Show how ½m⟨c²⟩ = (3/2)kT is obtained by combining pV = ⅓Nm⟨c²⟩ with pV = NkT.

Equating the two expressions gives ⅓Nm⟨c²⟩ = NkT, and cancelling N leaves ⅓m⟨c²⟩ = kT. Multiplying both sides by 3/2 gives ½m⟨c²⟩ = (3/2)kT.

Show that the total translational kinetic energy of a mole of monatomic gas is (3/2)RT.

Each molecule has a mean translational kinetic energy (3/2)kT, and a mole contains Nₐ molecules, so the total is (3/2)Nₐ kT. Since k = R/Nₐ, that product is (3/2)RT.

What does the equation ½m⟨c²⟩ = (3/2)kT tell you about the physical meaning of temperature?

The thermodynamic temperature of an ideal gas is directly proportional to the mean translational kinetic energy of its molecules, so temperature is a measure of that mean energy.

Eduqas A-Level Physics A420QS

Basic Physics

Topic overview SI base units and prefixes Checking equations for homogeneity Scalars and vectors Adding and resolving vectors Density Moments and the principle of moments Centre of gravity and equilibrium

Kinematics

Topic overview Displacement, velocity and acceleration Displacement-time graphs Velocity-time graphs and non-uniform acceleration The equations of uniformly accelerated motion Free fall, air resistance and terminal velocity Projectile motion

Dynamics

Topic overview Force and Newton's third law Free body diagrams and resultant force Newton's second law and F = ma Linear momentum Force as the rate of change of momentum Conservation of momentum in collisions

Energy Concepts

Topic overview Work done by a constant force Work done when the force is not along the motion Gravitational and elastic potential energy Kinetic energy and the work-energy relationship Conservation of energy Power, dissipative forces and efficiency

Circular Motion

Topic overview Period and frequency of rotation The radian as a measure of angle Angular velocity Centripetal acceleration and centripetal force Applying the circular motion equations

Vibrations

Topic overview Defining simple harmonic motion Frequency, period, amplitude and phase The displacement equation for simple harmonic motion Velocity in simple harmonic motion Displacement and velocity graphs The mass-spring oscillator The simple pendulum Energy in simple harmonic motion Free oscillation and damping Forced oscillation and resonance

Kinetic Theory

Topic overview The ideal gas equation of state Assumptions of the kinetic theory of gases The Avogadro constant, the mole and molar mass The kinetic theory pressure equation Molecular kinetic energy and temperature

Thermal Physics

Topic overview Internal energy Absolute zero and temperature Internal energy of an ideal monatomic gas Heat as energy in transit Thermal equilibrium Work done by a gas The first law of thermodynamics Specific heat capacity

Conduction of Electricity

Topic overview Electric charge and the coulomb Electric current and the ampere Conductors, insulators and charge carriers Drift of free electrons in a metal Deriving and using I = nAve

Resistance

Topic overview Potential difference and the volt Resistance, Ohm's law and the ohm I–V characteristics of a metal wire and a filament lamp Electrical power and energy transfer The microscopic origin of resistance Resistivity and the equation R = ρl/A Resistance and temperature Superconductivity and the transition temperature High temperature superconductors and their uses

D.C. Circuits

Topic overview Current at a junction and conservation of charge Potential difference round a circuit and conservation of energy Resistors in series Resistors in parallel The potential divider Sensing circuits with LDRs and thermistors emf and internal resistance

Capacitance

Topic overview The parallel plate capacitor and stored charge Capacitance and the farad The capacitance of a parallel plate capacitor Dielectrics The uniform field between the plates and E = V/d Energy stored in a capacitor Capacitors in series and in parallel Charging and discharging through a resistor

Solids Under Stress

Topic overview Hooke's law and the spring constant Stress, strain and the Young modulus Work done in deforming a solid Crystalline, amorphous and polymeric solids Metals under stress: dislocations and ductile fracture Brittle solids and rubber under stress

Electrostatic and Gravitational Fields of Force

Topic overview Electric field strength and Coulomb's law Gravitational field strength and Newton's law of gravitation Field lines and equipotential surfaces Electric potential and potential energy Gravitational potential and potential energy Resultant fields and potentials from several sources Comparing electric and gravitational fields

Using Radiation to Investigate Stars

Topic overview Stellar spectra and the origin of absorption lines Black bodies and the black body spectrum Wien's displacement law and stellar temperature Stefan's law, luminosity and stellar radius The inverse square law, stellar distance and multiwavelength astronomy

