What is the relationship between air temperature and air pressure?

Before answering this question, it will be helpful to
clarify some terms. When we say air, it generally refers to the mixture of gases that is
present in the earths atmosphere. This should no be mixed up with gases in general. Also
we need to realize that the air as it exists naturally in the atmosphere is impacted by
the uncontrolled influence of the environment in which it exists. However, the
conditions of air in a closed space can be controlled by an
experimenter.

Now coming to the main question. Air being a
mixture of gases, behaves in line with various laws of gases. One of this laws states
that all other things being equal, the pressure of a gas increases with increases in
temperature. This given a fixed quantity of air, contained in a space with fixed volume,
its pressure will increase with temperature. However as the air exists naturally in
atmosphere, there is no constraint on change of its volume. Therefore, the air will
expand with increasing temperature. This will cause the the density of air to reduce.
This in turn will cause the hot air to rise up in atmosphere. This upward movement of
air actually tends to create partial vacuum and thus reduce the air pressure at places
which are hotter. To fill this region of partial vacuum or low pressure cooler air from
adjacent regions will flow towards it.