IChO 2013 · Theory problem 5

Simple estimates of graphene properties

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Graphene is a two-dimensional, one atom thick carbon material (Fig. 1 a). Many layers of graphene stack together to form graphite (Fig. 1 b).

Figure 1

Fig. 1. (a) The structure of graphene. Spheres are carbon atoms. They are arranged in hexagons. The area of one carbon hexagon is \(5.16\cdot10^{-20}\ \mathrm{m^2}\). (b) Crystal lattice of graphite. Three graphene layers are shown.

Such atomic structure was long considered to be unstable. However, in 2004 Andrey Geim and Konstantin Novoselov have reported production of the first samples of this unusual material. This groundbreaking invention was awarded by Nobel prize in 2010.

Experimental studies of graphene are still restricted. Production of massive portions of the new substance still is a challenging synthetic problem. Many properties of graphene were estimated. Usually, there is not enough information for rigorous calculations, so we have to make assumptions and neglect unimportant factors. In this problem, you will estimate the adsorption properties of graphene.

5.1 Estimate the specific surface of graphene open for adsorption in units \(\mathrm{m^2\,g^{-1}}\). Consider that graphene plane is separated from any other solid or liquid substance.

The single layer of nitrogen molecules adsorbed on the outer surface of graphite is shown in Fig. 2. Assume that the same arrangement of nitrogen molecules is formed on a graphene surface.

Figure 2

Fig. 2. Nitrogen molecules \(\ce{N2}\) (grey circles) on the outer surface of graphite

5.2 How many grams of nitrogen can be adsorbed on 1 gram of graphene assuming that the graphene layer is placed onto the surface of a solid support? Estimate the volume occupied by these nitrogen molecules after the complete desorption from 1 g of graphene (pressure 1 bar, temperature 298 K).

Let us consider adsorption as a common chemical equilibrium

Figure 3

(\(\ce{A_{gas}}\) are molecules \(\ce{A}\) in the gaseous state, \(\ce{A_{ads}}\) are the same molecules on the surface)

with the equilibrium constant \(K\):

\[ K=\frac{n_{\ce{A_{ads}}}\;(\mathrm{mol\,m^{-2}})} {p_{\ce{A_{gas}}}\;(\mathrm{bar})} \]

(Such an assumption holds if a small number of molecules is adsorbed on the surface.)

Adsorption properties of graphene can be estimated from the data for adsorption on a regular three-dimensional graphite. The enthalpy of adsorption (\(\Delta H^\circ\) of reaction (1)) of any molecule \(\ce{A}\) on graphene is on average by 10 % less negative compared to that on graphite. On graphite, the adsorbed molecule is bound more strongly due to the interaction with the lower graphene layers in the lattice (Fig. 1 b) and hence the enthalpy of adsorption is more negative. The standard entropies of adsorption on graphene and graphite are assumed to be the same.

5.3 How many moles, \(n\), of \(\ce{CCl4}\) are adsorbed on 1 g of graphene at \(p(\ce{CCl4})=10^{-4}\ \mathrm{bar}\) if \(2.0\cdot10^{-7}\ \mathrm{mol}\) of \(\ce{CCl4}\) are adsorbed on \(1\ \mathrm{m^2}\) of graphite at \(p(\ce{CCl4})=6.6\cdot10^{-5}\ \mathrm{bar}\)? Assume that graphene is placed onto the surface of a solid support and the interaction of \(\ce{CCl4}\) with the support does not change the enthalpy of adsorption of \(\ce{CCl4}\) on graphene. The temperature in both cases is 293 K. \(\Delta H^\circ\) of adsorption of \(\ce{CCl4}\) on graphite is \(-35.1\ \mathrm{kJ/mol}\).

The graphene films are expected to be sensitive gas detectors. If \(10^9\) particles of a gas are adsorbed on \(1\ \mathrm{cm^2}\) of a graphene surface this is enough to measure an electrical resistivity change of the graphene layer and to detect the presence of a gas in the environment.

5.4 Determine the minimal content of ethane, \(\ce{C2H6}\), in the air (in mol. %) at atmospheric pressure (\(T=293\ \mathrm{K}\)) at which a graphene sensor will detect this gas. The known data for the adsorption of alkanes on graphite are shown in Fig 3. Assume that air doesn't affect the adsorption properties of ethane.

