Title 11. Thermometry and Radiation.pdf ✅

Thermometry and Radiation 223 11 Thermometry and Radiation QUICK LOOK Heat The energy associated with configuration and random motion of the atoms and molecules with in a body is called internal energy and the part of this internal energy which is transferred from one body to the other due to temperature difference is called heat. As it is a type of energy, it is a scalar. Dimension: 2 2 [ ] ML T − . Units: Joule (S.I.) and calorie (Practical unit) One calorie is defined as the amount of heat energy required to raise the temperature of one gm of water through 1°C (more specifically from 14.5°C to 15.5°C). As heat is a form of energy it can be transformed into others and vice-versa. e.g. Thermocouple converts heat energy into electrical energy, resistor converts electrical energy into heat energy. Friction converts mechanical energy into heat energy. Heat engine converts heat energy into mechanical energy. Here it is important that whole of mechanical energy i.e. work can be converted into heat but whole of heat can never be converted into work. When mechanical energy (work) is converted into heat, the ratio of work done (W) to heat produced (Q) always remains the same and constant, represented by J. W J Q = or W = JQ J is called mechanical equivalent of heat and has value 4.2 J/cal. J is not a physical quantity but a conversion factor which merely express the equivalence between Joule and calories. 1 calorie = 4.186 Joule ≃ 4.12 Joule Work is the transfer of mechanical energy irrespective of temperature difference, whereas heat is the transfer of thermal energy because of temperature difference only. Generally, the temperature of a body rises when heat is supplied to it. However the following two situations are also found to exist. When heat is supplied to a body either at its melting point or boiling point, the temperature of the body does not change. In this situation, heat supplied to the body is used up in changing its state. When the liquid in a thermos flask is vigorously shaken or gas in a cylinder is suddenly compressed, the temperature of liquid or gas gets raised even without supplying heat. In this situation, work done on the system becomes a source of heat energy. The heat lost or gained by a system depends not only on the initial and final states, but also on the path taken up by the process i.e. heat is a path dependent and is taken to be positive if the system absorbs it and negative if releases it. Temperature Temperature is defined as the degree of hotness or coldness of a body. The natural flow of heat is from higher temperature to lower temperature. Two bodies are said to be in thermal equilibrium with each other, when no heat flows from one body to the other. That is when both the bodies are at the same temperature. Temperature is one of the seven fundamental quantities with dimension [θ ]. It is a scalar physical quantity with S.I. unit kelvin. When heat is given to a body and its state does not change, the temperature of the body rises and if heat is taken from a body its temperature falls i.e. temperature can be regarded as the effect of cause “heat”. According to kinetic theory of gases, temperature (macroscopic physical quantity) is a measure of average translational kinetic energy of a molecule (microscopic physical quantity). Temperature ∝ kinetic energy 3 As 2 E RT   =     Although the temperature of a body can to be raised without limit, it cannot be lowered without limit and theoretically limiting low temperature is taken to be zero of the kelvin scale. Highest possible temperature achieved in laboratory is about 108K while lowest possible temperature attained is 10–8 K. Branch of physics dealing with production and measurement of temperatures close to 0K is known as cryogenics while that dealing with the measurement of very high temperature is called as pyrometry. Temperature of the core of the sun is 107 K while that of its surface is 6000 K.

Thermometry and Radiation 225 and volume respectively. The three coefficients of expansion are not constant for a given solid. Their values depend on temperature range in which they are measured. Figure: 11.3 Percentage change in length on heating α θ∆ ×100. Percentage change in area of heating β θ∆ ×100, percentage change in volume heating γ θ∆ ×100. α β γ : : 1 : 2 : 3, = Hence for same rise in temperature, Percentage change in area = 2 × percentage change in length and percentage change in volume = 3 × percentage change in length. Variation of Density with Temperature. Most substances expand when they are heated, i.e., volume of a given mass of a substance increases on heating, so the density should decrease 1 as V ρ     ∝   . m V ρ = or 1 V ρ ∝ ∴ 1 1 V V V V V V V V T T ρ ρ γ γ ′ = = = = ′ + ∆ + ∆ + ∆ (For a given mass) or 1 T ρ ρ γ ′ = + ∆ 1 ρ γ (1 ) T − = + ∆ = ρ γ (1 ) − ∆T [As γ is small ∴ using Binomial theorem] ∴ ρ ρ γ ' (1 ) = − ∆T Expansion of Liquid Liquids also expand on heating just like solids. Since liquids have no shape of their own, they suffer only volume expansion. If the liquid of volume V is heated and its temperature is raised by ∆θ then ' (1 ) V V L L = + ∆ γ θ [γL = coefficient of real expansion or coefficient of volume expansion of liquid] As liquid is always taken in a vessel for heating so if a