Thermal Physics Notes

Temperature

  • Temperature is a physical quantity related to our sense of hotness or coldness.

  • Heat flows from a higher temperature to a lower temperature.

  • It's thermodynamically incorrect to speak of the amount of heat a body contains.

Thermal Equilibrium

  • When two bodies, A and B, are at the same temperature and in thermal contact, there's no net flow of heat; heat flows from A to B and B to A equally.

  • This dynamic equilibrium is called thermal equilibrium.

Zeroth Law of Thermodynamics

  • If two bodies, A and B, are separately in thermal equilibrium with a third body, C, then A and B are in thermal equilibrium with each other.

  • The zeroth law is applied when taking temperature readings with a thermometer.

  • If a thermometer gives the same reading in two different fluids, those fluids are at the same temperature.

Thermometric Properties

  • Thermometric properties are measurable properties that vary uniformly with temperature and are used to measure temperature.

Examples:

  • Volume of a fixed mass of mercury

  • EMF generated in a thermocouple

Important Features:

  1. Single-Valued Function: The property should not give the same value for two different temperatures or two different values for the same temperature. It can be used in a region where it behaves as a single-valued function.

  2. Continuous Function: The property should be continuous and not discontinuous (e.g., during a change of state).

  3. Linear Function: An ideal thermometric property should exhibit a linear variation with temperature.

Measurement of Temperature

  • Consider a thermometric property X that varies linearly with temperature \theta.

  • If XL and XH are values of X at lower (\thetaL) and upper (\thetaH) fixed points, and X\theta is the value at an unknown temperature \theta, then: \frac{\theta - \thetaL}{\thetaH - \thetaL} = \frac{X\theta - XL}{XH - XL}
    \theta = \left( \frac{X\theta - XL}{XH - XL} \right) (\thetaH - \thetaL) + \theta_L

Celsius Temperature Scale

  • Lower fixed point (\theta_L): Melting point of pure ice at standard atmospheric pressure, defined as 0 °C.

  • Upper fixed point (\theta_H): Boiling point of pure water at standard atmospheric pressure, defined as 100 °C.

  • Formula:
    \theta = \frac{X\theta - XL}{XH - XL} \times 100 °C

Absolute Thermodynamic Temperature Scale (Kelvin)

  • Based on an ideal thermometric property (e.g., pressure of a fixed mass of ideal gas at constant volume).

  • Extrapolating the graph to where it cuts the temperature axis gives -273.15 °C.

  • A new scale, the thermodynamic scale, is defined with this point as absolute zero.

  • Unit of measurement: Kelvin (K).

  • Absolute temperature is denoted by T.

  • This scale is theoretical and independent of any particular substance's properties, identical to the scale based on the pressure variation of an ideal gas at constant volume.

Triple Point of Water

  • The temperature at which pure water, water vapor, and ice are in thermal equilibrium.

  • Defined as 273.16 K (0.01 °C).

  • Formula:
    \frac{T}{T{tr}} = \frac{XT}{X{tr}} T = \frac{XT}{X_{tr}} \times 273.16

Relationship Between T and θ

  • One division of Kelvin = One division of Celsius.

  • When considering temperature differences (\Delta \theta or \Delta T), Kelvin and °C values are equal.

  • However, when considering a single temperature, the values are different.

  • 0 °C = 273.15 K

  • T = \theta + 273.15

Worked Examples

  1. A thermometric property gives values of 5.0 and 20.0 at 0 °C and 100 °C respectively. If the property is 11.0, then:
    \theta = \frac{11.0 - 5.0}{20.0 - 5.0} \times 100 = 40 °C
    T = 40 + 273.15 = 313.15 K

  2. A thermometric property is 68.29 at the triple point of water. At 300 K:
    X_T = \frac{300}{273.16} \times 68.29 \approx 75

Thermometers

  • Various types exist, differing in accuracy, range, sensitivity, and response time.

