A Level · A2 · structured
A Level extension · Question 8
A Level extension · Original GioPhysics question with a detailed, mark-by-mark answer guide.
- Demand
- demanding
- Marks
- 12
- Topics
- 2
- Answer
- Complete
A sealed cylinder contains an ideal gas of volume 2.0 × 10⁻³ m³ at a pressure of 1.5 × 10⁵ Pa and a temperature of 290 K. The Boltzmann constant is 1.38 × 10⁻²³ J K⁻¹.
- (a)
Calculate Calculate the number of molecules of gas in the cylinder.
3 marks - (b)
Determine Determine the total kinetic energy of the molecules in the cylinder.
3 marks - (c)
Deduce The gas is now compressed slowly at a constant temperature of 290 K. During the compression, 120 J of work is done on the gas. Deduce the change in internal energy of the gas and the thermal energy transferred to or from it.
4 marks - (d)
Explain Explain why the internal energy of an ideal gas depends only on its temperature, whereas that of a real gas does not.
2 marks
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(a)
Calculate Calculate the number of molecules of gas in the cylinder.
List the given quantities with units, identify the required quantity, write the governing relationship before substituting, and keep extra digits until the final line so rounding does not distort the result.
- 1
Use the ideal-gas molecular relation pV = NkT for the sealed gas.
- 2
N = pV/kT = (1.5 × 10⁵ × 2.0 × 10⁻³) / (1.38 × 10⁻²³ × 290)
- 3
N = 7.5 × 10²²
(b)
Determine Determine the total kinetic energy of the molecules in the cylinder.
List the given quantities with units, identify the required quantity, write the governing relationship before substituting, and keep extra digits until the final line so rounding does not distort the result.
- 1
mean kinetic energy per molecule = (3/2)kT = 1.5 × 1.38 × 10⁻²³ × 290 = 6.0 × 10⁻²¹ J
- 2
total = N × 6.0 × 10⁻²¹
- 3
total = 4.5 × 10² J (450 J)
(c)
Deduce The gas is now compressed slowly at a constant temperature of 290 K. During the compression, 120 J of work is done on the gas. Deduce the change in internal energy of the gas and the thermal energy transferred to or from it.
List the given quantities with units, identify the required quantity, write the governing relationship before substituting, and keep extra digits until the final line so rounding does not distort the result.
- 1
the temperature is unchanged, so the mean kinetic energy per molecule is unchanged
- 2
the internal energy of an ideal gas is entirely kinetic, so ΔU = 0
- 3
first law: ΔU = q + w, so 0 = q + 120
- 4
q = −120 J: 120 J of thermal energy is transferred out of the gas to the surroundings
(d)
Explain Explain why the internal energy of an ideal gas depends only on its temperature, whereas that of a real gas does not.
State the outcome first, then link cause to effect with the relevant physical principle. Each link in the reasoning should be explicit enough to earn its own marking point.
- 1
an ideal gas is defined to have no forces between its molecules except during collisions, so there is no potential energy contribution and the internal energy is the total random kinetic energy alone
- 2
a real gas has intermolecular forces, so its internal energy also includes potential energy, which depends on the separation of the molecules and hence on the volume
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