24 20 The photograph shows a vase made of uranium glass. Uranium glass is radioactive. Uranium glass usually contains a maximum of 2% uranium. Uranium glass made in the early part of the 20th century can contain up to 25% uranium. A student carried out an investigation to determine the percentage of uranium in the glass. The student measured the count rate by placing a Geiger Muller (GM) tube against the vase at a single position. This value was used to calculate the decay rate for the whole vase. (a) (i) Show that the decay constant for uranium is about 5 × 10−18 s−1 half-life of uranium = 1.41 × 1017 s (2) .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. Turn over 25 (ii) Calculate the percentage of uranium, by mass, in the glass. area of GM tube window = 6.36 × 10−5 m2 surface area of vase = 0.0177 m2 background count rate = 525 counts in 10 minutes count rate when GM tube next to vase = 3623 counts in 5 minutes mass of vase = 149 g mass of uranium atom = 238 u (6) ................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................... Percentage of uranium = .................................................................... (iii) The uranium decays by emitting alpha particles. Criticise the method used to determine the percentage of uranium in the vase. (2) .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. 26 (b) A uranium nucleus decays to thorium by emission of an alpha particle. It can be assumed that all the energy of the decay is transferred to kinetic energy of the alpha particle. Calculate the speed of the emitted alpha particle. mass of uranium nucleus = 238.0003 u mass of thorium nucleus = 233.9942 u mass of alpha particle = 4.0015 u (5) ................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................... Speed of alpha particle = .................................................................... (Total for Question 20 = 15 marks) TOTAL FOR PAPER = 90 MARKS Every effort has been made to contact copyright holders to obtain their permission for the use of copyright material. Pearson Education Ltd. will, if notified, be happy to rectify any errors or omissions and include any such rectifications in future editions. 27 List of data, formulae and relationships Acceleration of free fall g = 9.81 m s−2 (close to Earth’s surface) Boltzmann constant k = 1.38 × 10−23 J K−1 Coulomb law constant k = 1 4πε0 = 8.99 × 109 N m2 C−2 Electron charge e = −1.60 × 10−19 C Electron mass me = 9.11 × 10−31 kg Electronvolt 1 eV = 1.60 × 10−19 J Gravitational constant G = 6.67 × 10−11 N m2 kg−2 Gravitational field strength g = 9.81 N kg−1 (close to Earth’s surface) Permittivity of free space ε0 = 8.85 × 10−12 F m−1 Planck constant h = 6.63 × 10−34 J s Proton mass mp = 1.67 × 10−27 kg Speed of light in a vacuum c = 3.00 × 108 m s−1 Stefan-Boltzmann constant σ = 5.67 × 10−8 W m−2 K−4 Unified atomic mass unit u = 1.66 × 10−27 kg Mechanics Kinematic equations of motion s =  (u + v)t 2 v = u + at s = ut + 1 2 at2 v2 = u2 + 2as Forces ∑F = ma g =  F m W = mg moment of force = Fx Momentum p = mv Work, energy and power ΔW = FΔs Ek = 1 2 mv2 ΔEgrav = mgΔh P = E t P =  W t efficiency = useful energy output total energy input efficiency =  useful power output total power input 28 Electric circuits Potential difference V =  W Q Resistance R = V I Electrical power and energy P = VI P = I 2R P = V 2 R W = VIt Resistivity R =  ρl A Current I = ΔQ Δt I = nqvA Materials Density ρ =  m V Stokes’ law F = 6πηrv Hooke’s law ΔF = kΔ x Young modulus Stress σ =  F A Strain ε = Δ x x E =  σ ε Elastic strain energy ΔEel = 1 2 FΔ x Waves and particle nature of light Wave speed v = f λ Speed of a transverse wave on a string v = T μ Intensity of radiation I =  P A Power of a lens P =  1 f P = P1 + P2 + P3 + … Thin lens equation 1 u  +  1 v  =  1 f Magnification for a lens m =  image height object height =  v u Diffraction grating nλ = d sin θ Refractive index n1 sin θ1 = n2 sin θ2 n = c v Critical angle sin C = 1 n Photon model E = h f Einstein’s photoelectric equation hf = ϕ + 1 2 mv2 max de Broglie wavelength λ = h p 29 Further mechanics Impulse FΔt = Δp Kinetic energy of a non-relativistic particle Ek =  p2 2m Motion in a circle v = ωr T = 2π ω F = ma =  mv2 r a = v2 r a = rω2 F = mrω2 Fields Coulomb’s law F =  Q1Q2 4πε0r2 Electric field strength E =  F Q E = Q 4πε0r2 E =  V d Electric potential V = Q 4πε0r Capacitance C = Q V Energy stored in a capacitor W = 1 2 QV W = 1 2 CV 2 W = 1 2 Q 2 C Capacitor discharge Q = Q0e−t/RC I = I0e−t/RC V = V0e−t/RC ln Q = ln Q0 −  t RC ln I = ln I0 −  t RC ln V = ln V0 −  t RC In a magnetic field F = BIl sin θ F = Bqv sin θ Faraday’s and Lenz’s laws E =  −d(Nϕ) dt Root-mean-square values Vr ms =  V0 √2 Ir ms =  I0 √2 30 Nuclear and particle physics In a magnetic field r =  p BQ Thermodynamics Heating ΔE = mcΔθ ΔE = LΔm Molecular kinetic theory 1 2 mác2ñ = 3 2 kT pV = 1 3 Nmác2ñ Ideal gas equation pV = NkT Stefan-Boltzmann law L = σAT 4 L = 4πr2σT 4 Wien’s law λmaxT = 2.898 × 10−3 m K Space Intensity I =  L 4πd 2 Redshift of electromagnetic radiation z = Δλ λ  ≈ Δf f  ≈ v c Cosmological expansion v = H0d Nuclear radiation Mass-energy ΔE = c2Δm Radioactive decay A = λN dN dt  = −λN λ =  ln 2 t½ N = N0 e−λt A = A0 e−λt Gravitational fields Gravitational force F = Gm1m2 r 2 Gravitational field strength g = Gm r 2 Gravitational potential Vgrav = −Gm r Oscillations Simple harmonic motion F = −k x a = −ω2x x = A cos ωt v = −Aω sin ωt a = ‒Aω2 cos ωt T = 1 f  =  2π ω ω = 2π f Simple harmonic oscillator T = 2π m k T = 2π l g 31 BLANK PAGE 32 BLANK PAGE