doPhysics · Thermal Physics 2
The First Law and Thermodynamic Processes
Heat, work and internal energy in one balance. Systems and states, the zeroth law, the work of a gas as an area on the p-V diagram, the first law \(Q = \Delta U + W\), the isochoric, isobaric, isothermal and adiabatic processes, polytropic processes, free expansion and cycles solved leg by leg.
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Why this chapter
Why this chapter exists
A pressure cooker whistles, a bicycle pump grows warm, a diesel engine fires with no spark, and the Himalaya keeps its snow through June. Each is a story of energy moving as heat and as work. This chapter keeps the accounts. It fixes what a system and its state are, and measures the work a gas does as an area on a graph. Then it states the first law, the conservation of energy with heat included. With it, every process of a gas becomes a short calculation. That holds at constant volume, pressure or temperature, with no heat at all, and round a whole cycle.
You will learn
By the end of this chapter you will
- Describe a system as open, closed or isolated, fix its state by p, V and T, state the zeroth law, and tell a quasi-static process from a sudden one.
- Find the work done by a gas as \(\int p\,dV\) and as an area on the p-V diagram, show that it depends on the path, and find the net work of a cycle.
- State the first law, \(Q = \Delta U + W\), with its sign convention; use \(U\) as a state function; derive \(C_p - C_v = R\) from it.
- Find \(Q\), \(W\) and \(\Delta U\) in isochoric, isobaric and isothermal processes of an ideal gas.
- Use \(pV^\gamma\), \(TV^{\gamma - 1}\) and \(p^{1 - \gamma}T^\gamma\) for adiabatic processes, explain why compression heats a gas, and find the adiabatic lapse rate.
- Use \(pV^x\) for polytropic processes with \(C = C_v + R/(1 - x)\), analyse a free expansion, and solve a cycle leg by leg.
Before you begin
Before you begin
From the chapter on kinetic theory you should know \(pV = nRT\), the internal energy \(U = \tfrac f2 nRT\), and \(C_v\), \(C_p\) and \(\gamma\) for monatomic and diatomic gases. From calorimetry you should know latent heat, and from Mechanics the work \(F\,d\) of a steady force. The barometric formula is used once, for the air on mountains.
Lessons
1
Systems, States and Processes
Thermodynamics first draws a boundary round the system. Its state is fixed by p, V and T, a point on a p-V diagram. The zeroth law makes temperature a property, and a slow, quasi-static process can be drawn as a line.
Systems and surroundingsState variables and equilibriumThe zeroth lawQuasi static processes
2
Work Done by a Gas
A gas does work \(W = \int p\,dV\) on its piston, the area under its path on the p-V diagram. The sign convention is fixed here. Two paths between the same states do different work, and a cycle does the work of the area it encloses.
Work is p dvWork as areaWork depends on the pathCycles and net work
3
The First Law of Thermodynamics
Internal energy belongs to the state; heat and work belong to the process. The first law, \(Q = \Delta U + W\), is energy conservation for a gas. It explains a whistling pressure cooker, splits the heat of boiling, and gives \(C_p - C_v = R\).
Internal energy is a state functionThe first lawEnergy budget of boilingMayers relation
4
Isochoric, Isobaric and Isothermal Processes
Three processes each hold one variable fixed. At constant volume all the heat raises \(U\); at constant pressure it splits between \(U\) and the work \(nR\Delta T\); at constant temperature it all becomes work, \(nRT\ln(V_2/V_1)\).
Isochoric processIsobaric processIsothermal process
5
Adiabatic Processes
With no heat exchanged, a gas pays for its work out of its own internal energy. It obeys \(pV^\gamma = \text{constant}\), its adiabat is steeper than the isotherm, compression heats it (a bicycle pump, a diesel engine), and rising air cools by about 9.8 K per kilometre.
