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Gas Laws (DP IB Physics)

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    Define pressure.

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  • Define pressure.

    Pressure is the force applied per unit area.

  • True or False?

    When an object is immersed in a fluid, forces act parallel to its surfaces to produce pressure.

    False.

    When an object is immersed in a fluid, forces act perpendicular to its surfaces to produce pressure.

  • How do gas particles exert a force on the walls of a container?

    Gas particles exert a force on the container by colliding with the walls. Over the area of a wall, this exerts a pressure.

  • Which person exerts a higher pressure on the ground, a 70 kg person standing on one foot or a 100 kg person standing on both feet? Assume their feet are the same size.

    The 70 kg person on one foot exerts a higher pressure. The 70 kg person exerts ~700 N per foot but the 100 kg person exerts ~500 N per foot, so the 70 kg person exerts a greater pressure.

  • Recall the equation for pressure.

    The equation for pressure is P space equals space F over A

    Where:

    • P = pressure, measured in pascals (Pa)

    • F = force, measured in newtons (N)

    • A = area measured in metres squared (m2)

  • True or False?

    Gas particles travelling at a greater speed exert a greater force on container walls.

    True.

    When gas particles collide with container walls at higher speeds, they exert a greater force and hence a greater pressure on the container.

  • True or False?

    The frequency of collisions between particles and container walls is increased, but their speed remains constant. The pressure exerted by the gas is unchanged.

    False.

    When gas particles collide with container walls more frequently, the average force is increased so the pressure increases.

  • Define one mole.

    One mole is the amount of substance that contains as many elementary entities as the number of atoms in 12 g of carbon-12.

  • How many atoms are in one mole of argon (Ar)? Argon is a monatomic gas.

    In one mole of argon there are 6.02 × 1023 atoms – this is the Avogadro constant.

  • True or False?

    There are 6.02 × 1023 atoms in one mole of gaseous carbon dioxide.

    False.

    In one mole of CO2 there are 6.02 × 1023 elementary elements, but each elementary element has three atoms so there are 1.86 × 1024 atoms in one mole of CO2.

  • Write the equation for the number of moles in a substance when the number of particles is known.

    The equation for the number of moles is n space equals space N over N subscript A

    Where:

    • n = number of moles (mol)

    • N= number of particles (particles may be atoms or molecules, depending on the substance)

    • N subscript A= Avogadro constant, measured in particles per mole (mol−1)

  • Write the equation for molar mass when number of moles and mass is known.

    The equation for molar mass is m subscript r space equals space m over n

    Where:

    • m subscript r= molar mass, measured in grams per mole (g mol−1)

    • m= mass of substance, measured in grams (g)

    • n= number of moles (mol)

  • True or False?

    There are fewer atoms in one mole of N2 than there are in one mole of C2H5OH.

    True.

    While both contain the same number of molecules, there are more atoms in a molecule of C2H5OH than in a molecule of N2.

  • How would the molar mass of H2O be determined if the relative atom masses of H and O are known?

    The molar mass in g mol−1 is the sum of the relative atomic masses of the atoms in the molecule.

  • What relation does an ideal gas obey?

    An ideal gas obeys the relation P V space proportional to space T or fraction numerator P V over denominator T end fraction space equals space constant

    Where:

    • P = pressure, measured in pascals (Pa)

    • V = volume, measured in metres cubed (m3)

    • T = temperature, measured in kelvin (K)

  • Define the meaning of an empirical law.

    An empirical law is formed based on observed data from experiments, rather than being derived from theory.

  • Write the equation for Boyle's law in terms of initial and final states.

    Boyle's law in terms of initial and final states is P subscript 1 V subscript 1 space equals space P subscript 2 V subscript 2 for a constant temperature.

    Where:

    • P subscript 1 = initial pressure, measured in pascals (Pa)

    • P subscript 2 = final pressure, measured in pascals (Pa)

    • V subscript 1 = is initial volume, measured in cubic metres (m3)

    • V subscript 2 = final volume, measured in cubic metres (m3)

  • What is the relationship shown by the graph below?

    Graph showing the relationship between volume and pressure. The curve starts high on the volume axis with a low pressure and volume decreases as pressure increases. The gradient begins steep and negative but becomes shallower.

    The graph shows that pressure is inversely proportional to volume.

  • Write the equation for Charles' law in terms of initial and final states.

    Charles' law in terms of initial and final states is V subscript 1 over T subscript 1 space equals space V subscript 2 over T subscript 2 for a constant pressure.

    Where:

    • T subscript 1 = initial temperature, measured in kelvin (K)

    • T subscript 2 = final temperature, measured in kelvin (K)

    • V subscript 1 = initial volume, measured in cubic metres (m3)

    • V subscript 2 = final volume, measured in cubic metres (m3)

  • True or False?

    The graph showing Charles' law is curved. The axes are volume and temperature (K).

    False.

