Thermodynamics notes

MDCAT Physics

Thermodynamics deals with heat, temperature, internal energy, work and the laws governing energy transfer in physical systems. These notes cover thermal equilibrium, heat capacity of gases, thermodynamic processes, the first law, Carnot engines and related gas laws.

Temperature, Thermal Equilibrium and Heat

Temperature is a measure of the degree of hotness of a body and is related to the average kinetic energy of its molecules. Two bodies are in thermal equilibrium when there is no net flow of heat between them. The zeroth law of thermodynamics states that if body A is in thermal equilibrium with body B, and body B is in thermal equilibrium with body C, then A and C are also in thermal equilibrium.

Heat is energy transferred from a body at higher temperature to a body at lower temperature because of a temperature difference. Heat is not stored as a substance. Its SI unit is joule (J). The temperature of a body can change when it receives or loses heat, but heat transfer can also change its state or do work.

If no heat enters or leaves a system, the process is adiabatic. An adiabatic process does not always have constant temperature. If the system also does no work, its internal energy and, for a fixed mass of an ideal gas, its temperature remain constant.

Temperature is measured in kelvin (K) in thermodynamics. The Celsius and Kelvin scales are related by T(K) = θ(°C) + 273.
Absolute zero is 0 K, which corresponds to approximately minus 273°C.
Heat flows naturally from a higher-temperature body to a lower-temperature body.
Thermal equilibrium means that the net heat flow between bodies is zero.
An isolated system exchanges neither heat nor work with its surroundings.
For an ideal gas, internal energy depends only on temperature.
Heating a gas in a fixed-volume container increases its pressure because the average force per impact on the container walls increases.
Heat capacity is the heat required to raise the temperature of a body by 1°C or 1 K.
Specific heat capacity is the heat required to raise the temperature of unit mass of a substance by 1°C or 1 K.
Q = mcΔT, where c is the specific heat capacity and C = Q/ΔT is heat capacity.

Internal Energy and Work

Internal energy is the total microscopic energy of a system. It includes the random translational, rotational and vibrational kinetic energies of molecules, together with intermolecular potential energy. For an ideal gas, intermolecular forces are neglected, so internal energy is a function of temperature only.

When a gas expands, it usually does work on its surroundings. When a gas is compressed, work is done on the gas. Using the FSc convention, W is work done by the system, so the first law is ΔU = Q - W.

For a small change at constant pressure, the work done by a gas is W = PΔV. More generally, work is the area under the pressure-volume graph. Expansion gives positive work by the gas, while compression gives negative work by the gas.

Internal energy is represented by U and its change by ΔU.
For an ideal gas, constant temperature means constant internal energy.
Expansion: ΔV is positive and work done by the gas is positive.
Compression: ΔV is negative and work done by the gas is negative in the convention W by the system.
At constant volume, ΔV = 0, so work done by the gas is zero.
For 2 moles expanding adiabatically with ΔU = minus 100 J, W = 100 J because Q = 0 and ΔU = -W.
If 100 J of work is done on a gas during adiabatic compression, W by the gas = minus 100 J and ΔU = 100 J.
Internal energy is a state function, but heat and work depend on the path followed.
The work done in a process is represented by the area under the P-V curve.

Molar Specific Heat of a Gas

Molar specific heat is the heat required to raise the temperature of one mole of a substance by 1 K. For gases, the molar heat capacity depends on the condition under which heating takes place. The two important values are molar heat capacity at constant volume, Cv, and at constant pressure, Cp.

At constant volume, the gas does no external work. Therefore, all supplied heat increases the internal energy. At constant pressure, the gas expands while being heated, so supplied heat increases internal energy and also does external work. Therefore Cp is greater than Cv.

For an ideal gas, the difference between molar heat capacities is given by Mayer’s relation, Cp - Cv = R. Here R is the universal gas constant, 8.31 J mol⁻¹ K⁻¹.

