Thermal Properties of Matter
Temperatures in different scales are related as follows:
⇒ \(\frac{\mathrm{C}-0}{100}=\frac{\mathrm{F}-32}{180}=\frac{\mathrm{K}-273.15}{100}\)
Or, \(\frac{\mathrm{C}}{5}=\frac{\mathrm{F}-32}{9}=\frac{\mathrm{K}-273}{5}\)
Temperature difference in Celsius scale = temperature difference in kelvin scale
i.e., (T2 – T1)0C = (T2 – T1)K
Linear Expansion Of Solids
w.k.t., Δl ∝ L0 and Δl ∝ ΔT
or, Δl – α L0ΔT
α = \(\frac{\Delta l}{\mathrm{~L}_0 \Delta \mathrm{T}}\)
Where Δl is the change in length, L0 is the original length, ΔT is the increase in temperature, and α is known as the coefficient of linear expansion.
Unit of α is 0C-1 or K-1.
Read And Learn More: NEET Physics Notes
Superficial/areal expansion of solids
ΔA = \(\beta \mathrm{A}_0 \Delta \mathrm{T}\)
β = \(\frac{\Delta A}{A_0 \Delta T}\)
Where ‘β’ is known as the coefficient of areal expansion.
Cubical or volume expansion of solids
ΔV = \(\gamma \mathrm{V}_0 \Delta \mathrm{T}\)
γ = \(\frac{\Delta \mathrm{V}}{\mathrm{V}_0 \Delta \mathrm{T}}\)
Where ‘γ’ is known as the coefficient of volume expansion.
α, β, and γ are related as follows
α = \(\frac{\beta}{2}=\frac{\gamma}{3}\) ⇒ α:β:γ = 1:2:3
The time period of oscillation of simple pendulum is given by,
∴ T = \(2 \pi \sqrt{\frac{\ell}{g}}\)
Fractional change in time period of a simple pendulum is given by,
∴ \(\frac{\Delta \mathrm{T}}{\mathrm{T}}=\frac{1}{2} \alpha \Delta \theta\)
In summer length of the pendulum of the clock will increase. Therefore, time period also increases. i.e., the clock will lose time.
Time lost by the clock in a day is given by
Δt = \(\left(\frac{1}{2} \alpha \Delta \theta\right) \times 86400\)
Pendulums are made of invar as their coefficient of linear expansion is very small.
Thermal Properties Of Matter
Thermal Stress
When a rod is fixed between two rigid walls and the temperature is increased, thermal stress gets developed in the rod. Thermal stress, \(\frac{\mathrm{F}}{\mathrm{A}}\)= YA α θΔ
Anomalous expansion of water: Generally, all materials expand on heating, and contract on cooling.
But as the temperature of the water is increased from 00C to 40C it contracts. This unusual behavior of water is called anomalous expansion.
For a given mass of water,
- Density is maximum at 4° C and
- Volume is minimal at 4° C
Heat
The amount of heat given to a body depends upon its mass (m), change in temperature (Δθ), and nature of material.
i.e., Q = mc Δθ, where c is the specific heat
The amount of heat required to raise the temperature of 1g of water from 14.50C to 15.50C is called one calorie.
1 cal = 4.18 J
Heat always flows from a body of higher temperature to a body at a lower temperature.
Specific heat capacity: The amount of heat required to increase unit mass of the substance by unit degree is called specific heat capacity.
c = \(\frac{\mathrm{Q}}{\mathrm{m} \Delta \theta} \mathrm{J} \mathrm{Kg}^{-1} \mathrm{~K}^{-1}\)
Heat Capacity
The amount of heat required to increase the temperature of given mass of the substance by unit degree is called its heat capacity.
Heat capacity = \(\frac{\mathrm{Q}}{\Delta \theta} \mathrm{JK}^{-1}\)
Molar specific heat: The amount of heat required to be given to increase the temperature of 1 mole of the substance by unit degree.
i.e., Molar specific heat = \(\frac{\mathrm{Q}}{\mu \Delta \theta}\)
Where μ is the number of moles of the substance.
Principle of calorimetry: When 2 bodies at different temperature are mixed, heat will be transferred from body at a higher temperature to a body at lower temperature till their temperature become equal.
i.e., Heat lost = Heat gained
Principle of calorimetry is in accordance with law of conservation of energy.
Water equivalent: When the heat capacity of a body is expressed in terms of mass of water, it is called water equivalent of the body.
Or
Water equivalent is the mass of water which when given same heat as the body, changes the temperature of water through the same range as that of the body, w = mc
Amount of heat supplied to an object to change its state is directly proportional to its mass, Q = mL
Where L is latent heat.
The latent heat of the fusion of ice is, Lf = 80 cal/g
Latent heat of vaporization of water is, Lv = 540 cal/g
Law of thermal conductivity: Consider a rod of length of ‘l’ and area of cross-section A whose faces are maintained at temperatures θ1 and θ2. The rod is insulated in order to avoid leakage of heat.
