Gas Laws

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Gas Pressure

Pressure is defined as the force applied per unit area: P = \frac{F}{A}.
The SI unit of pressure is the pascal (Pa), where 1 Pa = 1 N m⁻².
Gas pressure arises from the collisions of gas molecules with the walls of their container.
The force exerted by a gas on a surface acts perpendicular (at right angles) to that surface.
For a given force, a smaller area produces a larger pressure (e.g., a drawing pin or high heels).
When using P = \frac{F}{A}, ensure A is the cross-sectional area perpendicular to the force.

Diagram showing how increasing the temperature of a gas at constant volume leads to more frequent and energetic collisions with the container walls, thus increasing pressure.

Amount of Substance

The mole is the SI base unit for the amount of substance.
One mole is the amount of substance that contains as many elementary entities as there are atoms in 12 g of carbon-12.
This number is the Avogadro constant, N_A = 6.02 \times 10²³ mol⁻¹ (rounded to 3 s.f.).
The number of moles is given by n = \frac{N}{N_A}, where N is the number of particles.
The molar mass m_r is the mass of one mole of a substance, in g mol⁻¹: m_r = \frac{m}{n}.
For an element, the molar mass in grams is numerically equal to its relative atomic mass.
For a compound, add the relative atomic masses of all atoms in the formula to find the molar mass.

Gas Laws

An ideal gas obeys the relation PV \propto T, or equivalently \frac{PV}{T} = \text{constant}.
Boyle's law (constant temperature): P \propto \frac{1}{V}, so P₁V₁ = P₂V₂.
Charles's law (constant pressure): V \propto T, so \frac{V₁}{T₁} = \frac{V₂}{T₂}.
Gay-Lussac's (pressure) law (constant volume): P \propto T, so \frac{P₁}{T₁} = \frac{P₂}{T₂}.
For all gas law experiments, the mass and number of molecules of the gas are assumed constant.
On a P-V diagram: a horizontal line is constant pressure, a vertical line is constant volume, and a curve along an isotherm is constant temperature.
Always convert temperature to kelvin before using any gas law equation: T(\text{K}) = \θ(\text{°C}) + 273.

Gas particles and pressure

Ideal Gas Equation

The ideal gas equation combines the empirical gas laws: PV = nRT.
R is the molar gas constant, R = 8.31\ \text{J K}^-1\text{mol}^-1.
An alternative form using the number of molecules is PV = Nk_BT.
The Boltzmann constant is k_B = 1.38 \times 10^-23\ \text{J K}^-1, defined as k_B = \frac{R}{N_A}.
R relates to macroscopic quantities (volume, temperature), while k_B relates to the thermal energy of microscopic particles.
In PV = nRT, P is in Pa, V in m³, n in mol, and T in K.

Kinetic Theory of Gases

The kinetic theory models a gas as many identical molecules in continuous random motion at high speeds.
Assumptions include: molecular volume is negligible, collisions are elastic, and there are no intermolecular forces except during collisions.
Other assumptions: molecules obey Newton's laws, collision time is negligible, and external forces (e.g. gravity) are ignored.
Gas pressure is the average force per unit area from molecules colliding with the container walls.
A real gas approximates an ideal gas at low pressure, low density, and high temperature (well above its boiling point).
At high pressure or low temperature, real gases deviate from ideal behaviour because intermolecular forces and molecular volume become significant.

Derivation of the Kinetic Theory Equation

For one molecule in a cube of side l, the change in momentum per collision with a wall is \Delta p = -2mv.
The time between collisions with the same wall is \frac{2l}{v}, giving a force on the wall of F = \frac{mv²}{l}.
The pressure from one molecule is P = \frac{mv²}{l³}; for N molecules, P = \frac{Nmv²}{l³}.
In three dimensions, v_x² = \frac{1}{3}v², leading to P = \frac{Nmv²}{3V}.
Using density \rho = \frac{Nm}{V}, the equation becomes P = \frac{1}{3}\rho \overline{v²}.
This is the kinetic theory of gases equation, where \overline{v²} is the mean square speed.

Average Kinetic Energy of a Molecule

For an ideal monatomic gas, internal energy is entirely kinetic (no potential energy).
The average kinetic energy of one molecule is E_k = \frac{1}{2}m\overline{v²} = \frac{3}{2}k_BT.
A higher temperature means a greater average kinetic energy of the molecules.
The total internal energy of a monatomic gas is U = \frac{3}{2}Nk_BT or U = \frac{3}{2}nRT.
For a fixed volume, supplying heat increases internal energy and therefore temperature: U \propto T.
These equations apply only to monatomic gases (e.g. helium, neon, argon); diatomic molecules also have rotational energy.

