Guide 18-1b. Fundamentals of Thermodynamics

The usual assumptions made in doing thermodynamics problems are:

  • The processes are quasi-static. (system remains in equilibrium with its surroundings)
  • The processes are reversible.  (no dissipative forces)

Other important things to know:

  • P, V, T, and U are state variables. They are used to describe a gas at a particular point in time.
  • Q and W (Wse or Wes) are process variables. They only apply when the gas is taken by a process from one state to another state.

A key to solving thermal process problems is to know how to apply the fundamental relationships:

Relationship Equation Notes about use
Law of conservation of energy (in the form of the First Law of Thermodynamics) Q - Wse = ΔU

Generally applicable; can be used to solve for any of the three quantities given the other two

When the gas is ideal, U = 3nRT/2 can be used to calculate the internal energy.

Work can be calculated using the area under a graph of pressure vs. volume (see next item).
Graph of pressure vs. volume (area under the line is the work done by or on the gas)

P-V diagram

Graph 1

In applying this relationship, keep in mind the following:

  • The average pressure over the volume interval is used in the calculation.
  • The work done by the system on the environment is positive when Vf > Vi as in Graph 1.
  • The work done by the system on the environment is negative when Vf < Vi as in Graph 2.

Notes about the sign of W:

  • Our textbook uses W to represent work done by the system on the environment (Wse). The formula for work given in the text is W = PΔV. We will use the book's convention, but you are expected to always begin with Wse = PΔV.
  • As we have seen before, the AP exam uses W to represent work done by the environment on the system (Wes). As a result, the formula given on the AP exam for work is W = -PΔV. Note the negative sign.

 

Wse = Pave(Vf - Vi)

 

P-V diagram

Graph 2
Ideal gas law PV = nRT = NkT

Often assumed but doesn't have to be; assumed when interactions of the molecules of the gas are negligible

Units: P in pascals, V in meter3, T in kelvins, n in moles
N is the number of molecules.
Universal gas constant R = 8.31 J/molK
Boltzmann's constant k = 1.38E-23 J/K

Know your thermal processes!

Name of process Description P-V Graph What else we know about the process
Isochoric Constant volume Isochoric

Wse = 0 (area under the P-V graph = 0)

Since Wse = 0, the first law tells us that Q = ΔU.
Isobaric Constant pressure Isobaric Wse = PΔV    (P = Pave)
Isothermal Constant temperature Isothermal

Since ΔT = 0, ΔU = 0 and, by the first law, Q = Wse.

For an enclosed ideal gas, the equation of the line on the P-V graph is hyperbolic because P = (nRT)/V, where (nRT) is constant. The gas must be enclosed so that the number of moles remains constant.

Wse can be calculated using Wse = (nRT)ln(Vf/Vi).
Linear Pressure varies linearly with volume Linear Wse = PaveΔV = ½(Pi + Pf)ΔV
Adiabatic No heat transfer Adiabatic Since Q = 0, the first law tells us that Wse = -ΔU.

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