### Video Transcript

For an ideal gas, for the pressure
and temperature of the gas to remain constant, if the number of moles of the gas is
increased by a factor of three, by what factor must the volume of the gas
change?

Here, weβre considering an ideal
gas. That tells us the gas can be
described by the ideal gas law, which says that the pressure of an ideal gas
multiplied by its volume equals the number of moles of the gas times a constant
multiplied by the gasβs temperature. In this equation, the quantity π
is a constant; itβs called the molar gas constant. In our particular case, the
pressure π and the temperature π of the gas are also held constant.

As a next step, letβs take our
application of the ideal gas law and separate it so that constant values are on one
side and nonconstant values are on the other. We know that the pressure π, the
molar gas constant π
, and the temperature π are constant values. If we divide both sides of our
equation by the number of moles of the gas π times the pressure π, then on the
left, the pressure π cancels out, and on the right, the number of moles of gas
cancels out. We get then that π divided by π
is equal to π
times π over π. We now have all of our constant
values on the right side of this expression and all of the variables on the
left.

Since π
and π and π are all
constants individually, the ratio π
times π divided by π is also a constant. If we call this overall constant
π, then we can write π divided by π is equal to π.

In our situation, weβre told that
the number of moles of the gas, thatβs π, is increased by a factor of three. In other words, what was π now has
become three times π. We want to know by what factor the
volume π must change along with the change in π so that the ratio of these
quantities is still equal to the constant π. If π divided by π is equal to π,
then if π is multiplied by a factor of three, π must be as well. That way the new ratio three π
divided by three π is still equal to π divided by π, which is equal to π. We know then by what factor π must
correspondingly be multiplied.

For an ideal gas, where the
pressure and temperature of the gas are held constant, if the number of moles of the
gas is multiplied by a factor of three, then the volume must also be multiplied by a
factor of three.