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Carbon monoxide gas is produced in a pond by the decay of organic matter - Edexcel - A-Level Physics - Question 14 - 2023 - Paper 2
Question 14
Carbon monoxide gas is produced in a pond by the decay of organic matter. (a) A bubble of carbon monoxide rises at a steady speed through the still water of the pon... show full transcript
Worked Solution & Example Answer:Carbon monoxide gas is produced in a pond by the decay of organic matter - Edexcel - A-Level Physics - Question 14 - 2023 - Paper 2
Step 1
Show that the upthrust acting on the bubble is about 1.7 × 10⁻³ N.
Answer
To calculate the upthrust (buoyant force) on the bubble, we can apply Archimedes' principle. The upthrust is equal to the weight of the water displaced by the bubble.
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Calculate the volume of the bubble: Since the diameter is given, we can find the radius:
[ r = \frac{1.5 \text{ mm}}{2} = 0.00075 \text{ m} ]
The volume ( V ) of a sphere is given by:
[ V = \frac{4}{3} \pi r^3 ]
Substituting the radius:
[ V = \frac{4}{3} \pi (0.00075)^3 \approx 1.767 \times 10^{-9} \text{ m}^3 ]
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Calculate the weight of the displaced water: The weight of the displaced water can be calculated using the density of water and the volume:
[ ext{Weight} = \text{density} \times V \times g ]
where ( g \approx 9.81 ext{ m/s}^2 ) is the acceleration due to gravity.[ \text{Weight} = 997 \text{ kg/m}^3 \times 1.767 \times 10^{-9} \text{ m}^3 \times 9.81 \text{ m/s}^2 \approx 1.7 \times 10^{-3} ext{ N} ]
Thus, the upthrust acting on the bubble is approximately ( 1.7 \times 10^{-3} ext{ N} ).
Step 2
Calculate the steady speed at which the bubble rises.
Answer
To calculate the steady speed at which the bubble rises, we can use the formula for viscous drag, which must equal the upthrust for constant velocity.
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Use the formula for viscous drag:
[ F_d = 6 \pi \eta r v ]
Where:
- ( F_d ) = viscous drag force
- ( \eta = 0.0011 ext{ Pa s} ) (viscosity of water)
- ( r = 0.00075 ext{ m} ) (radius of the bubble)
- ( v ) = speed of the bubble
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Set the drag force equal to the upthrust:
[ 6 \pi \eta r v = 1.7 \times 10^{-3} ]
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Solve for speed:
[ v = \frac{1.7 \times 10^{-3}}{6 \pi (0.0011)(0.00075)} ]
After performing the calculation, we find the steady speed ( v ) of the bubble as it rises.
Step 3
Calculate the increase in the potential energy between the two atoms.
Answer
To find the increase in potential energy when the distance between the two atoms changes, we can use the formula for elastic potential energy stored in a spring:
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Calculate the increase in distance:
[ \Delta x = d_{final} - d_{initial} = 18 ext{ mm} - 12 ext{ mm} = 6 ext{ mm} = 0.006 ext{ m} ]
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Use the formula for elastic potential energy:
[ PE = \frac{1}{2} k (\Delta x)^2 ]
Where:- ( k = 1195 ext{ N/m} )
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Substitute values to calculate increase in potential energy:
[ PE = \frac{1}{2} (1195) (0.006)^2 ]
Once calculated, this will give the increase in potential energy between the two atoms.