liquid is heated, the vessel also gets heated and it also expands. ' (1 ) V V S S = + ∆ γ θ [γS = coefficient of volume expansion for solid vessel] So the change in volume of liquid relative to vessel. ' ' [ ] V V V L S L S − = − ∆ γ γ θ V V app app ∆ = ∆ γ θ [ app L S γ γ γ = − = Apparent coefficient of volume expansion for liquid] L S γ γ > 0 app γ > ∆ = Vapp positive Level of liquid in vessel will rise on heating. L S γ γ < 0 app γ < ∆ = Vapp negative Level of liquid in vessel will fall on heating. L S γ γ = 0 app γ = 0 ∆ = Vapp level of liquid in vessel will remain same. Effect of Temperature on up thrust The thrust on V volume of a body in a liquid of density σ is given by Th V g = σ Now with rise in temperature by ∆θ C°, due to expansion, volume of the body will increase while density of liquid will decrease according to the relations 1 S V V γ θ ′ = + ∆ and 1 L σ σ γ θ ′ = + ∆ So, the thrust will become ′ = ′σ ′gVhT ∴ (1 ) (1 ) S L Th V g Th V g σ γ θ σ γ θ ′ ′ ′ + ∆ = = + ∆ and apparent weight of the body Wapp = Actual weight – Thrust As S L γ γ < ∴ Th Th ′ < with rise in temperature thrust also decreases and apparent weight of body increases. Anomalous Expansion of Water: Generally matter expands on heating and contracts on cooling. In case of water, it expands on heating if its temperature is greater than 4°C. In the range 0°C to 4°C, water contracts on heating and expands on cooling, i.e., γ is negative. At 4°C, density of water is maximum while its specific volume is minimum. This behaviour of water in the range from 0°C to 4°C is called anomalous expansion. The anomalous behaviour of water arises due to the fact that water has three types of molecules, viz., H O(H O ) 2 2 2 and 2 3 (H O) having different volume per unit mass and at different temperatures their properties in water are different. Ice Water vapor Ice and water Phase change Liquid water Water and water vapor Phase change Boiling Q = mcLi Q = mc∆T Q = mc∆T Q = mcLv Q = mc∆T Constant heat input (cal) Temperature (°C) 100 – 20 0 Warming Melting Warming Warming
226 Quick Revision NCERT-PHYSICS Figure: 11.4 Expansion of Gases Gases have no definite shape, therefore gases have only volume expansion. Since the expansion of container is negligible in comparison to the gases, therefore gases have only real expansion. Coefficient of Volume Expansion: At constant pressure, the unit volume of a given mass of a gas, increases with 1°C rise of temperature, is called coefficient of volume expansion. V 1 V T α ∆ = × ∆ ∴ Final volume V V T ′ = + ∆ (1 ) α Coefficient of Pressure Expansion: P 1 P T β ∆ = × ∆ ∴ Final pressure P P T ′ = + ∆ (1 ) β For an ideal gas, coefficient of volume expansion is equal to the coefficient of pressure expansion. i.e. 1 273 1 − βα °== C Specific Heat Gram Specific Heat: When heat is given to a body and its temperature increases, the heat required to raise the temperature of unit mass of a body through 1°C (or K) is called specific heat of the material of the body. If Q heat changes the temperature of mass m by ∆T Specific heat Q c m T = ∆ . Units: Calorie/gm × °C (practical), J/kg × K (S.I.) Dimension: [L 2 T –2θ –1] Molar Specific Heat: Molar specific heat of a substance is defined as the amount of heat required to raise the temperature of one gram mole of the substance through a unit degree it is represented by (capital) C. By definition, one mole of any substance is a quantity of the substance, whose mass M grams is numerically equal to the molecular mass M. ∴ Molar specific heat = × M Gram specific heat or C M c = Q Q 1 C M m T T μ = = ∆ ∆ As and Q m c m T M μ   = =     ∆ ∴ Q C μ T = ∆ Units: calorie/mole × °C (practical); J/mole × kelvin (S.I.) Dimension: 2 2 1 1 [ ] ML T θ μ − − − Note Specific heat for hydrogen is maximum (3.5 / ) o cal gm C × and for water, it is 1 / cal gm C ×° . For all other substances, the specific heat is less than 1 / cal gm C ×° and it is minimum for radon and actinium ( 0.022 / ) − ×° ɶ cal gm C . Specific heat of a substance also depends on the state of the substance i.e. solid, liquid or gas. For example, cice = 0.5 cal/gm × °C (Solid), cwater = 1 cal/gm × °C (Liquid) and csteam = 0.47 cal/gm × °C (Gas) The specific heat of a substance when it melts or boils at constant temperature is infinite. As 0 Q Q C m T m = = = ∞ ∆ × [As ∆T = 0] The specific heat of a substance when it undergoes adiabatic changes is zero. As 0 0 Q C m T m T = = = ∆ ∆ [As Q = 0] Specific heat of a substance can also be negative. Negative specific heat means that in order to raise the temperature, a certain quantity of heat is to be withdrawn from the body. Example. Specific heat of saturated vapours. Specific Heat of Solids When a solid is heated through a small range of temperature, its volume remains more or less constant. Therefore specific heat of a solid may be called its specific heat at constant volume Cv . From the graph it is clear that at T = 0, Cv tends to zero With rise in temperature, Cv increases and becomes constant = 3R = 6 cal/mole × kelvin = 25 J/mole × kelvin at some particular temperature (Debye Temperature) For most of the solids, Debye temperature is close to room temperature. Figure: 11.5 Specific Heat vs. Temperature 3R Debye temp. T Cv Y 4 10 0.9997 1.9997 Density g/cm 3 Maximum density at 3.90° = 39.2°F Temperature °C 0