Quantum theory — Question 184

Original QCE Vault practice · 7 marks

Q184 · Practice questionComplex unfamiliar7 marks

QUESTION 184 (7 marks)

A clean metal has work function

2.60\times10^{-19}\,\mathrm J

. It is illuminated at frequency

5.50\times10^{14}\,\mathrm{Hz}

. Use

h=6.626\times10^{-34}\,\mathrm{J\,s}

m_e=9.1093835\times10^{-31}\,\mathrm{kg}

and

c=3.00\times10^8\,\mathrm{m\,s^{-1}}

. Assume one-photon emission.

Calculate the maximum kinetic energy and maximum speed of emitted electrons.

[3 marks]

Predict the effect of quadrupling the intensity at unchanged frequency on maximum kinetic energy and electron emission rate. Assume constant efficiency and no saturation.

[2 marks]

Determine the longest wavelength that can cause emission at threshold.

[2 marks]

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WORKED SOLUTION

Practice marking scheme

7 marks

ANSWER

(a)

E_{k,\max}=1.04\times10^{-19}\,\mathrm J

;

v_{\max}=4.79\times10^5\,\mathrm{m\,s^{-1}}

. (b) Maximum kinetic energy is unchanged; emission rate increases by a factor of four. (c)

765\,\mathrm{nm}

(approximately).

Worked solution

(a)

E_{k,\max}=hf-W=3.6443\times10^{-19}-2.6000\times10^{-19}=1.0443\times10^{-19}\,\mathrm J

v_{\max}=\sqrt{2E_{k,\max}/m_e}=4.78832\times10^5\,\mathrm{m\,s^{-1}}

(b) Each photon retains energy

hf

, so

E_{k,\max}

is unchanged. Four times the intensity at the same frequency means four times the photon arrival rate; under the stated conditions the emission rate also quadruples.

hc/\lambda_0=W

, so

\lambda_0=hc/W=7.64538\times10^{-7}\,\mathrm m=764.538\,\mathrm{nm}

. At the threshold the ideal maximum kinetic energy is zero.

Equivalent physically justified methods and consistent equivalent units accepted.

Displayed decimals are model answers; accept appropriate significant figures and consistent rounding from stated constants.

Part a: Calculate incident photon energy.

[1 mark]

Part a: Subtract the work function for maximum kinetic energy.

[1 mark]

Part a: Convert maximum kinetic energy to maximum speed.

[1 mark]

Part b: Link unchanged photon energy to unchanged maximum kinetic energy.

[1 mark]

Part b: Link increased photon rate to the specified emission-rate increase.

[1 mark]

Part c: Relate the threshold photon energy to the work function.

[1 mark]

Part c: Calculate the threshold wavelength.

[1 mark]

Practice question aligned to the current QCAA syllabus; review the worked solution and marking criteria.

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Quantum theory — Question 184

QUESTION 184 (7 marks)

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