Matter

 Module 5: Rise and the fall of the clockwork universe

5.2 Matter
5.2.1

Matter: Very simple

a)   Describe and explain:

I)                   Energy transfer producing a change in temperature and the concept of specific thermal capacity c

II)                The behavior of ideal gases

III)             Impulse F∆t = ∆p

IV)             The kinetic theory of ideal gases

V)                Temperature as proportional to average energy per particle [math]\text{average energy} = \frac{3}{2} kT ≈ kT[/math] as a useful approximation

VI)             Random walk of molecules in a gas: displacement in N steps related to [math]\sqrt{N}[/math].

b)   Make appropriate use of:

I)                   The terms: ideal gas, root mean square speed, absolute temperature, internal energy, Avogadro constant, Boltzmann constant, gas constant, mole

by sketching and interpreting:

II)                Relationships between p, V, N and T for an ideal gas

III)             Force–time graph for an interaction with area under line representing the impulse

c)    Make calculations and estimates involving:

I)                   Temperature and energy change using [math]∆E=mc∆θ[/math]

II)            [math]pV = NkT[/math]  where [math] N=nN_A [/math] and [math] Nk = nR[/math]

III)    [math]pV = \frac{1}{3} N m \overline{c^2}[/math]

d)   Demonstrate and apply knowledge and understanding of the following practical activities (HSW4):

I)                   Using an electrical method to find the specific thermal capacity of a metal block or liquid

II)                Using appropriate apparatus to investigate the gas laws including determining absolute zero

III)             Using apparatus to investigate the relationship of volume with pressure, measured either by pressure gauge or differential pressure monitor and data logger.

  • a) Describe and explain

  • I)  Energy Transfer Producing a Change in Temperature and the Concept of Specific Heat Capacity

  • ⇒ Energy Transfer and Temperature
  • When energy is transferred to or from a material, the internal energy of the material changes. This internal energy is stored in the motion of particles (kinetic energy) and their interactions (potential energy).
  • The result of energy transfer is typically a change in the temperature of the material, provided no phase change occurs.
  • ⇒  Specific Heat Capacity (c)
  • The specific heat capacity of a substance is defined as the amount of energy required to raise the temperature of 1 kg of the substance by 1°C or 1 K.
  • Figure 1 Specific heat capacity
  • ⇒ Formula:
  • [math]Q = mc∆T[/math]
  • Where:
  • – Q: Heat energy transferred (J).
  • – m: Mass of the substance (kg).
  • – c: Specific heat capacity ([math]J/kgK[/math]).
  • – ΔT: Temperature change (K or °C).
  • ⇒ Example:
  • To heat 2 kg of water ([math]C_{\text{water}} = 4200 \, \frac{J}{\text{kg} \cdot K}[/math]) by 10°C:
  • [math]Q = mc\Delta T \\
    Q = (2)(4200)(10^\circ C) \\
    Q = 84,000 \, J[/math]
  • Materials with high c (like water) require more energy to change temperature and are excellent for thermal storage.
  • Materials with low c (like metals) heat up or cool down quickly.

  • II) The Behavior of Ideal Gases

  • An ideal gas is a theoretical model where gas particles:
  • – Move randomly and rapidly.
  • – Do not interact except during elastic collisions.
  • – Are point masses with negligible volume compared to the container.
  • ⇒ Laws:
  • Boyle’s Law:
  • [math]\text{PV} = \text{constant (at constant temperature)}[/math]
  • – If pressure increases, volume decreases, and vice versa.
  • Figure 2 Boyle’s Law
  • Charles’s Law:
  • [math] \frac{V}{T} = \text{constant (at constant pressure)}[/math]
  • – Volume increases as temperature increases.
  • Figure 3 Charles’s Law
  • Pressure Law:
  • [math]\frac{P}{T} = \text{constant (at constant Volume)}[/math]
  • – Pressure increases as temperature increases.
  • Figure 4 Pressure Law
  • Combined Gas Law:
  • [math]\frac{PV}{T} = \text{Constant}[/math]
  • ⇒  Ideal Gas Equation:
  • [math]PV = nRT[/math]
  • Where:
  • – P: Pressure (Pa).
  • – V: Volume ([math]m^3[/math]).
  • – n: Number of moles.
  • – [math]R = 8.31 J/molK[/math]: Universal gas constant.
  • – T: Temperature (K).
  • ⇒ Example:
  • For 1 mole of gas at [math]300 \, \text{K} \text{ and } 1 \, \text{atm} = 1.01 \times 10^5 \, \text{Pa}[/math]
  • [math]V = \frac{nRT}{P} \\
    V = \frac{(1 \times 8.31 \times 300)}{(1.01 \times 10^5)} \\
    V = 0.0247 \, \text{m}^3[/math]
  • ⇒ Kinetic Theory of Gases:
  • The pressure of an ideal gas is related to the kinetic energy of particles:
  • [math]P = \frac{1}{3} \frac{N}{V} m \langle v^3 \rangle[/math]
  • Where:
  • – N: Number of particles.
  • – m: Mass of a particle.
  • – [math]\langle v^3 \rangle[/math]: Mean squared speed.

