what is the difference in volume-temperature relationship in the ff. situations?

Question

a. air is blown with mouth slightly open

b. air is blown with mouth wide open

how will you account for this?

Answer 1

Thermodynamically:
Case (a) can be modeled as a throttling process (isenthalpic, adiabatic) where pressure drops as air is blown out of the mouth. By Joule Thompson effect, the relationship for volume and temperature is: Joule Thompson coefficient= dT/dP =V*(T*alpha-1)/Cp, where dT/dP is constant.
alpha is thermal expansion coeff., Cp is heat capacity.
This process thus causes cooling.

Case (b) can be perhaps modeled as a constant pressure process (not very sure though). In that case use the corresponding gas law equation for real gases (air), roughly: V is directly proportional to T.

Answer 2

This is one of Aesop's Fables:

The Man and the Satyr

A MAN and a Satyr once drank together in token of a bond of
alliance being formed between them. One very cold wintry day, as
they talked, the Man put his fingers to his mouth and blew on
them. When the Satyr asked the reason for this, he told him that
he did it to warm his hands because they were so cold. Later on
in the day they sat down to eat, and the food prepared was quite
scalding. The Man raised one of the dishes a little towards his
mouth and blew in it. When the Satyr again inquired the reason,
he said that he did it to cool the meat, which was too hot. "I
can no longer consider you as a friend," said the Satyr, "a
fellow who with the same breath blows hot and cold."

With the mouth slightly open, the air is compressed. As it escapes from the mouth, its pressure falls, and so it cools.

With the mouth wide open, the breath comes out at body temperature, and feels warm.