19.8 - Interactive checkpoint: volume of an ideal gas
A quantity of an ideal gas occupies a
volume of 2.18 m^3 at 288 K, with an
absolute pressure of 1.45×10^5 Pa. If
its temperature increases to 298 K
and its pressure increases to
2.15×10^5 Pa, what is the volume?
Answer:Vf =m^3
19.9 - Kinetic energy and temperature
In this chapter, we have shown that by varying the speed of its molecules, we can affect
the pressure of a gas. We also have mentioned that the molecules’ speed increases
with temperature. Now we will get more specific about this relationship.
Rather than showing a relationship between speed and temperature, it proves more
productive to show a relationship between translational molecular kinetic energy and
temperature. To put it another way, we find it more useful to discuss the relationship
between the molecular speed squared and temperature.
The molecules that compose a gas move at a variety of speeds. As the temperature of
a gas increases, the average speed of its molecules increases, as does the molecules’
average kinetic energy. The kinetic theory of gases links the temperature of a gas to the
average kinetic energy of the molecules that make up the gas.
The relationship is quantified in Equation 1. The average translational kinetic energy of
a particle in an ideal gas equals (3/2)kT, with k being Boltzmann’s constant and T the
Kelvin temperature.
The equation for the kinetic energy of a particle has an interesting implication: The
kinetic energy of gas molecules is solely a function of temperature. It is independent of
other factors, such as pressure or volume. As the graph in Equation 1 illustrates, when
the temperature of a gas increases, the kinetic energy of the molecules that make it up
increases linearly.
Now we move from considering the kinetic energy of a particle to the kinetic energy of
an amount of an ideal monatomic gas. A monatomic gas is one whose particles are
single atoms. The internal kinetic energy of a monatomic gas is the sum of the
individual kinetic energies of all the atoms.
You see the equation for the internal energy of an ideal monatomic gas in Equation 2. It
can be derived from the first equation by multiplying by the number of atoms. The
kinetic energy of one molecule is (3/2)kT, as described above. For n moles of gas,
there are nNA atoms, so the total energy is (nNA)(3/2) kT. Since k equals R/NA, the
internal energy of n moles of monatomic gas equals (3/2)nRT.
This energy makes up all the internal energy of an ideal monatomic gas, because the
only form of energy of this type of gas is translational (linear) kinetic energy. There is no
potential energy in any ideal gas. Since there are no forces between the molecules,
there can be no energy due to position or configuration. The nature of monatomic
molecules (single atoms) means they have neither rotational nor vibrational energy.
Although we have focused on ideal monatomic gases, the internal energy and
temperature of ideal diatomic and polyatomic gases are also linearly related.
Kinetic energy of ideal gases
As temperature of gas increases:
·average molecular speed increases
·average translational KE of molecule
increases