  • III) Impulse and Momentum

  • ⇒ Momentum (p)
  • Momentum is the product of mass and velocity:
  • [math]p = mv [/math]
  • Where:
  • – p: Momentum ([math]kgm/s[/math]).
  • – m: Mass (kg)
  • – v: Velocity (m/s).
  • ⇒ Impulse ([math]F \Delta t [/math])
  • Impulse is the product of force and the time it acts. It equals the change in momentum ([math]\Delta p[/math]).
  • Formula:
  • [math]F \Delta t = \Delta p \\
    F \Delta t = m (v_f – v_i)[/math]
  • Where:
  • – F: Force (N).
  • – [math]\Delta t[/math]: Time during which the force acts (s).
  • ​- [math]v_f , v_i[/math]: Final and initial velocities.
  • ⇒  Example:
  • A 2 kg object accelerates from 3 m/s to 8 m/s over 2 seconds:
  • [math]F \Delta t = m (v_f – v_i) \\
    F \Delta t = 2 (8 – 3) \\
    F \Delta t = 10 \, \text{kg} \cdot \text{m/s}[/math]
  • ⇒ Applications:
  • Collisions: Impulse helps analyze collisions (elastic or inelastic) and determine changes in momentum.
  • Safety Devices: Airbags and crumple zones reduce F by increasing .

  • IV)  The Kinetic Theory of Ideal Gases

  • The kinetic theory of gases provides a microscopic explanation for the macroscopic properties of gases, such as pressure, temperature, and volume.
  • It describes the behavior of gases in terms of the motion of their particles, assuming the gas is ideal.
  • ⇒  Assumptions of the Kinetic Theory of Ideal Gases
  • Large Number of Particles:
  • – An ideal gas consists of a vast number of identical particles (atoms or molecules) in constant, random motion.
  • Negligible Volume:
  • – The particles occupy negligible space compared to the volume of the container.
  • No Intermolecular Forces:
  • – Particles do not exert forces on each other except during collisions, which are perfectly elastic.
  • Elastic Collisions:
  • – Collisions between particles and with the container walls conserve kinetic energy and momentum.
  • Random Motion:
  • – The particles move randomly in all directions, with velocities distributed according to statistical laws.
  • Time Between Collisions:
  • – The time a particle spends in free motion is much greater than the time it spends colliding.
  • Temperature Proportional to Kinetic Energy:
  • – The average kinetic energy of particles is directly proportional to the absolute temperature of the gas.