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1.Which of the following is the correct definition of pressure?
Easy
- AThe force applied per unit area
- BThe force applied per unit volume
- CThe energy transferred per unit area
- DThe force applied per unit mass
2.Which of the following is the correct definition of the mole?
Easy
- AThe amount of substance that contains as many elementary entities as there are atoms in 12 g of carbon-12
- BThe mass of a substance that contains 6.02 × 10²³ atoms
- CThe volume occupied by one gram of a gas at room temperature and pressure
- DThe number of particles in one gram of any substance
3.An ideal gas is one that obeys the relation pV ∝ T at all pressures, volumes and temperatures.
Easy
True or false?
4.Which of the following is a correct assumption of the kinetic theory of gases?
Easy
- AThe molecules collide elastically with each other and the walls of the container
- BThe molecules exert strong attractive forces on each other at all times
- CThe volume of the molecules is a significant fraction of the container volume
- DThe molecules move in ordered, parallel paths
5.Which of the following are assumptions of the kinetic model of an ideal gas? (Select all that apply)
Medium
- AThe molecules are in continuous random motion
- BThere are no intermolecular forces between molecules except during collisions
- CThe molecules have a range of different masses
- DThe volume of the molecules is negligible compared to the volume of the container
- EThe molecules collide inelastically with the walls
6.According to Charles's law, for a fixed mass of gas at constant pressure, the volume is directly proportional to which of the following?
Medium
- AThe thermodynamic temperature in kelvin
- BThe temperature in degrees Celsius
- CThe pressure of the gas
- DThe number of moles of gas
7.Gay-Lussac's law (the pressure law) states that for a fixed mass of gas at constant volume, the pressure is directly proportional to the thermodynamic temperature.
Easy
True or false?
8.Under which of the following conditions does a real gas best approximate an ideal gas?
Medium
- ALow pressure, low density, and temperature well above its boiling point
- BHigh pressure, high density, and temperature close to its boiling point
- CLow pressure, high density, and temperature close to its boiling point
- DHigh pressure, low density, and temperature well above its boiling point

