Heat and Thermodynamics Formula Sheet - NDA Physics Notes

Welcome to this comprehensive LibreTexts-style chapter guide covering Heat and Thermodynamics specifically tailored for NDA (National Defence Academy) aspirants. Heat and Thermodynamics forms a critical part of the Physics section in the General Ability Test (GAT) paper. This guide breaks down essential concepts, key mathematical relations, and exam-focused takeaways derived from actual NDA examination trends.

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1. Temperature Scales and Thermal Expansion

Temperature is a measure of the average kinetic energy of the molecules in a substance. Understanding inter-conversion between different temperature scales is vital for solving basic NDA Physics problems.

Temperature Conversions

To convert between Celsius (\(^{\circ}\text{C}\)), Kelvin (\(\text{K}\)), and Fahrenheit (\(^{\circ}\text{F}\)), use the following relations:

\[T(^{\circ}\text{C}) = T(\text{K}) - 273.15\] \[T(^{\circ}\text{F}) = \frac{9}{5}T(^{\circ}\text{C}) + 32\]

Thermal Expansion

When solids, liquids, or gases are heated, their dimensions increase. Expansion occurs in length (linear), area (superficial), and volume (cubical):

• Linear Expansion: \(\Delta L = \alpha L_0 \Delta T\)
• Area Expansion: \(\Delta A = \beta A_0 \Delta T\) where \(\beta \approx 2\alpha\)
• Volume Expansion: \(\Delta V = \gamma V_0 \Delta T\) where \(\gamma \approx 3\alpha\)

Here, \(\alpha\), \(\beta\), and \(\gamma\) are the coefficients of linear, superficial, and cubical expansion respectively.

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2. Heat, Specific Heat Capacity, and Calorimetry

Calorimetry is the science of measuring heat released or absorbed during a chemical or physical process.

Key Formulas

• Sensible Heat: The heat required to change temperature without altering the state: \[Q = mc\Delta T\] where \(m\) is mass, \(c\) is specific heat capacity, and \(\Delta T\) is temperature change.
• Latent Heat: The heat required for a phase transition (e.g., solid to liquid or liquid to gas) at constant temperature: \[Q = mL\] where \(L\) is the latent heat of fusion or vaporization.
• Heat Capacity (\(C\)): Total heat needed to raise the temperature of a body by \(1^{\circ}\text{C}\): \[C = mc = \frac{Q}{\Delta T}\]

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3. Mechanisms of Heat Transfer

Heat transfer occurs via three primary mechanisms: Conduction, Convection, and Radiation.

1. Conduction (Fourier's Law)

Direct transfer of heat through molecular collisions without bulk motion of the medium: \[H = \frac{dQ}{dt} = -kA \frac{dT}{dx}\]

Thermal resistance is given by \(R_{\text{th}} = \frac{L}{kA}\), where \(k\) is thermal conductivity, \(A\) is cross-sectional area, and \(L\) is length.

2. Convection (Newton's Law of Cooling)

Heat transfer by actual movement of fluid mass:

\[\frac{dQ}{dt} = hA(T - T_s)\]

where \(h\) is the convective heat transfer coefficient, \(A\) is surface area, and \(T_s\) is the surrounding temperature.

3. Radiation

Electromagnetic radiation emitted by a body due to its temperature.

• Stefan-Boltzmann Law: Total energy radiated by an ideal black body per unit area: \[E = \sigma e A T^4\] Net radiation rate in surroundings at temperature \(T_s\): \[E_{\text{net}} = \sigma e A (T^4 - T_s^4)\]
• Wien's Displacement Law: The wavelength corresponding to peak radiation emission is inversely proportional to absolute temperature: \[\lambda_{\text{max}} T = b\] where \(b \approx 2.898 \times 10^{-3} \text{ m K}\) is Wien's constant.

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4. Kinetic Theory of Gases (KTG)

Kinetic theory links microscopic properties of gas molecules to macroscopic observables like pressure and temperature.

Ideal Gas Equation

\[PV = nRT = N k_B T\]

where \(n\) is the number of moles, \(R\) is the universal gas constant, \(N\) is total number of molecules, and \(k_B\) is Boltzmann's constant.

Molecular Speeds & Kinetic Energy

• Root Mean Square (RMS) Speed: \[v_{\text{rms}} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3k_B T}{m}}\]
• Average Kinetic Energy per Molecule: \[E_{\text{avg}} = \frac{3}{2} k_B T\]

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5. Laws of Thermodynamics and Work Done

First Law of Thermodynamics

Statement of conservation of energy:

\[\Delta U = Q - W\]

where \(\Delta U\) is change in internal energy, \(Q\) is heat added to the system, and \(W\) is work done by the system.

Work Done by Gas

\[W = \int P dV\]

Mayer's Relation and Ratio of Specific Heats

\[C_p - C_v = R\] \[\gamma = \frac{C_p}{C_v}\]

where \(C_p\) is molar specific heat at constant pressure, \(C_v\) is molar specific heat at constant volume, and \(\gamma\) is the adiabatic index.

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6. NDA Exam Analysis: PYQs, Weightage, and Strategy

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7. People Also Ask (Frequently Asked Questions)