  • V)   Temperature and Average Energy per Particle

  • Temperature is directly proportional to the average kinetic energy of particles in an ideal gas.
  • ⇒  Average Energy per Particle
  • The kinetic energy (KE) of an individual gas particle is related to the temperature (T) by the Boltzmann constant (k):
  • [math]KE_{avg} = \frac{3}{2} k T[/math]
  • Where:
  • – [math]KE_{avg}[/math]: Average kinetic energy per particle (J).
  • – [math]k = 1.38 \times 10^{-23} \, \text{J/K}[/math]: Boltzmann constant.
  • – T: Absolute temperature (K).
  • ⇒ Interpretation:
  • Higher temperature means particles move faster, increasing their average kinetic energy.
  • For an ideal gas, the internal energy depends entirely on the temperature, as all the energy is kinetic (there are no intermolecular forces).
  • ⇒ Example Calculation:
  • – At T=300 K:
  • [math]KE_{\text{avg}} = \frac{3}{2} k T \\ KE_{\text{avg}} = \frac{3}{2} (1.38 \times 10^{-23}) (300) \\ KE_{\text{avg}} = 6.21 \times 10^{-21} \, \text{J}  [/math]
  • This value represents the average energy per particle. For a mole of gas particles:
  • [math]KE_{\text{avg, per mole}} = (6.022 \times 10^{23}) (6.21 \times 10^{-21}) \\ KE_{\text{avg, per mole}} = 3.74 \, \text{kJ/mol}[/math]
  • ⇒ Useful Approximation:
  • For rough calculations, the energy per particle can be approximated as:
  • [math]KE_{\text{avg}} \approx kT[/math]
  • This simplification is used when ​[math]\frac{3}{2}[/math] precision isn’t critical.

  • VI) Random Walk of Molecules in a Gas:

  • Molecules in a gas move randomly, colliding with each other and the walls of the container.
  • After N random steps, the net displacement of a molecule (R) is proportional to the square root of N:
  • [math]R \propto \sqrt{N}[/math]
  • ⇒ Random Walk Model
  • Each step has a fixed length (l) and random direction.
  • After N steps, the molecule’s root mean square displacement ([math]R_{\text{rms}} [/math]) is given by:
  • [math]R_{\text{rms}} = l \sqrt{N}[/math]
  • Figure 5 Random walk model
  • ⇒ Explanation of [math]\sqrt{N}[/math] Relationship
  • Since motion is random, the displacements in different directions tend to cancel out partially.
  • The net displacement (RRR) is smaller than the total path length ([math]L = Nl[/math])
  • Using statistical principles, the displacement follows:
  • [math]R_{\text{rms}} = l \sqrt{N} \\
    R_{\text{rms}} = \sqrt{\langle x^2 \rangle + \langle y^2 \rangle + \langle z^2 \rangle}[/math]
  • Where [math]\langle x^2 \rangle , \langle y^2 \rangle , \langle z^2 \rangle[/math] are proportional to N.
  • ⇒ Example Calculation:
  • If a gas molecule takes [math]N = 10^6 \text{ steps of } l = 10^{-8} \text{ m}[/math]:
  • [math]R_{\text{rms}} = l \sqrt{N} \\ R_{\text{rms}} = 10^{-8} \sqrt{10^6} \\ R_{\text{rms}} = 10^{-8} \times 10^3 \\ R_{\text{rms}} = 10^{-5} \text{ m}[/math]
  • – Thus, after [math]10^6[/math] steps, the net displacement is [math]10^{-5}m[/math], much smaller than the total path length of [math]10^{-2}m[/math].

  • b)   Make Appropriate Use of:


  • I) The terms: ideal gas, root mean square speed, absolute temperature, internal energy, Avogadro constant, Boltzmann constant, gas constant, mole