更多題目

9.Match each gas law with the quantity that is held constant.
Medium
- Boyle's law
- Charles's law
- Gay-Lussac's law
- Temperature
- Pressure
- Volume
10.Place the following steps of the kinetic theory derivation in the correct order.
Medium
- Find the change in momentum when a molecule hits a wall
- Calculate the number of collisions per second on a wall
- Find the change in momentum per second (force)
- Calculate the total pressure from N molecules
- Consider the effect of molecules moving in 3D space
11.The average kinetic energy of a molecule of an ideal gas is directly proportional to which of the following?
Medium
- AThe thermodynamic temperature of the gas
- BThe pressure of the gas
- CThe volume of the gas
- DThe number of moles of gas
12.The internal energy of an ideal monatomic gas is equal to the total kinetic energy of its molecules.
Easy
True or false?
13.The ideal gas constant R is defined as the product of which two constants?
Medium
- AAvogadro's constant and the Boltzmann constant
- BAvogadro's constant and the molar mass
- CThe Boltzmann constant and the molar mass
- DThe Boltzmann constant and the gas density
14.Which of the following statements about the ideal gas equation pV = nRT are correct? (Select all that apply)
Medium
- AT must be in kelvin
- BR is the ideal gas constant with value 8.31 J K^-1 mol^-1
- CpV = nRT applies to any gas under all conditions
- Dn is the number of moles of gas
- EpV = nRT can be written as pV = Nk_BT
15.A graph of pressure p against 1/V for a fixed mass of gas at constant temperature is plotted. What is the shape of the graph?
Medium
- AA straight line through the origin
- BA curve that decreases as 1/V increases
- CA horizontal straight line
- DA straight line with a non-zero intercept on the p-axis
16.The number of atoms in one mole of any substance is equal to the Avogadro constant, approximately 6.02 × 10²³.
Easy
True or false?
17.An ideal gas has a density of 2.0 kg m^-3 at a pressure of 1.5 × 10⁵ Pa. Calculate the mean square speed of the gas molecules.
Hard
- A2.25 × 10⁵ m² s^-2
- B1.5 × 10⁵ m² s^-2
- C4.5 × 10⁵ m² s^-2
- D3.0 × 10⁵ m² s^-2
18.Match each quantity with its correct SI unit.
Medium
- Pressure
- Volume
- Temperature
- Amount of substance
- Pascal (Pa)
- Cubic metre (m³)
- Kelvin (K)
- Mole (mol)
19.Arrange the following gas laws in order of increasing constant quantity: constant temperature, constant pressure, constant volume.
Medium
- Boyle's law
- Charles's law
- Gay-Lussac's law
20.A fixed mass of ideal gas undergoes a change where its pressure is doubled and its volume is halved. What happens to its temperature?
Hard
- AIt remains constant
- BIt doubles
- CIt halves
- DIt quadruples
21.Which of the following are true about real gases compared to ideal gases? (Select all that apply)
Hard
- AReal gases may have intermolecular forces
- BReal gases have non-negligible molecular volume at high densities
- CReal gases obey pV = nRT under all conditions
- DReal gases can condense into liquids at low temperatures
- EReal gases behave most ideally at high pressures and low temperatures
22.A gas cylinder contains 4.0 × 10²³ molecules of an ideal gas at a temperature of 300 K and a pressure of 2.0 × 10⁵ Pa. Calculate the volume of the cylinder. Use k_B = 1.38 × 10^-23 J K^-1.
Challenge
- A8.3 × 10^-3 m³
- B4.1 × 10^-3 m³
- C1.7 × 10^-2 m³
- D2.1 × 10^-3 m³
23.A fixed mass of ideal gas is taken through a cycle A → B → C → A on a p-V diagram. A → B is isothermal expansion, B → C is isobaric compression, C → A is isochoric heating. Which of the following describes the change in internal energy from A to B?
Challenge
- AInternal energy remains constant
- BInternal energy increases
- CInternal energy decreases
- DInternal energy first increases then decreases
24.The pressure of a gas is defined as the force applied per unit area. The equation for pressure is P = F/A. Which of the following is the SI unit of pressure?
Easy
- Anewton (N)
- Bpascal (Pa)
- Cjoule (J)
- Dkilogram (kg)
25.Which of the following are correct rearrangements or statements of the ideal gas equation? (Select all that apply)
Medium
- APV = nRT
- BPV = NkBT
- CP = nRT / V
- DT = PV / nR
- En = PVRT
26.In the kinetic theory of gases, the molecules are assumed to collide elastically with each other and with the walls of the container.
Easy
True or false?
27.Match each gas law to the relationship it describes for a fixed mass of gas.
Medium
- Boyle's law
- Charles's law
- Gay-Lussac's law
- P ∝ 1/V at constant T
- V ∝ T at constant P
- P ∝ T at constant V
28.Place the following steps in the correct order for calculating the number of gas particles using the ideal gas equation.
Medium
- Convert temperature to kelvin and pressure to pascals if necessary
- Calculate the number of moles n using n = PV / RT
- Calculate the number of particles N using N = nNA
