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Introduction to Thermodynamics: Heat Capacity

Shaun Williams, PhD

The First Law of Thermodynamics

Extensive and Intensive Parameters

In any system, energy, entropy, volume, and number of moles are extensive propteries. In any system, temperature and pressure and intensive properties.

Thermodynamic State

First Law of Thermodynamics

Inexact Differential, \(\dbar\)

Approximations and Assumptions


Types of Systems

Definitions of Terms Used for Thermodynamics

System Meaning
Open May exchange mass and energy with surroundings
Closed All \(n_i\) fixed (no mass exchange and no reactions)
Pure One chemical component (implied unless stated otherwise)
Isolated No mass or energy exchange with surroundings
Homogeneous One phase of matter
Heterogeneous More than one phase of matter
Container Meaning
Adiabatic Walls prevent heat flow
Diathermal Walls allow heat flow
Processes Meaning
Quasistatic System and surroundings always in equilibrium
Reversible System and surroundings at constant total entropy (\(\Delta S_T=0\))
Adiabatic No heat flow (\(q=0\))
Isentropic System at constant entropy (\(\Delta S=0\))
Isothermal System at constant temperature (\(\Delta T=0\))
Isobaric system at constant pressure (\(\Delta P=0\))
Isochoric System at constant volume (\(\Delta V=0\))
Isoenergetic System at constant energy (\(\Delta E=0\))

The Standard State

Mathematical Tools

Partial Derivatives of Energy

The Fundamental Equation


Legendre Transformation

Legendre Transforms - Thermodynamic Potentials

Derivatives From the Thermodynamic Potentials

Closed System Example

Mixed Second Derivatives

$$ \begin{align} \left[ \frac{\partial}{\partial V} \left( \frac{\partial E}{\partial S} \right)_{V,n} \right]_{S,n} &= \left[ \frac{\partial}{\partial S} \left( \frac{\partial E}{\partial V} \right)_{S,n} \right]_{V,n} = \left( \frac{\partial T}{\partial V} \right)_{S,n} \\ \left[ \frac{\partial}{\partial S} \left( \frac{\partial E}{\partial V} \right)_{S,n} \right]_{V,n} &= -\left( \frac{\partial P}{\partial S} \right)_{V,n} \\ \end{align} $$

Maxwell Relations

Volume Scaled Derivatives

Partition Functions

The Entropy

$$ \begin{align} \frac{E}{k_\mathrm{B}T} &= \ln \Omega - \ln Q(T) \\ k_\mathrm{B} \ln \Omega &= k_\mathrm{B} \ln Q(T) + \frac{E}{T} \\ S &= k_\mathrm{B} \ln Q(T) + k_\mathrm{B}T \left( \frac{\partial \ln Q(T)}{\partial T}\right)_{V,n} \end{align} $$

The Helmholtz Energy

$$ \begin{align} F =& E-TS \\ =& k_\mathrm{B}T^2 \left(\frac{\partial \ln Q(T)}{\partial T}\right)_{V,n} \\ & -T\left[ k_\mathrm{B} \ln Q(T) + k_\mathrm{B}T\left(\frac{\partial \ln Q(T)}{\partial T}\right)_{V,n} \right] \\ F =& -k_\mathrm{B}T \ln Q(T) \end{align} $$

The Pressure and the Chemical Potential

$$ \begin{align} P &= -\left( \frac{\partial F}{\partial V}\right)_{T,n} = k_\mathrm{B}T \left(\frac{\partial \ln Q}{\partial V}\right)_{T,n} \\ \mu &= \left(\frac{\partial F}{\partial n}\right)_{T,V} = -k_\mathrm{B}T \left(\frac{\partial \ln Q}{\partial n}\right)_{T,V} \end{align} $$

Heat Capacities

Heat Capacity

Measuring Heat Capacities

Different Types of Work

Work can be done by a gas pushing on a movable wall.
Work can be done by a sphere flattening out when it contacts a solid surface.
Work can be done when a chemical reaction occurs.

More on Heat Capacities

\(PV\) Work

Without \(PV\) Work

$$ \begin{align} \left(\frac{\partial E}{\partial T}\right)_{V,n} &= \left( \frac{T\partial S - P\partial V}{\partial T}\right)_{V,n} \\ &= \left(\frac{T\partial S}{\partial T}\right)_{V,n} \\ &= \left(\frac{\dbar q}{dT}\right)_{V,n} \\ &= C_V(T) \\ \text{So: }& C_V(T)=\left(\frac{\partial E}{\partial T}\right)_{V} \end{align} $$

For an Ideal Gas

Example 7.1

You have a constant-volume sample with a known quantity of mercury gas, a thermometer, and a calorimeter that allows you to add or remove measured amounts of energy from the sample. How could you use this apparatus to determine whether gas-phase mercury was in monatomic or diatomic form, that is, \(\chem{Hg(g)}\) or \(\chem{Hg_2(g)}\)? What numerical results would you look for in your measurements?