Orbits and the Wider Universe

Topic overview Kepler's laws of planetary motion Newton's law of gravitation applied to planets and satellites Deriving Kepler's third law for a circular orbit Using orbital data to find the mass of the central object Galactic rotation curves and dark matter The centre of mass of a two-body system The Doppler shift of spectral lines and radial velocity Masses in a double system from radial velocity data Hubble's law, the age of the universe and its fate

The Nature of Waves

Topic overview Progressive waves and energy transfer Transverse and longitudinal waves Polarisation Phase and antiphase Wave quantities and the wave equation Wave graphs and wavefronts

Wave Properties

Topic overview Diffraction at slits and obstacles Wavelength, slit width and the spread of the beam The principle of superposition Coherence and the conditions for interference Path difference and the interference conditions Young's double slit experiment The historical importance of Young's experiment The diffraction grating Comparing the grating with the double slit Stationary waves

Refraction of Light

Topic overview Refractive index and the speed of light Snell's law Snell's law and the wave model Total internal reflection and the critical angle Multimode optical fibres Multimode dispersion and monomode fibres

Photons

Topic overview Photons and the quantum of light Demonstrating the photoelectric effect The vacuum photocell and stopping potential Einstein's equation and the energy graph The electromagnetic spectrum and photon energies Line emission and line absorption spectra Atomic energy levels Ionisation energy from an energy level diagram Electron diffraction and the de Broglie relationship

Lasers

Topic overview Stimulated emission and coherence Population inversion Two, three and four level systems Pumping and the structure of a laser The semiconductor laser and its uses

Nuclear Decay

Topic overview Spontaneous decay and the nature of α, β and γ radiation Nuclear transformation equations and nuclide notation Distinguishing α, β and γ by ionisation, penetration and range Background radiation and gamma intensity with distance Activity, the becquerel and the decay constant The exponential decay law and half-life

Particles and Nuclear Structure

Topic overview The Rutherford alpha particle scattering experiment Coulomb repulsion in the plum pudding and Rutherford models Quarks and leptons: the first generation Antiparticles, annihilation and their symbols Hadrons: baryons, antibaryons and mesons Quark composition of the proton, neutron and pions The four fundamental interactions Conservation laws in particle reactions

Nuclear Energy

Topic overview Mass–energy equivalence and E = Δmc² Mass defect, binding energy and the unified atomic mass unit Calculating binding energy per nucleon The binding energy per nucleon curve Fission, fusion and the nuclear power debate

Magnetic Fields

Topic overview The motor effect and Fleming's left-hand rule F = BIl sinθ and measuring magnetic flux density The force on a moving charge, F = Bqv sinθ The Hall effect and the Hall probe Magnetic fields due to a straight wire and a solenoid Forces between current-carrying conductors Charged particle beams and accelerators

Electromagnetic Induction

Topic overview Magnetic flux, flux linkage and the weber Faraday's law and Lenz's law Applying the laws: emf from rate of change of flux linkage Emf induced in a linear conductor moving through a field The rotating coil and the a.c. generator

Option A: Alternating CurrentsOption A

Topic overview Electromagnetic induction in a rotating coil Frequency, period, peak and rms values Mean power dissipated in a resistor Using an oscilloscope to measure voltage and frequency Inductors in a.c. circuits and inductive reactance Capacitors in a.c. circuits and capacitive reactance Phasors, phase angle and impedance Resonance in the series RCL circuit and the Q factor

Option B: Medical PhysicsOption B

Topic overview The nature and properties of X-rays X-ray production and controlling the beam Attenuation of X-rays and half-value thickness X-ray imaging, fluoroscopy and radiography Computed tomography scanning Ultrasound generation and acoustic impedance Ultrasound scanning and the Doppler technique Magnetic resonance and MRI Radiation effects on living matter and dosimetry Radionuclide tracers, the gamma camera and PET

Option C: The Physics of SportsOption C

Topic overview Centre of gravity, stability and toppling Moments in muscle systems and other sporting contexts Impulse and Newton's second law in sport The coefficient of restitution Moment of inertia Angular acceleration, torque and angular momentum Conservation of angular momentum and rotational energy Projectiles, Bernoulli's equation and drag

Option D: Energy and the EnvironmentOption D

Topic overview Thermal equilibrium and the Earth's energy balance Wien's law, the Stefan-Boltzmann law and solar radiation Melting ice, density and rising sea levels Photovoltaic cells and wind power Water power, pumped storage, fission and fusion Fuel cells, thermal conduction and insulation

Mathematical Requirements (Appendix C)