- Write down the ideal gas equation PV = nRT
29.A fixed mass of ideal gas is taken through a cycle on a p–V diagram. Which of the following processes is represented by a horizontal line on the diagram?
Medium
- AIsothermal expansion
- BIsobaric compression
- CIsochoric heating
- DAdiabatic expansion
30.A fixed mass of ideal gas is compressed at constant temperature. The pressure increases. Which statement explains this in terms of molecular motion?
Medium
- AThe molecules move faster and hit the walls more often.
- BThe molecules hit the walls more frequently because they travel a shorter distance between collisions.
- CThe molecules become larger and occupy more of the container.
- DThe molecules collide less frequently but with greater force.
31.A fixed mass of ideal gas is heated at constant volume. The pressure increases. Which statement explains this in terms of molecular motion?
Medium
- AThe molecules hit the walls more often and with greater force because their average speed increases.
- BThe molecules hit the walls less often but with greater force.
- CThe molecules hit the walls more often but with less force.
- DThe molecules expand and take up more space, increasing the pressure.
32.A fixed mass of ideal gas is heated at constant pressure. The volume increases. Which statement explains this in terms of molecular motion?
Medium
- AThe molecules move faster and push the walls outwards more forcefully, so the volume expands until the pressure returns to its original value.
- BThe molecules move slower and the volume increases to keep the pressure constant.
- CThe molecules collide more often with the walls, so the volume must increase to reduce the collision rate.
- DThe molecules exert a greater force on the walls, so the volume increases to keep the temperature constant.
33.An ideal gas is taken through a cycle A → B → C → A on a p–V diagram. A → B is isothermal expansion, B → C is isobaric compression, C → A is isochoric heating. Which of the following correctly describes the changes in pressure, volume and temperature?
Medium
- AA → B: p decreases, V increases, T constant; B → C: p constant, V decreases, T decreases; C → A: V constant, p increases, T increases.
- BA → B: p increases, V increases, T increases; B → C: p constant, V decreases, T constant; C → A: V constant, p decreases, T decreases.
- CA → B: p decreases, V constant, T decreases; B → C: p constant, V increases, T increases; C → A: V constant, p constant, T constant.
- DA → B: p constant, V increases, T increases; B → C: p decreases, V constant, T decreases; C → A: V increases, p increases, T constant.
34.A fixed mass of ideal gas is taken through a cycle on a p–V diagram. Which of the following processes is represented by a vertical line on the diagram?
Medium
- AAn isochoric process
- BAn isobaric process
- CAn isothermal process
- DAn adiabatic process
35.A fixed mass of ideal gas is taken through a cycle on a p–V diagram. Which of the following processes is represented by a curved line along an isotherm?
Medium
- AAn isothermal process
- BAn isobaric process
- CAn isochoric process
- DAn adiabatic process
36.A fixed mass of ideal gas is taken through a cycle on a p–V diagram. Which of the following correctly describes the changes in pressure, volume and temperature for a curved line between isotherms?
Hard
- APressure, volume and temperature all change.
- BPressure and volume change, but temperature is constant.
- CPressure and temperature change, but volume is constant.
- DVolume and temperature change, but pressure is constant.
37.A graph of pressure p against 1/V for a fixed mass of gas at constant temperature is plotted. What is the shape of the graph at a higher temperature?
Hard
- AA straight line through the origin with a steeper gradient
- BA straight line through the origin with a shallower gradient
- CA straight line not through the origin
- DA curve passing through the origin
38.An ideal gas is kept at a constant temperature. Its initial pressure is 1.2 × 10⁵ Pa and its initial volume is 3.0 × 10^-3 m³. The volume is decreased to 1.5 × 10^-3 m³. Calculate the new pressure.
Medium
- A2.4 × 10⁵ Pa
- B6.0 × 10⁴ Pa
- C1.2 × 10⁵ Pa
- D3.6 × 10⁵ Pa
39.A fixed mass of ideal gas undergoes a change where its pressure is doubled and its volume is doubled. What happens to its temperature?
Hard
- AIt quadruples
- BIt doubles
- CIt halves
- DIt remains constant
40.A fixed mass of ideal gas is taken through a cycle A → B → C → A on a p–V diagram. A → B is isothermal expansion, B → C is isobaric compression, C → A is isochoric heating. Which of the following correctly describes the changes in pressure, volume and temperature?
Medium
- AA → B: p decreases, V increases, T constant; B → C: p constant, V decreases, T decreases; C → A: V constant, p increases, T increases.