Adiabatic processAdiabat steeper than isothermWhy compression heats a gasAdiabatic lapse rate
6
Polytropic Processes, Free Expansion and Cycles
One law, \(pV^x = \text{constant}\), covers all four processes, with work \((p_1V_1 - p_2V_2)/(x - 1)\) and heat capacity \(C_v + R/(1 - x)\). A gas rushing into a vacuum does no work and keeps its temperature. A cycle is solved leg by leg, and its \(\Delta U\) adds to zero.
Polytropic processesFree expansionSolving a cycle leg by leg
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- adiabat
- The curve \(pV^\gamma = \text{constant}\) followed by an ideal gas in a quasi-static adiabatic process. Through any state it is \(\gamma\) times as steep as the isotherm.
- adiabatic lapse rate
- The fall in temperature with height of rising dry air that expands without exchanging heat, \(Mg/C_p\), about \(9.8\,\text{K}\) per kilometre.
- adiabatic process
- A process in which no heat crosses the boundary of the system, \(Q = 0\). The work done by the system then equals the fall in its internal energy.
- cyclic process
- A process that returns a system to its starting state. Its \(\Delta U\) is zero, and its net work is the area enclosed on the \(p\)-\(V\) diagram.
- equation of state
- An equation that links the state variables of a system. For an ideal gas it is \(pV = nRT\).
- first law of thermodynamics
- The heat given to a system equals the rise in its internal energy plus the work done by it: \(Q = \Delta U + W\).
- free expansion
- The expansion of a gas into a vacuum inside a rigid insulated vessel. It does no work and takes no heat, so an ideal gas keeps its temperature.
- heat reservoir
- A body so large that it can give or take any amount of heat without a change in its temperature.
- indicator diagram
- A graph of pressure against volume for a gas. Each equilibrium state is a point on it, each quasi-static process a line, and the area under a line is the work done.
- internal energy
- The total energy of the random motion and arrangement of the molecules of a system, measured in the frame of the system as a whole. For an ideal gas, \(U = \tfrac f2 nRT\).
- irreversible process
- A process that cannot run backwards by itself, such as a free expansion or heat flowing from hot to cold.
- isobaric process
- A process at constant pressure. For an ideal gas the work is \(p\,\Delta V = nR\,\Delta T\) and the heat is \(nC_p\,\Delta T\).
- isochoric process
- A process at constant volume. The gas does no work, so all the heat given to it raises its internal energy.
- isolated system
- A system that exchanges neither matter nor energy with its surroundings. A sealed vacuum flask comes close to one.
- isotherm
- A curve of constant temperature on the \(p\)-\(V\) diagram. For an ideal gas it is the hyperbola \(pV = \text{constant}\).
- isothermal process
- A process at constant temperature. For an ideal gas \(\Delta U = 0\), so the heat taken in equals the work done, \(nRT\ln(V_2/V_1)\).
- molar heat capacity
- The heat needed to warm one mole of a substance by one kelvin in a given process, \(C = Q/(n\,\Delta T)\). For a polytropic process of an ideal gas, \(C = C_v + R/(1 - x)\).
- polytropic index
- The exponent \(x\) in \(pV^x = \text{constant}\).
- polytropic process
- A process that obeys \(pV^x = \text{constant}\). The isobaric, isothermal, adiabatic and isochoric processes have \(x = 0\), \(1\), \(\gamma\) and \(\infty\).
- quasi-static process
- A process slow enough that the system passes only through equilibrium states, so it can be drawn as a line on the \(p\)-\(V\) diagram.
- state function
- A quantity that depends only on the state of a system, not on how it got there, such as \(p\), \(T\) or \(U\). Heat and work are not state functions.
- state variable
- A measurable large-scale quantity that describes the equilibrium state of a system, such as pressure, volume, temperature or the number of moles.
- surroundings
- Everything outside a thermodynamic system that can exchange heat, work or matter with it.
- thermal equilibrium
- The condition of bodies in contact when no heat flows between them, because their temperatures are equal.
- thermodynamic system
- The part of the world chosen for study, such as a gas in a cylinder, separated from everything else by a boundary, real or imagined.
- zeroth law of thermodynamics
- If two bodies are each in thermal equilibrium with a third, they are in thermal equilibrium with each other. It makes temperature a property of a body.