Examples:

  1. Mercury-in-Glass Thermometer

  2. Constant Volume Gas Thermometer: Uses pressure of a fixed mass of gas at constant volume.

  3. Constant Pressure Gas Thermometer: Uses volume of a fixed mass of gas at constant pressure.

  4. Platinum Resistance Thermometer: Uses electrical resistance of platinum wire.

  5. Thermocouple Thermometer

Mercury-in-Glass Thermometer

  • Thermometric property: Length of mercury column in a glass capillary tube.

  • Range: -30 °C to 350 °C (up to 500 °C with gas above the mercury column).

Advantages:

  • Easy to use, portable, low cost.

  • Direct readings, quick heat transfer.

  • Opaque mercury is easily visible and doesn't wet the glass.

Disadvantages:

  • Errors due to non-uniform bore, temperature differences, vapor pressure, deformation of bulb, and limited accuracy.

Thermocouple Thermometer

  • Thermometric property: Thermoelectric EMF developed in a junction of two different metals.

  • Two junctions: one at 0 °C (cold junction) and the other at the temperature to be measured (hot junction).

  • Examples of metal pairs: Copper-iron, nickel-nichrome, platinum-platinum rhodium alloy.

  • Seeback effect: Developing an EMF across a junction of two different metals.

  • Range: -200 °C to 1400 °C.

Advantages:

  • Small thermal capacity, quick response, suitable for small bodies, easy to construct, direct measurements with calibrated millivoltmeter.

Disadvantages:

  • Difficulty using a potentiometer, maintaining one junction at 0 °C, non-linear behavior at high temperatures (>400 °C).

Thermistor

  • A device whose electrical resistance varies with temperature, used as a temperature sensor.

  • Two types:

    • Negative Temperature Coefficient (NTC): Resistance decreases with temperature.

    • Positive Temperature Coefficient (PTC): Resistance increases with temperature.

  • Most practical thermistors are NTC type.

Thermal Expansion of Solids and Liquids

Expansion of Solids

  • Molecules in solids vibrate, and the amplitudes increase with temperature.

  • Thermal expansion: Increase of volume with temperature.

Linear Expansion

  • Increase of length of an object with temperature.

  • Formula: \Delta l \propto l0 \Delta \theta \Delta l = \alpha l0 \Delta \theta

    • \alpha: Linear expansivity (fractional increase of length per unit rise of temperature), unit: K^{-1}.

    • l2 = l1 (1 + \alpha \theta)

Area Expansion

  • Increase of area of an object with temperature.

  • Formula: \Delta A = \beta A_0 \Delta \theta

    • \beta: Superficial expansivity (fractional increase of area per unit rise of temperature), unit: K^{-1}.

    • A2 = A1 (1 + \beta \theta)

Relationship Between β and α

  • \beta = 2\alpha

Volume Expansion

  • Increase of volume of an object with temperature.

  • Formula: \Delta V = \gamma V_0 \Delta \theta

    • \gamma: Volume expansivity (fractional increase of volume per unit rise of temperature), unit: K^{-1}.

    • V2 = V1 (1 + \gamma \theta)

Relationship Between γ and α

  • \gamma = 3\alpha

Expansion of Liquids

  • Liquids are kept in containers, so both the liquid and container expand upon heating.

  • Volume expansion is the only relevant expansion.

  • Apparent expansion: Observed expansion without considering the container's expansion.

  • Real expansion: Actual expansion of the liquid.

  • Real expansion = Apparent expansion + Expansion of container.

Expansivities

  • Real expansivity (\gamma{real}): \gamma{real} = \frac{\Delta V{real}}{V0 \Delta \theta}

  • Apparent expansivity (\gamma{apparent}): \gamma{apparent} = \frac{\Delta V{apparent}}{V0 \Delta \theta}

  • \gamma{real} = \gamma{apparent} + \gamma_{container}

  • \gamma{real} = \gamma{apparent} + 3\alpha_{container}

Variation of Density with Temperature

  • V1 \rho1 = V2 \rho2

  • \rho2 = \frac{\rho1}{1 + \gamma \theta}

  • Density of liquids generally decreases with increasing temperature but water behaves differently in a particular range of temperature.

