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Problem 3. An Experiment to Verify the Compressibility of Air [Total of 50%]
The lab room contains an apparatus that can be used to verify the compressibility of air.
The device consists of a tube that contains a fixed number of air molecules (NA). The tube is oriented vertically with a moveable piston on the top end. Although the mass of air in the tube is constant, the volume (V) varies when pressure is applied to the piston, pushing it inward. We can vary the pressure by placing known masses on a wire hanger attached to the piston. See the apparatus set up in the lab room.
The volume of the air is V=Ah, where A is the cross sectional area of the tube, and h is the height of the air column. We measure h from the bottom of the tube to the base of the piston using a scale written on the side of the tube.
2
The experiment consists of the following. We incrementally increase the pressure on the air in the tube by adding weights to the hanger. The mass of each weight is 1000g, and we can place from 0 to 5 weights on the hanger simultaneously. Each time we add a weight, we record the height of the air column. As the number of weights increases, the air in the tube compresses, as indicated by the height of the air column (h) becoming smaller.
The data below come from three trials of the experiment.
Extra Mass Trial 1 Trail 2 Trial 3
ΔM (grams) h (cm) h (cm) h (cm)
0 16.5 16.9 16.5
1000 13.0 12.9 13.0
2000 10.2 10.5 10.2
3000 8.8 8.8 8.8
4000 7.3 7.4 7.3
5000 6.4 6.5 6.4
As the extra added mass (~pressure) increases, the height of the air column (~volume of air) decreases. So at least qualitatively, the data show that air is compressible. But, how can the above data verify the quantitative theory of the compressibility of air? We need to cast the theory of compressibility in a mathematical form that can make use of the above data.
The pressure exerted on the top of the piston is equal to the weight added per unit area:
P = (M0 + ΔM)g/A [1]
Here M0 is the mass of the atmosphere above the piston plus the contribution from the mass of the piston and the empty wire hanger attached to it. The extra mass attached to the hanger is ΔM, g is the acceleration due to gravity, and A is the cross sectional area of the tube.
The Ideal Gas Law specifies the pressure of the air in the tube,
P=nkT [2]
where k is Boltzmann’s constant. Here we will assume that absolute temperature T is constant at the value that existed in the lab at the time the data were collected. The Ideal Gas Law expresses the compressibility of air; if pressure doubles, then the number density n doubles (if T is held fixed). The number density n is related to the total number of air molecules in the tube of volume V by:
n = NA/V = (NA/A)(1/h) [3]
The combination of Equations 1, 2, and 3 gives:
(M0 + ΔM)g/A = (NAkT/A)(1/h)
Some algebra gives:
ΔM = -M0 + (NAkT/g) (1/h) [4]
3
Equation 4 says that if we plot ΔM as a function of 1/h, the result should be a straight line.
(a) [30%] Plot the values of ΔM versus values of 1/h using the data on page 2, and include this graph as part of your answer. Is the resulting graph consistent with the theory of a compressible gas?
(b) [20%] Describe how you would estimate the number of air molecules contained in the tube (NA). You do not need to do the calculation, but you should explain how you would go about it, including the information that you would need.
Problem 3. An Experiment to Verify the Compressibility of Air [Total of 50%]
The lab room contains an apparatus that can be used to verify the compressibility of air.
The device consists of a tube that contains a fixed number of air molecules (NA). The tube is oriented vertically with a moveable piston on the top end. Although the mass of air in the tube is constant, the volume (V) varies when pressure is applied to the piston, pushing it inward. We can vary the pressure by placing known masses on a wire hanger attached to the piston. See the apparatus set up in the lab room.
The volume of the air is V=Ah, where A is the cross sectional area of the tube, and h is the height of the air column. We measure h from the bottom of the tube to the base of the piston using a scale written on the side of the tube.
2
The experiment consists of the following. We incrementally increase the pressure on the air in the tube by adding weights to the hanger. The mass of each weight is 1000g, and we can place from 0 to 5 weights on the hanger simultaneously. Each time we add a weight, we record the height of the air column. As the number of weights increases, the air in the tube compresses, as indicated by the height of the air column (h) becoming smaller.
The data below come from three trials of the experiment.
Extra Mass Trial 1 Trail 2 Trial 3
ΔM (grams) h (cm) h (cm) h (cm)
0 16.5 16.9 16.5
1000 13.0 12.9 13.0
2000 10.2 10.5 10.2
3000 8.8 8.8 8.8
4000 7.3 7.4 7.3
5000 6.4 6.5 6.4
As the extra added mass (~pressure) increases, the height of the air column (~volume of air) decreases. So at least qualitatively, the data show that air is compressible. But, how can the above data verify the quantitative theory of the compressibility of air? We need to cast the theory of compressibility in a mathematical form that can make use of the above data.
The pressure exerted on the top of the piston is equal to the weight added per unit area:
P = (M0 + ΔM)g/A [1]
Here M0 is the mass of the atmosphere above the piston plus the contribution from the mass of the piston and the empty wire hanger attached to it. The extra mass attached to the hanger is ΔM, g is the acceleration due to gravity, and A is the cross sectional area of the tube.
The Ideal Gas Law specifies the pressure of the air in the tube,
P=nkT [2]
where k is Boltzmann’s constant. Here we will assume that absolute temperature T is constant at the value that existed in the lab at the time the data were collected. The Ideal Gas Law expresses the compressibility of air; if pressure doubles, then the number density n doubles (if T is held fixed). The number density n is related to the total number of air molecules in the tube of volume V by:
n = NA/V = (NA/A)(1/h) [3]
The combination of Equations 1, 2, and 3 gives:
(M0 + ΔM)g/A = (NAkT/A)(1/h)
Some algebra gives:
ΔM = -M0 + (NAkT/g) (1/h) [4]
3
Equation 4 says that if we plot ΔM as a function of 1/h, the result should be a straight line.
(a) [30%] Plot the values of ΔM versus values of 1/h using the data on page 2, and include this graph as part of your answer. Is the resulting graph consistent with the theory of a compressible gas?
(b) [20%] Describe how you would estimate the number of air molecules contained in the tube (NA). You do not need to do the calculation, but you should explain how you would go about it, including the information that you would need.
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