Thermal Properties Of Matter
In steady state the amount of heat flowing from one face to the other face in time ‘t’ is given by,
Q = \(\frac{\mathrm{KA}\left(\theta_1-\theta_2\right) \mathrm{t}}{l}\)
Where ‘K’ is the thermal conductivity of material of the rod.
If the rod has variable cross-section, then
⇒ \(\frac{\mathrm{dQ}}{\mathrm{dt}}=-\mathrm{KA} \frac{\mathrm{d} \theta}{\mathrm{dx}}\)
The equation Q = \(\frac{\mathrm{KA}\left(\theta_1-\theta_2\right) \mathrm{t}}{l}\) can be written as,
⇒ \(\mathrm{H}-\frac{\mathrm{Q}}{\mathrm{t}}=\frac{\mathrm{KA} \Delta \theta}{l}\)
Or \(\mathrm{H}=\frac{\Delta \theta}{\left(\frac{l}{\mathrm{KA}}\right)}\)
Which is analogous to, I = \(\frac{\mathrm{V}}{\mathrm{R}}\)
∴ Thermal resistance is given by \(\mathrm{R}_{\mathrm{t}}=\frac{l}{\mathrm{KA}}\)
Series combination of metallic rods:
Equivalent thermal resistance is given by, RS = R1 + R2 + …… + Rn
Equivalent thermal conductivity is given by,
⇒ \(\mathrm{K}_{\mathrm{s}}\) = \(\frac{\mathrm{n}}{\frac{1}{\mathrm{~K}_1}+\frac{1}{\mathrm{~K}_2}+\ldots+\frac{1}{\mathrm{~K}_{\mathrm{a}}}}\)
For two rods, \(\mathrm{K}_{\mathrm{s}}=\frac{2 \mathrm{~K}_1 \mathrm{~K}_2}{\mathrm{~K}_1+\mathrm{K}_2}\)
Parallel combination of metallic rods: Equivalent thermal resistance is given by,
⇒ \(\frac{1}{R_s}=\frac{1}{R_1}+\frac{1}{R_2}+\ldots \ldots+\frac{1}{R_2}\)
Equivalent thermal conductivity is given by,
K = \(\frac{\mathrm{K}_1 \mathrm{~A}_1+\mathrm{K}_2 \mathrm{~A}_2+\ldots+\mathrm{K}_{\mathrm{a}} \mathrm{A}_{\mathrm{a}}}{\mathrm{A}_1+\mathrm{A}_2+\ldots+\mathrm{A}_{\mathrm{a}}}\)
For ‘n’ slabs of equal area, K \(\frac{\mathrm{K}_1+\mathrm{K}_2+\ldots+\mathrm{K}_{\mathrm{a}}}{\mathrm{n}}\)
For two slabs of equal area, K = \(\frac{\mathrm{K}_1+\mathrm{K}_2}{2}\)
When ‘Q’ amount of energy falls on a surface, then a portion of energy gets reflected (Qr), a portion of energy gets transmitted (Qt) and a portion of energy gets absorbed (Qa) by the surface.
i.e., Q = Qa + Qt + Qr
1 = \(\frac{Q_{\mathrm{a}}}{Q}+\frac{Q_t}{Q}+\frac{Q_r}{Q}\)
Where, a = \(\frac{\mathrm{Q}_2}{\mathrm{Q}}\) is called absorptance.
t = \(\frac{Q_t}{Q}\) is called transmittance.
r = \(\frac{Q_f}{Q}\) is called reflectance.
Thermal Properties Of Matter
Stefan’s Law
Total power radiated by an object is given by, P = e σ AT4
Where 0 is called Stefan’s constant cr = 5.67×10 Wm K, e is emissivity and T is absolute temperature.
For a perfect black body, e = 1,
∴ P = σ AT4
Newton’s law of cooling: The rate of fall in temperature of a body is directly proportional to the temperature difference between the body and surroundings. (The temperature difference should not exceed 400C)
i.e„ \(\frac{\mathrm{d} \theta}{\mathrm{dt}}=-\mathrm{k}\left(\theta-\theta_0\right)\)
Where θ is the temperature of the body and θ0 is the temperature of the surrounding.
During the experiment if θ changes from θ1 to θ2, then
⇒ \(\frac{\mathrm{d} \theta}{\mathrm{dt}}=\mathrm{k}\left[\frac{\theta_1+\theta_2}{2}-\theta_0\right]\)
Wien’s Displacement Law
According to Wien’s law λmT = b = constant
Where λm is the wavelength corresponding to maximum energy emission and b is known as Wien’s constant b = 2.93 x 10-3 mK
Solar constant: It is the rate at which energy reaches the earth’s surface from the sun.
In SI units,
Solar constant, S = 1388Wm-2