    The graph representing Charles' law is a straight line passing through the origin. The axes are volume and temperature (K).

  • Write the equation for Gay-Lussac's law in terms of initial and final states.

    Charles' law in terms of initial and final states is P subscript 1 over T subscript 1 space equals space P subscript 2 over T subscript 2 for a constant volume.

    Where:

    • T subscript 1 = initial temperature, measured in kelvin (K)

    • T subscript 2 = final temperature, measured in kelvin (K)

    • P subscript 1 = initial pressure, measured in pascals (Pa)

    • P subscript 2 = final pressure , measured in pascals (Pa)

  • True or False?

    The graph below shows a directly proportional relationship.

    Graph showing the relationship between pressure and temperature in degrees Celsius, with a green line starting at -273°C and increasing upward.

    False.

    Gay-Lussac's law states that pressure is directly proportional to temperature in kelvin. This graph shows temperature in degrees Celsius and does not pass through the origin, so is not directly proportional.

  • Write the ideal gas equation when the number of moles is known.

    The ideal gas equation for a known number of moles is P V space equals space n R T

    Where:

    • P = pressure, measured in pascals (Pa)

    • V = volume, measured in metres cubed (m3)

    • n = number of moles measured in moles (mol)

    • R = gas constant, measured in joules per kelvin per mole (J K−1 mol−1)

    • T = temperature, measured in kelvin (K)

  • Write the ideal gas equation when the number of particles is known.

    The ideal gas equation for a known number of moles is P V space equals space N k subscript B T

    Where:

    • P = pressure, measured in pascals (Pa)

    • V = volume, measured in metres-cubed (m3)

    • N = number of particles

    • k subscript B = Boltzmann constant, measured in joules per kelvin (J K−1)

    • T = temperature, measured in kelvin (K)

  • True or False?

    The ideal gas law equation can be derived from the empirical gas laws for constant pressure and constant volume.

    False.

    The ideal gas law equation can be derived from the empirical gas laws for constant pressure, constant volume and constant temperature.

  • How are the Boltzmann constant, gas constant and Avogadro constant related?

    The Boltzmann constant, gas constant and Avogadro constant are related by k subscript B space equals space R over N subscript A

    Where:

    • k subscript B = Boltzmann constant, measured in joules per kelvin (J K−1)

    • R = gas constant, measured in joules per kelvin per mole (J K−1 mol−1)

    • N subscript A = Avogadro constant, measured in particles per mole (mol−1)

  • What is the ideal gas equation for a known number of particles when temperature is the subject?

    The ideal gas equation for a known number of particles with temperature as the subject is T space equals space fraction numerator P V over denominator N k subscript B end fraction

    Where:

    • P = pressure, measured in pascals (Pa)

    • V = volume, measured in metres-cubed (m3)

    • N = number of particles

    • k subscript B = Boltzmann constant, measured in joules per kelvin (J K−1)

    • T = temperature, measured in kelvin (K)

  • How can the motion of particles in a gas be described?

    Particles in a gas move:

    • randomly

    • at high speeds

  • What two properties of gases does the kinetic theory of gases connect?

    The kinetic theory of gases connects:

    • the microscopic properties of particles, e.g. mass and speed

    • the macroscopic properties of particles, e.g. pressure and volume

  • True or False?

    Kinetic theory assumes that there is a small gravitational force between particles.

    False.

    Kinetic theory assumes that no forces act between particles other than when they collide.

  • What are three assumptions made about forces in kinetic theory?

    Any three of the following assumptions about forces made in kinetic theory:

    1. The molecules obey Newton's laws of motion

    2. There are no forces between the molecules except during collisions

    3. External forces (e.g. gravity) are ignored

    4. Each particle exerts a force on the wall of the container with which it collides.

  • What are three assumptions about the quantity and size of molecules in kinetic theory?

    Any three of the following assumptions about molecules in kinetic theory:

    1. A gas consists of identical molecules in a container

    2. Molecules have negligible volume and are point particles

    3. The number of molecules of gas in a container is very large so the average behaviour is considered

  • What are three assumptions made about collisions in kinetic theory?

    Three of the following assumptions about collisions in kinetic theory:

    1. The molecules collide elastically

    2. There are no forces between the molecules except during collisions

    3. The time of a collision between molecules is negligible compared to the time between collisions

    4. Each particle exerts a force on the wall of the container with which it collides.

  • True or False?

    Real gases obey the ideal gas law.

    False.

    Real gases do not obey the ideal gas law as the assumptions of kinetic theory are not always valid.

  • What are the conditions for a gas to be considered an ideal gas?

    The conditions for a gas to be ideal are:

    • the gas pressure is low

    • the gas density is low

    • the temperature is sufficiently higher than the boiling point of the substance

  • Show that the change in momentum for a particle of mass, m, and initial speed, plus v, is negative 2 m v when it collides with a container wall.

