Problem 1

Gold nanoparticles modified with two kinds of single-stranded nucleic acid fragments a and b aggregate in the presence of the target nucleic acid a'b' from the coronavirus, causing a colour change of the solution from red to blue within minutes. Gold nanoparticles are composed of closely packed gold atoms with the density of solid gold,

\rho = 19.3

g cm

^{-3}

. (a) Calculate the number of gold atoms

N

in a spherical gold nanoparticle of diameter 30.0 nm (

M(\ce{Au}) = 197.0

g mol

^{-1}

N_A = 6.022\times 10^{23}

mol

^{-1}

). (b) 5.2 mg of

\ce{HAuCl4*3H2O}

(

M = 394

g mol

^{-1}

) was completely converted into uniform spherical gold nanoparticles of diameter 30.0 nm in 100.0 mL of solution; the absorbance of the red solution measured at 530 nm in a 1 cm cuvette was 0.800. Calculate the molar extinction coefficient (per mole of nanoparticles) at 530 nm.

Step 1 of 3: Atoms per nanoparticle

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N = \dfrac{\rho\,\frac{4}{3}\pi r^3}{M(\ce{Au})}\,N_A = \dfrac{19.3\times 1.414\times 10^{-20}}{197.0}\times 6.022\times 10^{23} = 8.3\times 10^{5}\ \text{atoms}

Analysis

The sphere of radius 15.0 nm has volume $V = \tfrac{4}{3}\pi(15.0\times 10^{-7}\text{ cm})^3 = 1.414\times 10^{-20}$ cm $^3$ , mass $2.73\times 10^{-19}$ g, i.e. $1.39\times 10^{-21}$ mol of Au or $8.3\times 10^{5}$ atoms per nanoparticle.

Common pitfall. Convert nm to cm before applying the density in g cm

^{-3}