Heat and work

Internal Energy of a Classical ideal gas

l “Classical” means Equipartition Principle applies: each molecule has average energy ½ kT per in thermal equilibrium.


At room temperature, for most gases:

monatomic gas (He, Ne, Ar, …)

3 translational modes (x, y, z)


diatomic molecules (N2, O2, CO, …)

3 translational modes (x, y, z)

+ 2 rotational modes (wx, wy)



Internal Energy of a Gas

A pressurized gas bottle (V = 0.05 m3), contains helium gas (an ideal monatomic gas) at a pressure p = 1×107 Pa and temperature T = 300 K. What is the internal thermal energy of this gas?



Changing the Internal Energy

l U is a “state” function — depends uniquely on the state of the system in terms of p, V, T etc.

(e.g. For a classical ideal gas, U = NkT )

l There are two ways to change the internal energy of a system:

WORK done by the system on the environment

Wby = -Won

HEAT is the transfer of thermal energy into the system from the surroundings


Work and Heat are process energies, not state functions.


Work Done by An Expanding Gas

The expands slowly enough to maintain thermodynamic equilibrium.


Increase in volume, dV


+dV Positive Work (Work is done by the gas)

dV Negative Work (Work is done on the gas)



+dV Positive Work (Work is

done by the gas)

Energy leaves the system

and goes to the environment

dV Negative Work (Work is

done on the gas)

Energy enters the system

from the environment.

clip_image019Total Work Done


To evaluate the integral, we must know how the pressure depends (functionally) on the volume.

Pressure as a Function of Volume


Work is the area under the curve of a PV-diagram.

Work depends on the path taken in “PV space.”

The precise path serves to describe the kind of process that took place.

Different Thermodynamic Paths


Work done by a Gas

l When a gas expands, it does work on its environment

l Consider a piston with cross-sectional area A filled with gas. For a small displacement dx, the work done by the gas is:



dWby = F dx = pA dx = p (A dx)= p dV

l We generally assume quasi-static processes (slow enough that p and T are well defined at all times):

This is just the area under the p-V curve clip_image030

clip_image032Note that the amount of work needed to take the system from one state to another is not unique! It depends on the path taken.

What is Heat

l Up to mid-1800’s heat was considered a substance — a “caloric fluid” that could be stored in an object and transferred between objects. After 1850, kinetic theory.

l A more recent and still common misconception is that heat is the quantity of thermal energy in an object.

l The term Heat (Q) is properly used to describe energy in transit, thermal energy transferred into or out of a system from a thermal reservoir …


(like cash transfers into and out of your bank account)

Q is not a “state” function — the heat depends on the process, not just on the initial and final states of the system

Sign of Q : Q > 0 system gains thermal energy

Q < 0 system loses thermal energy

The work done depends on the initial and final states and the path taken between these states.

BUT, the quantity Q – W does not depend on the path taken; it depends only on the initial and final states.

Only Q – W has this property. Q, W, Q + W,Q – 2W, etc. do not.

So we give Q – W a name: the internal energy.


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