    The change in momentum is straight capital delta p space equals space m straight capital delta v space equals space m open parentheses v subscript f space minus space v subscript i close parentheses

    After rebounding elastically, speed is negative v in the negative direction.

    Therefore, change in momentum is straight capital delta p space equals space m straight capital delta v space equals space m open parentheses open parentheses negative v close parentheses space minus space v close parentheses space equals space minus 2 m v

  • Show that, for a particle with speed, v, crossing a container of width, L, the time between collisions with a particular wall is fraction numerator 2 L over denominator v end fraction.

    To find the time between collisions:

    • The time between collisions is given by straight capital delta t space equals space distance over speed

    • Between collisions, the particle travels the width of the container and back, 2 L

    • The speed is v so straight capital delta t space equals space fraction numerator 2 L over denominator v end fraction

  • True or False?

    If the time between collisions with a container wall for a particle is fraction numerator 2 L over denominator v end fraction and the change in momentum is negative 2 m v then the force on the container wall is negative fraction numerator m v squared over denominator L end fraction.

    False.

    Force is rate of change of momentum, so fraction numerator negative 2 m v over denominator fraction numerator 2 L over denominator v end fraction end fraction space equals space minus fraction numerator m v squared over denominator L end fraction is the force on the particle. The particle exerts an equal and opposite force of plus fraction numerator m v squared over denominator L end fraction on the wall.

  • State the kinetic theory of gases equation.

    The kinetic theory of gases equation is P space equals space 1 third rho v squared

    Where:

    • P = pressure, measured in pascals (Pa)

    • rho = density of the gas, measured in kilograms per metre-cubed (kg m−3)

    • v = average speed of particles, measured in metres per second (m s−1)

  • True or False?

    When a single gas particle is assumed to travel only in the x direction, the square of its speed is only one third of the particle's actual speed in three dimensions.

    True.

    The squared speed in the x direction, v subscript x squared only accounts, on average, for one third of the square of the overall speed if the particle moves in all directions, so v subscript x squared space equals space 1 third v squared

  • State the kinetic energy equation.

    The kinetic energy equation is E subscript k space equals space 1 half m v squared

    Where:

    • E subscript k = kinetic energy, measured in joules (J)

    • m = mass, measured in kilograms (kg)

    • v = speed, measured in metres per second (m s−1)

  • How are the following three equations combined to give E subscript k space equals space 3 over 2 k subscript B T?

    1. P space equals space fraction numerator N m v squared over denominator 3 V end fraction

    2. P V space equals space N k subscript B T

    3. E subscript k space equals space 1 half m v squared

    To obtain the equation E subscript k space equals space 3 over 2 k subscript B T:

    • Combine equations 1 and 3: P space equals space fraction numerator N m v squared over denominator 3 V end fraction space equals space fraction numerator N space cross times space 2 E subscript k over denominator 3 V end fraction

    • Rearrange this and equate to equation 2: 2 over 3 N E subscript k space equals space P V and P V space equals space N k subscript B T

    • 2 over 3 up diagonal strike N E subscript k space equals space up diagonal strike N k subscript B T

    • Rearrange to give: E subscript k space equals space 3 over 2 k subscript B T

  • In terms of the Boltzmann constant, what is the total internal energy of N particles of an ideal gas?

    Total internal energy of an ideal gas is U space equals space 3 over 2 N k subscript B T

    Where:

    • U = total internal energy, measured in joules (J)

    • N = number of ideal gas particles

    • k subscript B = Boltzmann constant, measured in joules per kelvin (J K−1)

    • T = temperature, measured in kelvin (K)

  • True or False?

    Total internal energy and total kinetic energy are not equal for a real gas.

    True.

    In an ideal gas, total internal energy is total kinetic energy. In a real gas, however, internal energy is the sum of all kinetic and potential energies which arise from intermolecular interactions.

  • Why is it not possible to state that internal energy is proportional to temperature for H2 gas?

    This relationship only applies to monatomic gases. H2 is diatomic so also has rotational kinetic energy.

  • A low pressure sample of monatomic helium gas has an initial temperature of 450 K and a final temperature of 1800 K.

    By what factor has its internal energy increased?

    For an ideal gas, internal energy is proportional to temperature. Temperature has increased by a factor of 4 so the internal energy has also increased by a factor of 4.

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      Kinematics

    2. Topic 2

      Forces & Momentum

    3. Topic 3

      Work, Energy & Power

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      Rigid Body Mechanics

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      Galilean & Special Relativity

    1. Topic 1

      Thermal Energy Transfers

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      Greenhouse Effect

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      Gas Laws

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      Thermodynamics

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      Current & Circuits

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      Simple Harmonic Motion

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      Wave Model

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      Wave Phenomena

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      Standing Waves & Resonance

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      Doppler Effect

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      Structure of the Atom

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      Quantum Physics

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      Measurements in Physics

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      Processing Uncertainties

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      Scalars & Vectors