Figure 4
Figure 5

Fig. 3
Thermodynamic properties for adsorption of alkanes on a graphite surface.
(a) \(\ln K\ [\mathrm{mol\,m^{-2}\,bar^{-1}}]\) as a function of \(\ln M\) (\(M\) - molar mass of alkane in \(\mathrm{g\,mol^{-1}}\));
(b) \(\Delta H^\circ\) of adsorption as a function of \(\ln M\).
Linear dependences are assumed in both cases

Official English text of the problem.

Answers

  • 5.1: \(\boxed{2.59\times10^3\ \mathrm{m^2\,g^{-1}}}\), counting both exposed faces.
  • 5.2: Nitrogen, \(\ce{N2}\), SMILES N#N: \(\boxed{0.389\ \mathrm{g}}\) adsorbed per gram of supported graphene; after desorption, \(\boxed{0.344\ \mathrm{L}}\) at 1 bar and 298 K.
  • 5.3: Carbon tetrachloride, \(\ce{CCl4}\), SMILES ClC(Cl)(Cl)Cl: \(\boxed{9.3\times10^{-5}\ \mathrm{mol}}\) adsorbed.
  • 5.4: Ethane, \(\ce{C2H6}\), SMILES CC: approximately \(\boxed{6.6\times10^{-4}\ \mathrm{mol.\%}}\), or about \(7\) ppm by mole, subject to graphical reading precision.

Solution

Use \(N_{\mathrm A}=6.022\times10^{23}\ \mathrm{mol^{-1}}\), \(R=8.314\ \mathrm{J\,mol^{-1}\,K^{-1}}=0.08314\ \mathrm{L\,bar\,mol^{-1}\,K^{-1}}\), and the approximate molar masses \(M(\ce{C})=12.0\), \(M(\ce{N2})=28.0\), and \(M(\ce{C2H6})=30.0\ \mathrm{g\,mol^{-1}}\).

5.1

Each carbon hexagon has six vertices, and each carbon atom belongs to three adjacent hexagons. Thus, neglecting sheet edges, one hexagon accounts for

\[ \frac{6}{3}=2\text{ carbon atoms}. \]

Its corresponding mass is

\[ m_{\mathrm{hex}}=\frac{2M(\ce{C})}{N_{\mathrm A}}. \]

Both faces of the isolated graphene sheet are accessible, so the adsorption area corresponding to one hexagon is \(2S\). Therefore,

\[ a_{\mathrm{both}} =\frac{2S}{2M(\ce{C})/N_{\mathrm A}} =\frac{SN_{\mathrm A}}{M(\ce{C})}. \]

Substitution gives

\[ a_{\mathrm{both}} =\frac{(5.16\times10^{-20}\ \mathrm{m^2}) (6.022\times10^{23}\ \mathrm{mol^{-1}})} {12.0\ \mathrm{g\,mol^{-1}}} =\boxed{2.59\times10^3\ \mathrm{m^2\,g^{-1}}}. \]

For subsequent parts involving a solid support, only one face is accessible:

\[ a_{\mathrm{one}}=\frac{a_{\mathrm{both}}}{2} =1.295\times10^3\ \mathrm{m^2\,g^{-1}}. \]

5.2

The repeating arrangement in Fig. 2 places one nitrogen molecule over every three carbon hexagons. Since each hexagon accounts for two carbon atoms, this means

\[ 1\text{ adsorbed }\ce{N2}\text{ molecule per }6\text{ carbon atoms}. \]

Only the upper face of supported graphene is accessible. Consequently, for \(1\ \mathrm{g}\) of graphene,

\[ n(\ce{N2}) =\frac{1}{6}\frac{1.00\ \mathrm{g}} {12.0\ \mathrm{g\,mol^{-1}}} =1.389\times10^{-2}\ \mathrm{mol}. \]

The adsorbed nitrogen mass is

\[ m(\ce{N2}) =n(\ce{N2})M(\ce{N2}) =(1.389\times10^{-2})(28.0) =\boxed{0.389\ \mathrm{g}}. \]

After complete desorption, treating the nitrogen as an ideal gas,

\[ V=\frac{nRT}{p} =\frac{(1.389\times10^{-2}\ \mathrm{mol}) (0.08314\ \mathrm{L\,bar\,mol^{-1}\,K^{-1}}) (298\ \mathrm{K})} {1\ \mathrm{bar}} =\boxed{0.344\ \mathrm{L}}. \]