  • ⇒ Ideal Gas
  • Definition:
  • – A hypothetical gas that perfectly follows the ideal gas law:
  • [math]PV = nRT[/math]
  • OR
  • [math]PV = NKT[/math]
  • Uses:
  • – Modeling gas behavior in low-pressure and high-temperature environments where intermolecular forces are negligible.
  • – Example: Estimating the pressure of oxygen gas in a tank.
  • ⇒ Root Mean Square (RMS) Speed
  • Definition:
  • – A measure of the average speed of gas molecules, calculated as:
  • [math]v_{\text{rms}} = \sqrt{\frac{3kT}{m}}[/math]
  • OR
  • [math]v_{\text{rms}} = \sqrt{\frac{3RT}{M}}[/math]
  • – k: Boltzmann constant ([math]1.38 × 10^{-23} J/K[/math]).
  • – m: Mass of a single molecule.
  • – M: Molar mass of the gas.
  • Uses:
  • – Predicting the speed of gas molecules at a given temperature
  • – Example: Calculating the speed of nitrogen molecules at room temperature (300 K).
  • ⇒ Absolute Temperature
  • Definition:
  • – Temperature measured on the Kelvin scale, directly proportional to the average kinetic energy of particles.
  • [math]KE_{\text{avg}} = \frac{3}{2} k T[/math]
  • Uses:
  • – Describing the energy of gas particles.
  • – Example: Understanding why gas pressure increases with temperature in a closed system.
  • ⇒ Internal Energy
  • Definition:
  • – Total kinetic energy of all particles in an ideal gas:
  • [math]U = \frac{3}{2} k T[/math]
  • Uses:
  • – Calculating the energy required to change the temperature of a gas.
  • – Example: Determining the energy increase when a gas is heated.
  • ⇒ Avogadro Constant ([math]N_A[/math])
  • Definition:
  • – The number of particles in one mole of a substance:
  • [math]N_A = 6.022 \times 10^{23} \text{ particles/mol}[/math]
  • Uses:
  • – Converting between macroscopic and microscopic quantities.
  • Example: Calculating the number of oxygen molecules in 1 mole of oxygen gas.
  • ⇒ Boltzmann Constant (k)
  • Definition:
  • – Relates the energy of individual particles to temperature:
  • [math]k = 1.38 × 10^{-23} J/K[/math]
  • Uses:
  • – Relating temperature to molecular energy.
  • – Example: Estimating the kinetic energy of a single hydrogen atom at a given temperature.
  • ⇒ Gas Constant (R)
  • Definition:
  • – Relates pressure, volume, and temperature of a gas:
  • [math]R = 8.31 J/molK[/math]
  • Uses:
  • – Used in the ideal gas law.
  • – Example: Determining the pressure of 2 moles of helium gas in a container of 10 liters at 300 K.
  • ⇒ Mole
  • Definition:
  • – A quantity representing [math]6.022 × 10^{23}[/math] particles of a substance.
  • Uses:
  • – Counting large numbers of particles.
  • – Example: Converting the mass of CO₂ into the number of molecules.

  • II) Relationships Between P,V,N,T for an Ideal Gas

  • The ideal gas law links these quantities:
  • [math]PV = nRT[/math]
  • Or
  • [math]PV = NkT[/math]
  • ⇒ Relationships:
  • Pressure and Volume (P–V) at constant T (Boyle’s Law):
  • [math]P ∝ \frac{1}{V}[/math]
  • – Pressure decreases as volume increases.
  • – Graph: A hyperbolic curve (inverse relationship).
  • Figure 6 Hyperbolic curve
  • Volume and Temperature (V–T) at constant P (Charles’s Law):
  • [math]V ∝ T[/math]
  • – Volume increases with temperature.
  • – Graph: A straight line passing through the origin.
  • Figure 7 A straight line passing through the origin
  • Pressure and Temperature (P–T) at constant V:
  • [math]P ∝ T[/math]
  • – Pressure increases with temperature.
  • – Graph: A straight line passing through the origin.
  • Pressure, Volume, and Temperature (Combined Gas Law):
  • [math]\frac{PV}{T} = \text{constant}[/math]
  • ⇒ Interpretation of Graphs:
  • Pressure-Volume (P–V): Shows how gas compresses or expands.
  • Volume-Temperature (V–T): Demonstrates thermal expansion.
  • Pressure-Temperature (PPP–TTT): Explains how heating increases pressure.

  • III) Force–Time Graph and Impulse

  • Impulse Definition:
  • – Impulse is the change in momentum ([math] \Delta p[/math]) caused by a force acting over time:
  • [math]\text{Impulse} = F \Delta t \\
    \text{Impulse} = \Delta p[/math]
  • ⇒ Force-Time Graph:
  • The area under the curve of a force-time graph represents the impulse:
  • [math]\text{Area} = \text{Impulse}[/math]
  • ⇒ Cases:
  • Constant Force:
  • – Graph: A rectangle.
  • – Impulse:
  • [math]\text{Impulse} = FΔt[/math]
  • Variable Force:
  • – Impulse:
  • [math]\text{Impulse} = \int F \, dt[/math]
  • ⇒ Applications:
  • Car Crash Analysis: Calculating the impulse needed to stop a car.
  • Rocket Thrust: Determining momentum change due to variable thrust.

