15.2. Equation of state
A subsection of Physics, 9702, through 15. Ideal gases
Listing 10 of 19 questions
State what is meant by an ideal gas. A fixed amount of helium gas is sealed in a container. The helium gas has a pressure of 1.10 × 105 Pa, and a volume of 540 cm3 at a temperature of 27 °C. The volume of the container is rapidly decreased to 30.0 cm3. The pressure of the helium gas increases to 6.70 × 106 Pa and its temperature increases to 742 °C, as illustrated in . initial state final state No thermal energy enters or leaves the helium gas during this process. Show that the helium gas behaves as an ideal gas. The first law of thermodynamics may be expressed as ΔU = q + W. Use the first law of thermodynamics to explain why the temperature of the helium gas increases. 1.10 × 105 Pa 540 cm3 27 °C 6.70 × 106 Pa 30.0 cm3 742 °C The average translational kinetic energy EK of a molecule of an ideal gas is given by EK = 3 2 kT where k is the Boltzmann constant and T is the thermodynamic temperature. Calculate the change in the total kinetic energy of the molecules of the helium gas. change in kinetic energy = J The mass of nitrogen gas in another container is 24.0 g at a temperature of 27 °C. The gas is cooled to its boiling point of –196 °C. Assume all the gas condenses to a liquid. For this change the specific heat capacity of nitrogen gas is 1.04 kJ kg–1 K–1. The specific latent heat of vaporisation of nitrogen is 199 kJ kg–1. Determine the thermal energy, in kJ, removed from the nitrogen gas. energy = kJ
9702_m23_qp_42
THEORY
2023
Paper 4, Variant 2
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State what is meant by an ideal gas. The product of pressure p and volume V of an ideal gas of density ρ at temperature T is given by the expressions and where N is the number of molecules and k is the Boltzmann constant. State the meaning of the symbol . Deduce that the mean kinetic energy EK of the molecules of an ideal gas is given by the expression EK =  kT. In order for an atom to escape completely from the Earth’s gravitational field, it must have a speed of approximately 1.1 × 104m s–1 at the top of the Earth’s atmosphere. Estimate the temperature at the top of the atmosphere such that helium, assumed to be an ideal gas, could escape from the Earth. The mass of a helium atom is 6.6 × 10–27kg. temperature = K Suggest why some helium atoms will escape at temperatures below that calculated in . p =  ρ pV = NkT, Use
9702_s05_qp_4
THEORY
2005
Paper 4, Variant 0
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State what is meant by an ideal gas. Two cylinders A and B are connected by a tube of negligible volume, as shown in . tap T tube cylinder A cylinder B 2.5 × 103 cm3 1.6 × 103 cm3 4.9 × 105 Pa 3.4 × 105 Pa 300 K Initially, tap T is closed. The cylinders contain an ideal gas at different pressures. Cylinder A has a constant volume of 2.5 × 103 cm3 and contains gas at pressure 3.4 × 105 Pa and temperature 300 K. Show that cylinder A contains 0.34 mol of gas. Cylinder B has a constant volume of 1.6 × 103 cm3 and contains 0.20 mol of gas. When tap T is opened, the pressure of the gas in both cylinders is 3.9 × 105 Pa. No thermal energy enters or leaves the gas. Determine the final temperature of the gas. temperature = K By reference to work done and change in internal energy, suggest why the temperature of the gas in cylinder A has changed.
9702_s13_qp_41
THEORY
2013
Paper 4, Variant 1
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State what is meant by an ideal gas. Two cylinders A and B are connected by a tube of negligible volume, as shown in . tap T tube cylinder A cylinder B 2.5 × 103 cm3 1.6 × 103 cm3 4.9 × 105 Pa 3.4 × 105 Pa 300 K Initially, tap T is closed. The cylinders contain an ideal gas at different pressures. Cylinder A has a constant volume of 2.5 × 103 cm3 and contains gas at pressure 3.4 × 105 Pa and temperature 300 K. Show that cylinder A contains 0.34 mol of gas. Cylinder B has a constant volume of 1.6 × 103 cm3 and contains 0.20 mol of gas. When tap T is opened, the pressure of the gas in both cylinders is 3.9 × 105 Pa. No thermal energy enters or leaves the gas. Determine the final temperature of the gas. temperature = K By reference to work done and change in internal energy, suggest why the temperature of the gas in cylinder A has changed.
9702_s13_qp_43
THEORY
2013
Paper 4, Variant 3
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State what is meant by the Avogadro constant NA, the mole. A container has a volume of 1.8 × 104 cm3. The ideal gas in the container has a pressure of 2.0 × 107 Pa at a temperature of 17 °C. Show that the amount of gas in the cylinder is 150 mol. Gas molecules leak from the container in at a constant rate of 1.5 × 1019 s−1. The temperature remains at 17 °C. In a time t, the amount of gas in the container is found to be reduced by 5.0%. Calculate the pressure of the gas after the time t, pressure = Pa the time t. t = s
9702_s16_qp_42
THEORY
2016
Paper 4, Variant 2
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State two of the basic assumptions of the kinetic theory of gases. An ideal gas has amount of substance n. The gas is initially in state X, with pressure 2p and volume V. The gas is cooled at constant volume to state Y, with pressure p. The gas is then heated at constant pressure to state Z, with volume 2V. Finally, the gas returns at constant temperature to state X. Determine an expression for the temperature T of the gas in state X, in terms of n, p and V. Identify any other symbols that you use. On , sketch the variation with volume of pressure for the gas as the gas undergoes the three changes. The state X is labelled. Label states Y and Z. pressure volume V 2V p 2p X During the change of state from Y to Z, the increase in internal energy of the gas is U. During the change of state from Z to X, the work done on the gas is W. Complete Table 4.1 to indicate, for each of the three changes of state, the increase in internal energy of the gas, the thermal energy transferred to the gas and the work done on the gas, in terms of p, V, U and W. Table 4.1 change increase in internal energy of gas thermal energy transferred to gas work done on gas X to Y Y to Z +U Z to X +W
9702_s23_qp_41
THEORY
2023
Paper 4, Variant 1
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State what is meant by an ideal gas. State the temperature, in degrees Celsius, of absolute zero. temperature = °C A sealed vessel contains a mass of 0.0424 kg of an ideal gas at 227 °C. The pressure of the gas is 1.37 × 105 Pa and the volume of the gas is 0.640 m3. Calculate: the number of molecules of the gas in the vessel number of molecules = the mass of one molecule of the gas mass = kg the root-mean-square (r.m.s.) speed v of the molecules of the gas. v = m s–1 The gas in is now cooled gradually to absolute zero. On , sketch the variation with thermodynamic temperature T of the r.m.s. speed of the molecules of the gas. r.m.s. speed v T / K
9702_s23_qp_42
THEORY
2023
Paper 4, Variant 2
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Questions Discovered
19