Topic overview Units, prefixes and standard form Significant figures and arithmetic means Uncertainty and how uncertainties combine Ratios, proportionality and order of magnitude estimates Rearranging and solving physics equations Logarithms, exponentials and decay Resolving vectors, Pythagoras and trigonometry Radians, small angles, areas and volumes Straight-line graphs, gradients and intercepts Tangents, areas under curves and logarithmic plots

How Science Works

Topic overview Hypotheses, predictions and scientific questions Evidence, experimental method and evaluating data Models and theories, and their limits Correlation and causation Peer review, the scientific community, ethics and society

Kinetic Theory

Molecular kinetic energy and temperature

3 things to know
Component 1

Overview

Recall questions on ½m⟨c²⟩ = (3/2)kT and what it says about the meaning of temperature, its derivation from the kinetic theory equation and the gas law, worked calculations of mean kinetic energy and root mean square speed, and comparing two gases at one temperature.

What you need to know

Answer each one before you reveal it.

Your recallNot started

Key term

State the equation linking the mean translational kinetic energy of a gas molecule to the thermodynamic temperature, defining every symbol and its unit.

Key term

Why does the relationship ½m⟨c²⟩ = (3/2)kT refer to translational kinetic energy only?

Key term

State the total translational kinetic energy of one mole of a monatomic ideal gas.

Questions from this subtopic

Each one with the model answer underneath.

1
State the equation linking the mean translational kinetic energy of a gas molecule to the thermodynamic temperature, defining every symbol and its unit.
Model answer
The equation is ½m⟨c²⟩ = (3/2)kT, where m is the mass of one molecule in kg, ⟨c²⟩ is the mean square speed in m² s⁻², k is the Boltzmann constant in J K⁻¹ and T is the thermodynamic temperature in K.
2
Show how ½m⟨c²⟩ = (3/2)kT is obtained by combining pV = ⅓Nm⟨c²⟩ with pV = NkT.
Model answer
Equating the two expressions gives ⅓Nm⟨c²⟩ = NkT, and cancelling N leaves ⅓m⟨c²⟩ = kT. Multiplying both sides by 3/2 gives ½m⟨c²⟩ = (3/2)kT.
3
State the total translational kinetic energy of one mole of a monatomic ideal gas.
Model answer
The total translational kinetic energy of one mole is (3/2)RT, where R is the molar gas constant and T is the thermodynamic temperature in K.
4
Show that the total translational kinetic energy of a mole of monatomic gas is (3/2)RT.
Model answer
Each molecule has a mean translational kinetic energy (3/2)kT, and a mole contains Nₐ molecules, so the total is (3/2)Nₐ kT. Since k = R/Nₐ, that product is (3/2)RT.
5
What does the equation ½m⟨c²⟩ = (3/2)kT tell you about the physical meaning of temperature?
Model answer
The thermodynamic temperature of an ideal gas is directly proportional to the mean translational kinetic energy of its molecules, so temperature is a measure of that mean energy.
6
Calculate the mean translational kinetic energy of a gas molecule at 300 K.
Model answer
The energy is 1.5 × 1.38 × 10⁻²³ × 300 = 6.21 × 10⁻²¹ J. Dividing 6.21 × 10⁻²¹ J by 1.5 × 1.38 × 10⁻²³ J K⁻¹ returns 300 K.

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Molecular kinetic energy and temperature

Overview

What you need to know

State the equation linking the mean translational kinetic energy of a gas molecule to the thermodynamic temperature, defining every symbol and its unit.

Why does the relationship ½m⟨c²⟩ = (3/2)kT refer to translational kinetic energy only?

State the total translational kinetic energy of one mole of a monatomic ideal gas.

Questions from this subtopic

State the equation linking the mean translational kinetic energy of a gas molecule to the thermodynamic temperature, defining every symbol and its unit.

Show how ½m⟨c²⟩ = (3/2)kT is obtained by combining pV = ⅓Nm⟨c²⟩ with pV = NkT.

State the total translational kinetic energy of one mole of a monatomic ideal gas.

Show that the total translational kinetic energy of a mole of monatomic gas is (3/2)RT.

What does the equation ½m⟨c²⟩ = (3/2)kT tell you about the physical meaning of temperature?

Calculate the mean translational kinetic energy of a gas molecule at 300 K.

Other ways to work this

What is inside Levo

The 3-step method, run for you

Cram a whole paper in one sitting

Blank Blitz makes facts automatic

Build It gets the exam shape right

Test It catches misconceptions

Brain Breaks between sessions

Every question in Kinetic Theory, on a schedule.