpler way:
Consequently, since , we have
It can been seen that the deviation from a more exact value of 19.9 mm amounts to 0.1 mm, i.e. the relative error .
Application 2
A solid body floats in a liquid at a temperature t = 0^{o }C and is completely submerged in it at 50^{o} C. What fraction d of volume of the body is submerged in the liquid at 0^{o} C ifÂ Â = 0.3 xÂ 10^{-5} K^{-2} and of the liquid, Â = 8.0 xÂ 10^{-5} K^{-1}?
Solution:
In both the cases the weight of the body will be balanced by the force of buoyancy on it.
AtÂ t_{o} = 0 ^{o}C, the buoyancy is Â Â Â Â Â Â (1)
where V_{o} is the volume of the body and Â is the density of the liquid at t_{o} = 0^{o}C. At t = 50 ^{o}C, the volume of the body becomes Â and the density of the liquid is . The buoyancy in this case is
Â Â Â Â Â Â Â Â (2)
Equating the right-hand sides of equation (1) and (2), we get
Specific Heat and Heat Capacity
If a quantity of heat Q produces a change in temperature DT in a body, its heat capacity is defined as
Heat capacityÂ
The SI unit of heat capacity is JK^{-1}.
The quantity of heat Q required to produce a change in temperature DT is also proportional to the mass m of the sample.
where C is called the specific heat of the substance.
Specific heat may be defined as the heat capacity per unit mass.
It is sometimes convenient, especially in the case with gases, to deal with the number of moles n of a substance rather than its mass. Then,
where C_{m} is the molar specific heat, measured in J/mol-KÂ (or cal/mol-K)
C_{m} = Mc
The specific heat of a substance usually varies with temperature.
The specific heat changes abruptly when the substance transforms from solid to liquid, or from liquid to gas. It also depends on the conditions under which the heat is supplied. For example, the specific heat of a gas kept at constant pressure C_{p} is different from its specific heat at constant volume C_{v}. For air, C_{v} = 0.17 cal/g-K and C_{p} = 0.24 cal/g-K For solids and liquids the difference is generally small, and in practice C_{p} is usually measured.
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