Constant Pressure Heat Capacity

Heating at Constant Pressure vs Constant Volume

When heat is added to a constant volume system, the particles move faster but they are still as close to each other as originally. When heat is added to a constant pressure system, the particles move faster but spread out.

How Are \(C_P\) and \(C_V\) Related?

How are \(C_P\) and \(C_V\) Related?

Ideal Gas

Example 7.2

Use the definition of the coefficient of thermal expansion to find the relationship between \(C_P\) and \(C_V\) at the temperature at maximum density (TMD) for any substance.

Heat Capacities and Real Gases

Heat Capacities of Several Substances

In general, as temperature increases, the heat capacities of all substances increase, some substances increase to a greater extent than others.

More on Heat Capacities of Real Gases

Example 7.3

The molar heat capacity at constant pressure of \(\chem{CCl_4}\) is \(83.30\,\mathrm{J\,K^{-1}\,mol^{-1}}\) at \(298\,\mathrm{K}\). If we heated a \(1.00\,\mathrm{mol}\) sample of \(\chem{CCl_4}\) gas with \(1.00\,\mathrm{J}\), starting near \(298\,\mathrm{K}\), roughly what percent of the heat would go into rotation, and what percent into vibration? Assume that the temperature change is small enough that the heat capacity may be treated as constant.

For a Linear Diatomic Molecule

Example 7.4

The vibrational constant of \(\chem{{}^{35}Cl_2}\) is \(560.5\,\mathrm{cm^{-1}}\). Estimate the value of \(C_{Pm}\) for \(\chem{{}^{35}Cl_2}\) at \(298.15\,\mathrm{K}\).

Idealized Solid

In a solid, Einstein envisioned that each particle of the solid is independently vibrating harmonically.

Einstein Solid

The Einstein Heat Capacity

Success of the Einstein Solid

Phonon Modes in Crystals

Groups of particles in a solid can move in a collective motion, called a phonon.

Debye Heat Capacity

The Debye Heat Capacity Equation

Comparison of Heat Capacities

Below 300 K there is a bit of a difference between the Debye value and the Einstein value of the heat capacities.
As the temperature increases, the Debye and Einstein heat capacities become very close to the same value.

Specific Heat Capacities

Heat Capacities of Selected Solids

Solid \(\mathcal{M} \\ (\mathrm{g\,mol^{-1}})\) \(C_{Pm} \\ (\mathrm{J\,K^{-1}\,mol^{-1}})\) \(c_{P} \\ (\mathrm{J\,K^{-1}\,g^{-1}})\) Solid \(c_{P} \\ (\mathrm{J\,K^{-1}\,g^{-1}})\)
Lithium 6.94 24.8 0.516 Hematite
Beryllium 9.01 16.4 0.113 Pyrex glass 0.84
Carbon (graphite) 12.01 8.53 0.059 Gypsum 1.09
Carbon (diamond) 12.01 6.11 0.043 Loose wool 1.26
Sodium 22.99 28.2 1.23 Paper 1.34
Silicon 28.09 20.0 0.75 White pine 2.5
Sulfur 32.07 22.6 0.71 Paraffin wax 2.9
Arsenic 74.92 24.6 0.33
Tantalum 180.9 25.4 0.14
Uranium 238.0 27.7 0.12

Heat Capacities of Monatomic Liquids

Molar Heat Capacities of Simple Liquids and Gases

Substance \(T_b \\ (\mathrm{K})\) \(C_{Vm}\mathrm{(liq)} \\ (\mathrm{J\, K^{-1}\, mol^{-1}})\) \(C_{Vm}\mathrm{(gas)} \\ (\mathrm{J\, K^{-1}\, mol^{-1}})\) \(\text{Difference} \\ (\mathrm{J\, K^{-1}\, mol^{-1}})\)
\(\chem{N_2}\) 77.4 54.5 29.3 25.2
\(\chem{Ar}\) 87.3 40.6 14.5 26.1
\(\chem{CH_4}\) 111.7 51.6 27.3 24.3
\(\chem{Xe}\) 165.1 37.3 12.4 24.9

Population of Different Phases in the Potential

The vibrational levels in a Lennard-Jones potential stack up as the energy increases. Within the well you effectively have a solid. At higher energies you have a liquid and then a gas.