Calculating moles per cubic meter and liters per mole
Calculate: (a) the number of moles in
of gas at STP, and (b) the number of liters of gas per mole.
Strategy and Solution
(a) We are asked to find the number of moles per cubic meter, and we know from
[link] that the number of molecules per cubic meter at STP is
. The number of moles can be found by dividing the number of molecules by Avogadro’s number. We let
stand for the number of moles,
(b) Using the value obtained for the number of moles in a cubic meter, and converting cubic meters to liters, we obtain
Discussion
This value is very close to the accepted value of 22.4 L/mol. The slight difference is due to rounding errors caused by using three-digit input. Again this number is the same for all gases. In other words, it is independent of the gas.
The (average) molar weight of air (approximately 80%
and 20%
is
Thus the mass of one cubic meter of air is 1.28 kg. If a living room has dimensions
the mass of air inside the room is 96 kg, which is the typical mass of a human.
The density of air at standard conditions
and
is
. At what pressure is the density
if the temperature and number of molecules are kept constant?
The best way to approach this question is to think about what is happening. If the density drops to half its original value and no molecules are lost, then the volume must double. If we look at the equation
, we see that when the temperature is constant, the pressure is inversely proportional to volume. Therefore, if the volume doubles, the pressure must drop to half its original value, and
A very common expression of the ideal gas law uses the number of moles,
, rather than the number of atoms and molecules,
. We start from the ideal gas law,
and multiply and divide the equation by Avogadro’s number
. This gives
Note that
is the number of moles. We define the universal gas constant
, and obtain the ideal gas law in terms of moles.
Ideal gas law (in terms of moles)
The ideal gas law (in terms of moles) is
The numerical value of
in SI units is
In other units,
You can use whichever value of
is most convenient for a particular problem.
Calculating number of moles: gas in a bike tire
How many moles of gas are in a bike tire with a volume of
a pressure of
(a gauge pressure of just under
), and at a temperature of
?
Strategy
Identify the knowns and unknowns, and choose an equation to solve for the unknown. In this case, we solve the ideal gas law,
, for the number of moles
.
Solution
1. Identify the knowns.
2. Rearrange the equation to solve for
and substitute known values.
Discussion
The most convenient choice for
in this case is
because our known quantities are in SI units. The pressure and temperature are obtained from the initial conditions in
[link] , but we would get the same answer if we used the final values.