Anomalous Expansion of Water

  • Water's volume increases with temperature only above 4 °C. Between 0 °C and 4 °C, volume decreases with increasing temperature.

  • The density of water is maximum at 4 °C.

Gas Laws

  • The physical state of a gas is indicated by volume (V), pressure (p) and temperature (T).

  • Gas laws explain the behavior of a constant mass of gas between two states using relationships between V, p, and T.

Perfect Gas Assumptions

  1. Intermolecular attractive forces are negligibly small.

  2. The volume of gas molecules is negligibly small compared with the gas volume.

Boyle’s Law

  • The pressure of a fixed mass of a gas is inversely proportional to its volume at constant temperature.

  • p \propto \frac{1}{V}

  • pV = k

  • p1V1 = p2V2

Charles’ Law

  • The volume of a fixed mass of a gas is directly proportional to its absolute temperature at constant pressure.

  • V \propto T

  • \frac{V}{T} = k

  • \frac{V1}{T1} = \frac{V2}{T2}

The Pressure Law

  • Pressure of a fixed mass of gas is directly proportional to its absolute temperature at constant volume.

  • p \propto T

  • \frac{p}{T} = k

  • \frac{p1}{T1} = \frac{p2}{T2}

Avogadro’s Hypothesis

  • Equal volumes of all gases at the same temperature and pressure contain the same number of molecules.

Mole

  • The amount of a substance containing the number of atoms equal to that contained in 0.012 kg of the carbon-12 isotope.

Avogadro Number (N_A)

  • The number of molecules in one mole of a gas.

  • N_A = 6.02 \times 10^{23} \text{mol}^{-1}

  • At standard temperature (0 °C) and normal pressure (760 mm Hg), the volume of a mole of any gas is 22.4 liters.

Molar Mass

  • The mass of one mole of a gas.

Equation of State of a Gas

  • Relates pressure, volume, and temperature for a fixed mass of gas.

  • \frac{p1V1}{T1} = \frac{p2V2}{T2}

  • \frac{pV}{T} = constant

Ideal Gas Equation

  • For one mole of a gas: pV = RT

  • “R” is the universal gas constant.

  • For 'n' moles of the gas: pV = nRT

  • R = 8.31 \text{ J mol}^{-1} \text{K}^{-1}

  • If the mass of the gas is 'm' and the molar mass is 'M': pV = \frac{m}{M} RT \rho = \frac{m}{V} = \frac{pM}{RT}

    • \rho is the density of the gas.

Dalton’s Law of Partial Pressures

  • For a mixture of non-reacting gases in a closed volume, the total pressure is the sum of the partial pressures exerted by each gas.

  • p = pA + pB + p_C

Kinetic Theory of Gases

  • Relates pressure and volume of a gas using microscopic properties of gas molecules.

Assumptions

  1. Gas molecules behave as perfectly elastic spheres.

  2. The volume of gas molecules is negligible compared with the volume of the container.

  3. Intermolecular attractive forces are negligibly small.

  4. The time of collision of a molecule with the wall is negligibly small compared with the time between collisions.

Expression

  • pV = \frac{1}{3} mN \overline{c^2}

    • p = pressure of the gas

    • V = Volume of the gas

    • m = mass of a gas molecule

    • N = total number of gas molecules

    • \overline{c^2} = mean square velocity of molecules

Mean Square Speed (\overline{c^2})

  • The mean of the squares of the speeds of gas molecules.

Root Mean Square Speed (\sqrt{\overline{c^2}})

  • The square root of the mean square speed

Relation between Density and Root Mean Square Speed

  • \rho = \frac{Nm}{V}

  • \overline{c^2} = \frac{3p}{\rho}

Relation between Temperature and Root Mean Square Speed

  • \overline{c^2} = \frac{3RT}{M}

Kinetic Energy and Temperature

  • \frac{1}{2} m \overline{c^2} = \frac{3}{2} kT

    • k is Boltzmann constant (k = 1.38 \times 10^{-23} \text{ J K}^{-1})

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