Heat Capacity

Definition and Basic Concepts

Heat Capacity Defined

Heat capacity (C): amount of heat (q) required to raise temperature (T) of a substance by 1 kelvin (K) or 1 degree Celsius (°C) at constant pressure or volume.

Fundamental Relation

Mathematically: C = q / ΔT where ΔT is temperature change; q is heat absorbed or released.

Extensive Property

Heat capacity depends on amount of substance: doubles mass, doubles heat capacity.

Types of Heat Capacity

Heat Capacity at Constant Volume (C_V)

Defined when volume is held constant; no work done by expansion; internal energy changes only.

Heat Capacity at Constant Pressure (C_P)

Defined when pressure is held constant; includes work done by expansion; generally greater than C_V.

Specific Heat Capacity

Heat capacity per unit mass; intensive property; units J·g^-1·K^-1.

Molar Heat Capacity

Heat capacity per mole; units J·mol^-1·K^-1; useful for chemical thermodynamics.

Units and Dimensions

SI Units

Heat capacity units: joules per kelvin (J·K^-1).

Specific Heat Units

Joules per gram kelvin (J·g^-1·K^-1), or calories per gram degree Celsius (cal·g^-1·°C^-1).

Molar Heat Capacity Units

Joules per mole kelvin (J·mol^-1·K^-1).

Dimensional Analysis

Heat capacity dimension: energy / temperature; M·L²·T^-2·Θ^-1.

Measurement Techniques

Calorimetry

Direct measurement of heat exchanged using calorimeters; types include bomb and coffee cup calorimeters.

Adiabatic Calorimetry

Measures temperature change with minimal heat loss; high precision for heat capacity determination.

Differential Scanning Calorimetry (DSC)

Measures heat flow difference between sample and reference as temperature varies; useful for phase transitions.

Experimental Considerations

Ensure thermal equilibrium; correct for heat losses; calibrate instruments accurately.

Thermodynamic Significance

Relation to Internal Energy and Enthalpy

C_V = (∂U/∂T)_V, C_P = (∂H/∂T)_P; U = internal energy, H = enthalpy.

Thermodynamic Identities

C_P - C_V = nR for ideal gases; reflects work done during expansion.

Entropy and Heat Capacity

Heat capacity influences entropy changes: dS = C dT / T for reversible processes.

Heat Capacity and Phase Changes

Heat capacity changes drastically near phase transitions; latent heat dominates.

Application in Calorimetry

Heat Transfer Calculations

q = C × ΔT used to quantify heat absorbed or released in reactions.

Determination of Specific Heat

Known mass and temperature change allow calculation of specific heat.

Reaction Enthalpy Measurements

Heat capacities enable conversion from temperature change to energy changes.

Example: Metal Sample

Metal mass heated; temperature recorded; specific heat inferred from heat exchange with water.

Temperature Dependence

Non-constant Heat Capacity

Heat capacity varies with temperature; not strictly linear.

Empirical Models

Heat capacity often expressed as polynomial: C = a + bT + cT² + ...

Debye Model

Low temperature behavior of solids; C ∝ T³ below Debye temperature.

Einstein Model

Quantized vibrational modes; approximates heat capacity of solids at intermediate temperatures.

Molecular Basis

Degrees of Freedom

Atoms possess translational, rotational, vibrational modes contributing to heat capacity.

Equipartition Theorem

Each quadratic degree of freedom contributes ½ R to molar heat capacity.

Quantum Effects

At low temperatures, vibrational modes freeze out; heat capacity decreases.

Polyatomic vs Diatomic

More complex molecules have higher heat capacities due to additional vibrational modes.

Heat Capacity vs Specific Heat

Definitions

Heat capacity: total heat needed for temperature change; specific heat: per unit mass or mole.

Intensive vs Extensive

Heat capacity: extensive; specific heat: intensive.

Conversions

Specific heat (c) = Heat capacity (C) / mass (m) or moles (n).

Practical Usage

Specific heat used for comparisons; heat capacity used for total energy calculations.

Heat Capacity of Gases

Ideal Gas Behavior

C_P and C_V related by C_P - C_V = R.

Monatomic Gases

C_V ≈ 3/2 R; C_P ≈ 5/2 R; degrees of freedom: 3 translational.

Diatomic Gases

Include rotational and vibrational modes; higher heat capacities.

Real Gas Deviations

Interactions cause deviations from ideal heat capacity values.

Heat Capacity in Materials Science

Metals

Heat capacity influenced by electron and lattice vibrations; linear term at low T.

Polymers

Heat capacity varies with chain flexibility and phase.

Insulators

Lattice vibrations dominate heat capacity; Debye model applicable.

Application in Thermal Management

Materials selected based on heat capacity for thermal storage and insulation.

Practical Examples and Calculations

Example 1: Calculating Heat Required

Calculate q to raise 100 g water from 25°C to 75°C; c = 4.18 J·g^-1·°C^-1.

q = m × c × ΔTq = 100 g × 4.18 J/g°C × (75 - 25)°Cq = 100 × 4.18 × 50 = 20,900 J

Example 2: Determining Specific Heat of Metal

Metal of mass 50 g heated to 100°C placed in 200 g water at 20°C; final temp 25°C; c_water = 4.18 J/g°C.

Heat lost by metal = heat gained by waterm_metal × c_metal × (100 - 25) = m_water × c_water × (25 - 20)50 × c_metal × 75 = 200 × 4.18 × 5c_metal = (200 × 4.18 × 5) / (50 × 75)c_metal ≈ 1.11 J/g°C

Table: Heat Capacities of Common Substances

References

• Atkins, P., Physical Chemistry, 10th ed., Oxford University Press, 2014, pp. 152-176.
• Laidler, K.J., Meiser, J.H., Physical Chemistry, 3rd ed., Benjamin/Cummings, 1999, pp. 123-140.
• McQuarrie, D.A., Statistical Mechanics, University Science Books, 2000, pp. 210-235.
• Smith, J.M., Van Ness, H.C., Abbott, M.M., Introduction to Chemical Engineering Thermodynamics, 7th ed., McGraw-Hill, 2005, pp. 45-60.
• Callen, H.B., Thermodynamics and an Introduction to Thermostatistics, 2nd ed., Wiley, 1985, pp. 78-95.

Introduction

Heat capacity quantifies heat required to change a substance's temperature. Essential in thermochemistry, it links energy transfer and temperature variation. Different forms exist: molar, specific, at constant volume and pressure. Applications span calorimetry, materials science, and thermodynamics.

"Heat capacity is a fundamental thermodynamic property characterizing a substance's ability to absorb heat without a significant change in temperature." -- Peter Atkins