# Ideal gas equation The ideal gas equation relates pressure, temperature, volume and the amount of gas.

Key Concepts

$$pV=nRT$$

Since the number of moles is linked to the number of molecules...

$$n={N\over N_A} \Rightarrow pV={N\over N_A}RT$$

$$R = N_Ak \Rightarrow pV=NkT$$

Essentials

Kinetic energy and temperature, $$\bar{E_k}={3\over2}kT$$

This equation can often get overlooked but is really quite extraordinary. It links the microscopic concept of the average kinetic energy of individual molecules of gas with the temperature of the gas as a whole.

It means that you could measure the temperature of the room you are in the moment and calculate the 'root mean square speed' ($$v_{rms}$$) of the air molecules (best to assume just nitrogen and oxygen as they are similar in size and mass!):

$$\Rightarrow {1\over2}m\bar{v^2}={3\over2}kT$$

$$v_{rms}=\sqrt {3kT\over m}$$

This equation also enables us to calculate the internal energy, U, of a gas (recall that we assume that potential energy is zero):

$$U=\sum E_p+\sum E_k=\sum E_k$$

$$\Rightarrow U=N{3\over 2} kT$$

PVT graphs P, V and T can be plotted on a 3-D graph. In practice, however, we draw just p and V with the different temperatures represented by isotherms.

Real Gases Real gas molecules take up space and are attracted to each other. If the gas is at low pressure, moderate temperature and low density, then the ideal gas law will hold.

Summary

Test Yourself

Use quizzes to practise application of theory.

Exam-style Questions

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