  • c) Make calculations and estimates involving:

  • I) Temperature and Energy Change Using [math]\Delta E = mc \Delta \theta[/math]

  • ⇒ Formula:
  • [math]\Delta E = mc \Delta \theta[/math]
  • Where:
  • – ΔE: Change in thermal energy (J).
  • – m: Mass of the substance (kg).
  • – c: Specific heat capacity ([math]\frac{J}{\text{kg} \cdot {}^\circ C}[/math]).
  • – [math]\Delta \theta[/math]: Temperature change (°C).
  • ⇒ Explanation:
  • This formula calculates the energy required to heat or cool a substance by a given temperature.
  • ⇒ Example Calculation:
  • Substance: Water ([math]c = 4200 \frac{J}{\text{kg} \cdot {}^\circ C}[/math]).
  • Mass: 2 kg
  • Temperature change:[math]\Delta \theta = 20 \cdot {}^\circ C[/math]
  • [math]\Delta E = mc \Delta \theta \\
    \Delta E = (2)(4200)(20) \\
    \Delta E = 168,000 \; J[/math]
  • Interpretation: Heating 2 kg of water by 20°C requires 168 kJ of energy.

  • II. Gas Laws: Using [math]pV = NkT[/math]

  • Formula:
  • [math]pV = NkT[/math]
  • Where:
  • – p: Pressure (Pa).
  • – V: Volume ([math]m^3[/math]).
  • – N: Number of particles.
  • – k=[math]1.38 \times 10^{-23} \; \frac{J}{K}[/math]: Boltzmann constant.
  • – T: Temperature (K).
  • Linking N to n:
  • [math]N = nN_A[/math]
  • where:
  • – n: Number of moles.
  • – [math]N_A = 6.022 \times 10^{23} \; \frac{\text{particles}}{\text{mol}}[/math]: Avogadro constant.
  • Thus, the ideal gas law can also be expressed as:
  • [math]pV =nRT[/math]
  • – Where [math]R = N_A k = 8.31 \; \frac{J}{\text{mol} \cdot K}[/math] is the gas constant.
  • ⇒ Example Calculation:
  • Given:
  • – p=100,000 Pa
  • – V=0.01[math]m^3[/math]
  • – T=300K,
  • – N is unknown.
  • [math]N = \frac{pV}{kT} \\
    N = \frac{(100{,}000)(0.01)}{(1.38 \times 10^{-23})(300)} \\
    N = 2.41 \times 10^{23} \; \text{particles}[/math]

  • III. Average Kinetic Energy: Using[math]pV = \tfrac{1}{3} N m \overline{c^2}[/math]

  • Formula:
  • [math]pV = \tfrac{1}{3} N m \overline{c^2}[/math]
  • Where:
  • [math]\overline{c^2}[/math]: Mean square speed of gas molecules.
  • – m: Mass of one gas particle.
  • From the kinetic theory:
  • [math]\frac{1}{2} m \overline{c^2} = \frac{3}{2} kT \\
    \overline{c^2} = \frac{3kT}{m}[/math]
  • ⇒ Example Calculation:
  • Given:
  • – T=300 K
  • – [math]m = 4.65 \times 10^{-26} \; \text{kg}[/math](mass of one nitrogen molecule).
  • [math]\overline{c^2} = \frac{3kT}{m} \\
    \overline{c^2} = \frac{3(1.38 \times 10^{-23})(300)}{4.65 \times 10^{-26}} \\
    \overline{c^2} = 2.67 \times 10^{5} \; \frac{m^2}{s^2}[/math]
  • The root mean square (RMS) speed is:
  • [math]v_{\text{rms}} = \sqrt{\overline{c^2}} \\
    v_{\text{rms}} = \sqrt{2.67 \times 10^{5}} \\
    v_{\text{rms}} = 516 \; \frac{m}{s}[/math]

  • d) Demonstrate and apply knowledge and understanding of the following practical activities (HSW4):

  • I) Using an Electrical Method to Find the Specific Thermal Capacity of a Metal Block or Liquid