At constant volume, Qv = nCvΔT.
At constant pressure, Qp = nCpΔT.
For one mole of an ideal gas, Cp - Cv = R.
For n moles, the difference in total heat capacities is nR.
Cp is greater than Cv because a gas heated at constant pressure also performs expansion work.
For a monatomic ideal gas, Cv = 3R/2 and Cp = 5R/2.
For a diatomic ideal gas with vibrational modes neglected, Cv = 5R/2 and Cp = 7R/2.
For a monatomic gas heated slowly at constant pressure, W = 2Q/5 because W = nRΔT and Q = nCpΔT = 5nRΔT/2.
For 3 moles of a monatomic ideal gas, n(Cp - Cv) = 3R.
The ratio γ is defined as γ = Cp/Cv.

Degrees of Freedom and Internal Energy of Gases

The energy distribution of a gas can be understood using degrees of freedom. A monatomic molecule has three translational degrees of freedom. A diatomic molecule has three translational and two rotational degrees of freedom when vibrational modes are neglected.

For an ideal gas with f active degrees of freedom, the internal energy is U = f nRT/2. This relation is useful for mixtures because the total internal energy is the sum of the internal energies of the individual gases.

Oxygen is diatomic, so its vibrational modes are neglected in the usual FSc treatment and it has f = 5. Argon is monatomic and has f = 3.

A monatomic ideal gas has U = 3nRT/2.
A diatomic ideal gas with vibrations neglected has U = 5nRT/2.
For a gas mixture, total internal energy equals the sum of the internal energies of all gases.
For 2 moles of O2 and 4 moles of Ar, U = 2(5RT/2) + 4(3RT/2) = 11RT.
The molar heat capacity of a monatomic gas at constant volume is 3R/2.
The molar heat capacity of a diatomic gas at constant volume, without vibrations, is 5R/2.
Vibrational modes increase the number of active degrees of freedom when they are included.
The internal energy of an ideal gas mixture depends on the temperature and the amount and type of gases present.

Thermodynamic Processes

A thermodynamic process describes a change from one state of a system to another. The common processes are isothermal, adiabatic, isobaric and isochoric. Each name states the quantity kept constant.

For an ideal gas, the equation of state is PV = nRT. Boyle’s law is the special case for constant temperature. If pressure is doubled at constant temperature, volume becomes one half of its original value.

A reversible process occurs through a continuous succession of equilibrium states and can be reversed without leaving a net change in the system and surroundings. In practice, perfectly reversible processes are idealisations. Isothermal compression carried out very slowly with negligible friction is a reversible process.

Isothermal process: temperature remains constant, so for an ideal gas ΔU = 0.
For an isothermal ideal-gas process, Q = W.
Boyle’s law: PV = constant when temperature and amount of gas are constant.
Adiabatic process: Q = 0, so ΔU = -W.
Adiabatic expansion causes the gas to cool because the gas does work at the expense of internal energy.
Adiabatic compression causes the gas to heat because work is done on the gas.
Isobaric process: pressure remains constant and W = PΔV.
Isochoric process: volume remains constant and W = 0.
For an adiabatic process, PV^γ = constant.
For a monatomic gas, γ = 5/3. If its volume is reduced to 1/8, P2/P1 = (V1/V2)^γ = 8^(5/3) = 32.

First Law of Thermodynamics

The first law is the law of conservation of energy applied to thermodynamic systems. Heat supplied to a system is used partly to increase its internal energy and partly to perform work by the system.

With W representing work done by the system, the first law is ΔU = Q - W. Some books define W as work done on the system. With that convention, the same law is written ΔU = Q + W. The physical result is the same if the sign convention is used consistently.

For an adiabatic process, Q = 0. Therefore ΔU = -W by the system. Thus, work done by the system equals the decrease in internal energy, while work done on the system equals the increase in internal energy.

First law using work done by the system: ΔU = Q - W.
First law using work done on the system: ΔU = Q + W.
At constant volume, W = 0, so ΔU = Q.
For an isothermal ideal-gas process, ΔU = 0, so Q = W.
For an adiabatic process, Q = 0 and ΔU = -W by the system.
During adiabatic expansion, the internal energy decreases and the temperature falls.
During adiabatic compression, the internal energy increases and the temperature rises.
When a high-pressure gas in a balloon suddenly expands after bursting, the process is approximately adiabatic and the gas cools.
The first law does not determine the direction in which a process naturally occurs.