- BA → B: p increases, V increases, T increases; B → C: p constant, V decreases, T constant; C → A: V constant, p decreases, T decreases.
- CA → B: p decreases, V constant, T decreases; B → C: p constant, V increases, T increases; C → A: V constant, p constant, T constant.
- DA → B: p constant, V increases, T increases; B → C: p decreases, V constant, T decreases; C → A: V increases, p increases, T constant.
41.State the name of the gas law that describes the relationship between pressure and volume at constant temperature.
Easy
42.Complete the equation for an ideal gas using the Boltzmann constant.
Easy
pV = N ____ T
43.An ideal gas is kept at a constant temperature. Its initial pressure is 1.2 × 10⁵ Pa and its initial volume is 3.0 × 10^-3 m³. The volume is decreased to 1.5 × 10^-3 m³. Calculate the new pressure of the gas.
Medium
44.Complete the kinetic theory equation relating pressure to density and mean square speed.
Medium
p = (1/3) ρ ____
45.A fixed mass of gas at constant volume has an initial pressure of 1.0 × 10⁵ Pa at 300 K. The temperature is increased to 450 K. Calculate the new pressure.
Medium
46.A gas has a temperature of 27 °C and a pressure of 1.5 × 10⁵ Pa. It occupies a volume of 0.04 m³. Calculate the number of moles of gas. Use R = 8.31 J K^-1 mol^-1.
Hard
47.Complete the equation for the average kinetic energy of a molecule of an ideal gas.
Medium
E____k = (3/2) ____ T
48.A container holds 3.0 moles of an ideal monatomic gas at a temperature of 400 K. Calculate the total internal energy of the gas. Use R = 8.31 J K^-1 mol^-1.
Hard
49.A gas has a temperature of –55°C and a pressure of 0.5 MPa. It occupies a volume of 0.02 m³. Calculate the number of gas particles. Use R = 8.31 J K⁻¹ mol⁻¹ and NA = 6.02 × 10²³ mol⁻¹. Give your answer to 2 significant figures.
Medium
50.An ideal gas has a density of 4.5 kg m⁻³ at a pressure of 9.3 × 10⁵ Pa. Calculate the mean square speed of the gas molecules. Give your answer to 2 significant figures.
Medium
51.A fixed mass of ideal gas occupies a volume of 2.5 × 10^-3 m³ at a pressure of 1.2 × 10⁵ Pa. It is compressed to a volume of 1.0 × 10^-3 m³ at constant temperature. Calculate the new pressure in Pa.
Medium
52.A fixed mass of gas at constant volume has an initial pressure of 1.0 × 10⁵ Pa at 300 K. The temperature is increased to 450 K. Calculate the new pressure in Pa.
Medium
53.State what is meant by an ideal gas.
Medium
54.Explain, in terms of molecular motion, why the pressure of a fixed mass of gas increases when its volume is decreased at constant temperature.
Hard
55.State three assumptions of the kinetic model of an ideal gas.
Hard
56.Describe how the ideal gas constant R is defined and state its value with units.
Hard
57.Explain why the pressure of a gas increases when it is heated at constant volume, in terms of molecular motion.
Challenge
58.An ideal gas is kept at a constant temperature. Its initial pressure is 1.2 × 10⁵ Pa and its initial volume is 3.0 × 10^-3 m³. The volume is decreased to 1.5 × 10^-3 m³. Calculate the new pressure of the gas.
Medium
- A240000 Pa
- B288000 Pa
- C216000 Pa
- D264000 Pa
59.A fixed mass of gas at constant volume has an initial pressure of 1.0 × 10⁵ Pa at 300 K. The temperature is increased to 450 K. Calculate the new pressure.
Medium
- A150000 Pa
- B135000 Pa
- C165000 Pa
- D180000 Pa
60.A gas has a temperature of 27 °C and a pressure of 1.5 × 10⁵ Pa. It occupies a volume of 0.04 m³. Calculate the number of moles of gas. Use R = 8.31 J K^-1 mol^-1.
Hard
- A3.4 mol
- B1.4 mol
- C4.4 mol
- D2.4 mol
61.A container holds 3.0 moles of an ideal monatomic gas at a temperature of 400 K. Calculate the total internal energy of the gas. Use R = 8.31 J K^-1 mol^-1.
Hard
- A17950 J
- B14958 J
- C16454 J
- D13462 J
62.A gas has a temperature of –55°C and a pressure of 0.5 MPa. It occupies a volume of 0.02 m³. Calculate the number of gas particles. Use R = 8.31 J K⁻¹ mol⁻¹ and NA = 6.02 × 10²³ mol⁻¹. Give your answer to 2 significant figures.
Medium
- A2.97e+24 particles
- B3.63e+24 particles
- C3.96e+24 particles
- D3.3e+24 particles
63.An ideal gas has a density of 4.5 kg m⁻³ at a pressure of 9.3 × 10⁵ Pa. Calculate the mean square speed of the gas molecules. Give your answer to 2 significant figures.
Medium
- A620000 m² s⁻²
- B682000 m² s⁻²
- C744000 m² s⁻²
- D558000 m² s⁻²
64.A fixed mass of ideal gas occupies a volume of 2.5 × 10^-3 m³ at a pressure of 1.2 × 10⁵ Pa. It is compressed to a volume of 1.0 × 10^-3 m³ at constant temperature. Calculate the new pressure in Pa.
Medium
- A300000 Pa
- B270000 Pa
- C330000 Pa
- D360000 Pa
65.A fixed mass of gas at constant volume has an initial pressure of 1.0 × 10⁵ Pa at 300 K. The temperature is increased to 450 K. Calculate the new pressure in Pa.
Medium
- A165000 Pa
- B180000 Pa
- C135000 Pa
- D150000 Pa

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