  • ⇒ Aim:
  • To determine the specific thermal capacity (ccc) of a metal block or a liquid by measuring the energy supplied electrically and the temperature change of the substance
  • Apparatus:
  • – Metal block (with embedded heater) or container with liquid.
  • – Electric heater.
  • – Power supply.
  • – Ammeter and voltmeter.
  • – Stopwatch.
  • – Thermometer or temperature probe.
  • – Insulation material.
  • Figure 8 Specific thermal capacity
  • ⇒ Procedure:
  • Setup:
  • – Place the metal block on insulation material, or place the liquid in a well-insulated container
  • – Insert the heater and thermometer into the block or liquid.
  • Record Initial Measurements:
  • – Measure the initial temperature of the block or liquid.
  • – Record the initial readings on the ammeter and voltmeter to calculate power.
  • Heat the Substance:
  • – Turn on the power supply to the heater.
  • – Start the stopwatch and heat the block or liquid for a fixed time interval (e.g., 5 minutes).
  • Monitor Temperature:
  • – Continuously monitor and record the temperature until the heating period ends
  • – Ensure consistent stirring (for liquids) to distribute heat evenly.
  • Calculate Energy Supplied:
  • – Use:
  • [math]E = IVt[/math]
  • –  Where:
  • – I: Current (A).
  • – V: Voltage (V).
  • – t: Time (s).
  • Determine Specific Heat Capacity:
  • – Use the formula:
  • [math]c = \frac{E}{m \Delta \theta}[/math]
  • Where:
  • – E: Energy supplied (J).
  • – m: Mass of the block or liquid (kg).
  • – [math]\Delta \theta[/math]: Temperature change (°C).
  • ⇒ Sources of Error:
  • Heat loss to the surroundings.
  • Inconsistent temperature measurement (especially in liquids).
  • Inaccurate current or voltage readings.
  • ⇒ Improvements:
  • Use insulation to minimize heat loss.
  • Use a data logger for precise temperature readings.
  • II) Using Appropriate Apparatus to Investigate the Gas Laws and Determine Absolute Zero
  • ⇒ Aim:
  • To investigate the relationships between pressure, volume, and temperature of a gas, and determine the value of absolute zero.
  • Experiment 1: Relationship Between Pressure and Temperature (Constant Volume)
  • ⇒ Apparatus:
  • Sealed gas container (e.g., a gas syringe or a round-bottomed flask).
  • Pressure sensor.
  • Water bath and heater.
  • Thermometer or temperature probe.
  • Data logger.
  • Figure 9 Investigate the Gas Law and determine absolute zero
  • ⇒ Procedure:
  • Setup:
  • – Place the sealed gas container in the water bath.
  • – Connect the pressure sensor to measure gas pressure.
  • Heat the Gas:
  • – Gradually heat the water bath and record the temperature and pressure at regular intervals.
  • – Ensure the gas volume remains constant (sealed container).
  • Plot Results:
  • – Plot P (pressure) against T (temperature in °C).
  • – Extrapolate the graph to find the temperature at which pressure equals zero. This corresponds to absolute zero (−273 °C).
  • Experiment 2: Relationship Between Volume and Temperature (Constant Pressure)
  • o   Apparatus:
  • – Gas syringe or container with movable piston.
  • – Water bath and heater.
  • – Thermometer or temperature probe.
  • ⇒ Procedure:
  • Setup:
  • – Place the gas syringe in the water bath.
  • – Ensure constant pressure by using a movable piston.
  • Heat the Gas:
  • – Gradually heat the water bath and record the gas volume and temperature.
  • Plot Results:
  • – Plot V (volume) against T (temperature in °C).
  • – Extrapolate to find the temperature where volume equals zero.

  • III) Using Apparatus to Investigate the Relationship Between Volume and Pressure

  • ⇒ Aim:
  • To investigate Boyle’s law (​[math]p \propto \frac{1}{V}[/math]) by measuring the relationship between gas pressure and volume at constant temperature.
  • ⇒ Apparatus:
  • Gas syringe or cylinder with piston.
  • Pressure sensor or gauge
  • Data logger.
  • ⇒ Procedure:
  • Setup:
  • – Connect the gas syringe to the pressure sensor.
  • – Ensure the temperature remains constant throughout the experiment (e.g., by performing the experiment quickly or using a constant-temperature environment).
  • Adjust Volume:
  • – Gradually decrease the volume of gas in the syringe by pushing the piston.
  • – Record the corresponding pressure for each volume.
  • Plot Results:
  • – Plot P (pressure) against [math] \frac{1}{V}[/math]​
  • – The graph should produce a straight line, confirming Boyle’s law:
  • [math]p \propto \frac{1}{V}[/math]
  • Figure 10 Investigate the relationship between pressure and volume
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