Heat Engines, Carnot Engine and Efficiency

A heat engine operates in a cycle. It absorbs heat Qh from a hot reservoir, converts part of this energy into work W, and rejects heat Qc to a cold reservoir or sink. Since the engine returns to its initial state after one cycle, its net change in internal energy is zero.

The efficiency of an engine is the ratio of useful work output to heat absorbed from the hot reservoir. A Carnot engine is an ideal reversible engine operating between two temperatures. Its efficiency depends only on the temperatures of the hot and cold reservoirs, not on the working substance.

A frictionless engine cannot reach 100% efficiency unless the sink temperature is 0 K. Absolute zero cannot be reached in practice, so a practical heat engine cannot be perfectly efficient.

For a heat engine, W = Qh - Qc.
Efficiency η = W/Qh = 1 - Qc/Qh.
Carnot efficiency: η = 1 - Tc/Th, where temperatures are in kelvin.
Carnot efficiency depends only on the temperatures of the hot and cold reservoirs.
The hot reservoir temperature must be greater than the cold reservoir temperature.
The efficiency is highest when Tc/Th is smallest.
For temperature pairs 40 K and 20 K, the efficiency is greater when 20 K is the cold temperature and 40 K is the hot temperature than for pairs with a larger Tc/Th ratio.
A sink at 0 K would give η = 1, or 100 percent, according to the Carnot equation.
A reversible engine has no friction and operates through infinitesimally small temperature differences.

Joule-Thomson Effect and Thermal Radiation

The Joule-Thomson effect is the change in temperature of a real gas when it undergoes sudden expansion through a porous plug, throttle or narrow opening without performing useful external work. The process is approximately throttling and occurs at constant enthalpy. It is different from the reversible adiabatic expansion of a gas in a piston.

Thermal radiation is electromagnetic radiation emitted by a body because of its temperature. A black body is an ideal absorber and emitter. The total energy radiated per unit time is proportional to the fourth power of its absolute temperature, according to Stefan’s law.

Care must be taken to use the Kelvin temperature in all radiation and Carnot equations. Celsius values cannot be substituted directly into these formulas.

The Joule-Thomson effect is based on the sudden expansion of gases.
A real gas may cool or warm during Joule-Thomson expansion, depending on its initial temperature and gas properties.
For a black body, total radiated power is proportional to T^4.
If the absolute temperature of a black body is reduced to half, its total radiated power becomes (1/2)^4 = 1/16.
Stefan’s law: P = σAT^4 for an ideal black body.
In the complete form for net radiation, P = σA(T^4 - T0^4), where T0 is the surrounding temperature.
Temperature must be expressed in kelvin in Stefan’s law.
A black body absorbs all incident radiation and is also an ideal emitter.

Key terms

Thermodynamics: The study of heat, temperature, work and energy transformations in macroscopic systems.
Thermal equilibrium: The state in which bodies have the same temperature and there is no net heat flow between them.
Heat: Energy transferred between bodies because of a temperature difference.
Temperature: A measure related to the average kinetic energy of the particles of a body.
Internal energy: The total microscopic kinetic and potential energy of the particles of a system.
Heat capacity: The heat required to raise the temperature of a body by one kelvin.
Specific heat capacity: The heat required to raise the temperature of unit mass of a substance by one kelvin.
Molar heat capacity: The heat required to raise the temperature of one mole of a substance by one kelvin.
Adiabatic process: A process in which no heat enters or leaves the system.
Isothermal process: A process carried out at constant temperature.
Isobaric process: A process carried out at constant pressure.
Isochoric process: A process carried out at constant volume.
Reversible process: An ideal process that can be reversed without leaving a net change in the system and surroundings.
First law of thermodynamics: The law of energy conservation expressed for a thermodynamic system as ΔU = Q - W when W is work done by the system.
Mayer’s relation: The relation Cp - Cv = R for one mole of an ideal gas.
Specific heat ratio: The ratio γ = Cp/Cv of the molar heat capacities of a gas.
Carnot engine: An ideal reversible heat engine operating between a hot reservoir and a cold reservoir.
Joule-Thomson effect: The temperature change of a real gas during sudden throttling expansion.
Black body: An ideal body that absorbs all incident radiation and emits radiation according to its temperature.

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