Heat and Work

Energy is the most fundamental dynamical concept in physics. It takes many forms (kinetic, electrostatic, gravitational, chemical, nuclear) and can be converted from one to another, but the total amount of energy in the universe never changes: the conservation of energy. Thermodynamics tracks two ways energy enters or leaves a system.

Heat is any spontaneous flow of energy from one object to another caused by a difference in temperature, by conduction (molecular contact), convection (bulk motion of a gas or liquid, as warmer material expands and rises), or radiation (electromagnetic waves, mostly infrared at room temperature, visible from a light-bulb filament or the Sun). Traditionally, 1 cal = 4.184 J raises one gram of water by 1 °C, and a food Calorie is a kilocalorie. Work is any other transfer of energy, where some agent actively puts energy into the system. Both are energy in transit. For the total energy content \(U\) of a system,

the first law of thermodynamics, with \(Q\) the heat added to the system and \(W\) the work done on it, all in joules.

Push a piston of area \(A\) inward by \(\Delta x\) with force \(F = PA\), and the work done on the gas is \(W = F\Delta x = PA\,\Delta x = -P\,\Delta V\). This requires the pressure to be uniform, so the piston must move slowly enough for the gas to keep equilibrating: a quasistatic process. When the pressure changes along the way,

which is minus the area under the curve on a \(PV\) diagram. For an ideal gas, \(PV = Nk_{\mathrm{B}}T\) and \(U = \tfrac{f}{2}Nk_{\mathrm{B}}T\), so \(\Delta U = \tfrac{f}{2}\Delta(PV)\). Two special cases stand out: isothermal compression, so slow that the temperature doesn’t change (\(\Delta T = 0\), so \(\Delta U = 0\)), and adiabatic compression, so fast that no heat escapes (\(Q = 0\)):

Build a process step by step below. One mole of gas; choose a kind of step, set where it ends, and apply it. The shaded area is the work, red when work is done on the gas and blue when the gas does work.

Adiabatic compression is so fast that no heat escapes, but not so fast that the process stops being quasistatic. Real compressions lie somewhere between isothermal and adiabatic. With \(Q = 0\), \(dU = \tfrac{f}{2}Nk_{\mathrm{B}}\,dT = -P\,dV = -\tfrac{Nk_{\mathrm{B}}T}{V}\,dV\), and integrating,

The adiabatic exponent \(\gamma > 1\) makes an adiabat steeper than an isotherm: compressing adiabatically connects a low-temperature isotherm to a high-temperature one, as the explorer above shows. In terms of pressure, \(dT/dP = \tfrac{2}{f+2}\,T/P\) (Problem 1.40).

Combine this with the barometric equation \(dP/dz = -(mg/k_{\mathrm{B}}T)P\): air rising adiabatically expands and cools at the dry adiabatic lapse rate,

for nitrogen (\(f = 5\)). Ground heated by sunlight warms the air above it. If the actual temperature falls with height faster than this, a rising parcel of air stays warmer and less dense than its surroundings even after its adiabatic cooling, so it keeps rising: convection, which stirs the lowest 10 to 15 km of the atmosphere, the troposphere.

The heat capacity is the heat needed per degree of temperature rise, an extensive quantity; per unit mass it is the specific heat:

This is ambiguous: it depends on whether work is being done. With no work (constant volume), \(C_V = (\partial U/\partial T)_V\), the “energy capacity”: 1 cal/°C = 4.2 J/°C for a gram of water. At constant pressure, the object also does work as it expands:

The last two hold for an ideal gas whose energy is all in quadratic degrees of freedom (with \(f\) independent of temperature, which isn’t true in general). Per mole, a monatomic gas has \(C_V = \tfrac32 R = 12.5\ \mathrm{J/K}\), and a solid at high temperature has \(f = 6\), so \(C_V = 3R\) (the rule of Dulong and Petit). Heat one mole by 10 K both ways and compare:

During a phase transformation such as melting ice or boiling water, heat goes in without raising the temperature at all, so \(C = Q/\Delta T \to \infty\). What matters instead is the latent heat \(L\) to convert the substance completely, or per unit mass \(\ell = L/m\), by convention at a constant pressure of 1 atm: 333 J/g (about 80 cal/g) to melt ice, and 2260 J/g (about 540 cal/g) to boil water. Compare 100 cal/g to warm water from 0 °C to 100 °C. Add heat to one gram of ice at −20 °C:

To handle constant-pressure processes conveniently, define the enthalpy

the energy needed to create the system out of nothing and put it into its environment, including the work \(PV\) to push the atmosphere out of the way. If the system were annihilated, \(U + PV\) is what you could extract, the atmosphere doing work \(PV\) as it collapses to fill the vacuum. At constant pressure, \(\Delta H = \Delta U + P\,\Delta V\), and the first law \(\Delta U = Q - P\,\Delta V + W_{\text{other}}\) becomes

so compression and expansion work can be forgotten; \(C_P\) is the “enthalpy capacity,” as \(C_V\) is the energy capacity. Enthalpy changes are tabulated for phase changes, chemical reactions, ionization, and dissolving. Forming water, \(\mathrm{H_2} + \tfrac12\mathrm{O_2} \to \mathrm{H_2O}\), has \(\Delta H = -286\ \mathrm{kJ}\) per mole: burning a mole of hydrogen releases 286 kJ. Where does it come from?

Problem 1.55. Heat capacities are normally positive, but systems held together by gravity, such as stars and star clusters, are an exception. For two equal masses \(m\) orbiting their center of mass at distance \(r\) each, \(Gm^2/4r^2 = mv^2/r\), so

This holds on average for any system bound by mutual gravitational attraction: a special case of the virial theorem, \(2\langle U_{\text{kinetic}}\rangle = n\langle U_{\text{potential}}\rangle\) for a potential \(\propto r^n\) (here \(n = -1\); for \(n = 2\), the harmonic potential, it is equipartition again). So \(U_{\text{total}} = -U_{\text{kinetic}}\). Model a star as a gas of particles interacting only gravitationally, with \(U_{\text{kinetic}} = \tfrac32 k_{\mathrm{B}}T\) per particle:

Add energy, and the system gets colder: the orbit widens, the kinetic energy decreases, and the potential energy rises by even more. Try it with the pair below.