5.3

For the adsorption equilibrium

\[ \ce{CCl4(g) <=> CCl4(ads)}, \]

let \(\Gamma\) denote the adsorbed amount per unit area. The given equilibrium relation is

\[ \Gamma=Kp. \]

The graphite data give

\[ K_{\mathrm{graphite}} =\frac{2.0\times10^{-7}\ \mathrm{mol\,m^{-2}}} {6.6\times10^{-5}\ \mathrm{bar}} =3.03\times10^{-3}\ \mathrm{mol\,m^{-2}\,bar^{-1}}. \]

The adsorption enthalpy on graphene is 10% less negative:

\[ \Delta H^\circ_{\mathrm{graphene}} =0.90(-35.1) =-31.59\ \mathrm{kJ\,mol^{-1}}. \]

Because the adsorption entropies are equal,

\[ \Delta G^\circ_{\mathrm{graphene}} -\Delta G^\circ_{\mathrm{graphite}} =\Delta H^\circ_{\mathrm{graphene}} -\Delta H^\circ_{\mathrm{graphite}} =+3.51\ \mathrm{kJ\,mol^{-1}}. \]

Using consistent standard states for both surfaces,

\[ \ln\frac{K_{\mathrm{graphene}}}{K_{\mathrm{graphite}}} =-\frac{3510}{8.314\times293} =-1.441. \]

Thus,

\[ K_{\mathrm{graphene}} =(3.03\times10^{-3})e^{-1.441} =7.17\times10^{-4}\ \mathrm{mol\,m^{-2}\,bar^{-1}}. \]

The weaker adsorption on graphene correctly gives the smaller equilibrium constant. At the specified pressure,

\[ \Gamma_{\mathrm{graphene}} =(7.17\times10^{-4})(10^{-4}) =7.17\times10^{-8}\ \mathrm{mol\,m^{-2}}. \]

One gram of supported graphene exposes \(1.295\times10^3\ \mathrm{m^2}\), so

\[ n=\Gamma_{\mathrm{graphene}}A =(7.17\times10^{-8})(1.295\times10^3) =\boxed{9.3\times10^{-5}\ \mathrm{mol}}. \]

5.4

For ethane,

\[ \ln M=\ln(30.0)=3.40. \]

Reading Fig. 3 at this abscissa gives approximately

\[ \ln K_{\mathrm{graphite}}\approx-12.0, \qquad \Delta H^\circ_{\mathrm{graphite}} \approx-22\ \mathrm{kJ\,mol^{-1}}. \]

Here \(K\) is expressed in the units specified by the graph. Applying the same enthalpy correction as in 5.3,

\[ \Delta H^\circ_{\mathrm{graphene}} \approx0.90(-22) =-19.8\ \mathrm{kJ\,mol^{-1}}, \]

and hence

\[ \ln\frac{K_{\mathrm{graphene}}}{K_{\mathrm{graphite}}} =-\frac{2200}{8.314\times293} =-0.903. \]

Therefore,

\[ \ln K_{\mathrm{graphene}}\approx-12.903, \]

\[ K_{\mathrm{graphene}} \approx2.49\times10^{-6}\ \mathrm{mol\,m^{-2}\,bar^{-1}}. \]

The detection threshold is \(10^9\) molecules per square centimetre. Since \(1\ \mathrm{cm^2}=10^{-4}\ \mathrm{m^2}\),

\[ \Gamma_{\min} =\frac{10^9}{(10^{-4}\ \mathrm{m^2})N_{\mathrm A}} =1.66\times10^{-11}\ \mathrm{mol\,m^{-2}}. \]

Using \(\Gamma=Kp\), the minimum ethane partial pressure is

\[ p_{\min} =\frac{1.66\times10^{-11}} {2.49\times10^{-6}} \approx6.67\times10^{-6}\ \mathrm{bar}. \]

For an ideal gas mixture, mole fraction equals partial pressure divided by total pressure. Taking atmospheric pressure as \(1.01325\ \mathrm{bar}\),

\[ x_{\min}=\frac{p_{\min}}{P} \approx6.58\times10^{-6}. \]

Thus the minimum molar percentage is

\[ \boxed{100x_{\min}\approx6.6\times10^{-4}\ \mathrm{mol.\%}}, \]

equivalent to approximately \(7\) ppm by mole.

The solution was checked against the official solution.