The trick in all the cases below is to "isolate" the fixed variables from those which are still varied. Then, the latter can be related to each other by initial and final states.
In doing these, we shall put in the appropriate equation and then describe the result. All results will be based on the ideal gas law,
The trick in all the cases below is to "isolate" the fixed variables from those which are still varied. Then, the latter can be related to each other by initial and final states.
| (a) | Halve the Kelvin temperature while holding the pressure constant | Here the volume would be halved.
This is easily seen from
The strategy employed here is to "isolate" those things which are kept constant from those which undergo change. Note that Tf = 0.5Ti. This can be more fully stated, perhaps by doing the appropriate substitution, viz.,
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| (b) | Increase the amount of gas by 1/4 while holding the temperature and pressure constant | We employ this strategy here by
putting T and P on one side (along with R).
This gives
The ratio of the numbers of moles
is 1.25 and this means that the volume would increase by 25% (or 1/4).
Here, you can plug in the numbers yourself!
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| (c) | Decrease the pressure by 75% at constant T | Here, n and T are
constant and we thus isolate them. The procedure in this case is
Decreasing the pressure by 75% would reduce it to 25% (1/4) of its initial value. Thus the volume would increase by a factor of 4. That is
As long as you understand that a
75% decrease means that 25% remains, you should be OK. Some folks
are helped out by the following way of saying this: Always think
in terms of what's there. In this case, what is "there"
is 100% - 75% = 25%. (If this confuses you, don't read this.)
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| (d) | Double the Kelvin temperature and double the pressure | In this instance, we isolate just
n
since, with P and T varying, V must also vary.
Thus,
In this case, there is no change at all! As you can see, this sort of thing is rather simple!
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In doing the problems above, we used several different types of reasoning. But all this was based on the simple manipulation of the ideal gas law into the form needed for the particular situation.Problem 9.46: Which sample contains the most molecules: 1.0 L of O2 at STP, 1.00 L of air at STP, or 1.00 L of H2 at STP?
All of these contain the same number of molecules. The reasoning here is that, according to the ideal gas law, pressure is exerted by the number of molecules regardless of what type of molecules they may be. That is, all ideal gases have the same properties and it is irrelevant whether a sample of gas is a pure gas or a mixture. So, at STP, T, V, and P are constant. Thus,Problem 9.47: Which sample contains more molecules: 2.50 L of air at 50°C and 750 mm Hg pressure or 2.16 L of CO2 at -10°C and 765 mm Hg pressure?
must be the same since all three samples have the same temperature, pressure, and volume. (It should go beyond saying that the same number of moles means the same number of molecules.
In this one, we need to do the actual calculations. We treat the two cases separately. Note that unit substitutions are done automatically to get pressures in atm and temperatures in K.Problem 9.49: A compressed air tank carried by scuba divers has a volume of 8.0 L and a pressure of 140 atm at 20°C. What is the volume of air in the tank (in liters) at STP?Case 1: Air
Case 2: Carbon Dioxide
By explicit calculation, we see that there are more molecules of carbon dioxide.
Here, the only thing which stays constant is n. Thus, we can writeProblem 9.52: The matter in interstellar space consists almost entirely of hydrogen atoms at a temperature of 100K and a density of approximately 1atom/cm3. What is the gas pressure in mm Hg?
Of course, you could just use the combined gas law directly (the last two parts of the equations on the first line), but doing it this way makes sure that you understand what is going on. It is generally better to derive all needed equations from the ideal gas law until you are really adept at this!
There are various ways of doing this. For instance, you could calculate n directly and then plug into the ideal gas equation. However, it is better to do this algebraically (or at least more fulfilling) and use the resulting equation. That is how we shall do it. The number of moles of atoms per cm3 is given byProblem 9.55: A small cylinder of helium gas used for filling balloons has a volume of 2.30 L and a pressure of 13,800 kPa at 25°C. How many balloons can you fill if each one has a volume of 1.5 L and a pressure of 1.25 atm at 25°C?
Here, N is he number of particles in the same volume. Thus, we can manipulate the ideal gas equation as follows:
This is a very hard vacuum and is better than we can obtain in the laboratory with even the best pumps!
Again, let's be clever and do this all in one fell swoop! What we do here is find the total volume that the tank delivers at 1.25 atm and 25°C (298.15K). We then take that total volume and divide by the volume of each balloon to get our answer. We shall do this in two steps, but algebraically. The advantage to doing it this way is that you have solved this problem for any time you are filling balloons with a tank of He (any tank).To start, we note that n and T constant here. (We could go even more general and let T vary, but we'll assume that the temperature is about the same each time we do this (it is left as an exercise for you to make it more general). So,
Then, we can calculate the number of balloons we can fill by simply dividing by the volume per balloon. All we have to do is derive the equation and put in our numbers (being careful, of course, with our units). Here is our result.
This is quick, elegant, and not really all that difficult. Note that the net pressure in the ballons is 0.25 atm (0.25 atm in excess of the atmospheric pressure). When dealing with gases we always use the total pressure in the gas laws.
