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 This article uses material from the Wikipedia article The Investigation of Nanoparticle from Sol-Gel Method, that was deleted or is being discussed for deletion, which is released under the Creative Commons Attribution-ShareAlike 3.0 Unported License.
This article contains about the derivative of radius particle (colloid, sol, and gel)formulation which also has ever been written by Einstein. It is just continuing from his writing. Sol-gel process is the colloidal suspension from inorganic materials which is included in metal alkoxides to the solid forms such as ceramic and glass. This process can establish a nanoparticle because be affected by Brown motion[1]. Brown motion can be occured to the sol-gel formation due to changing in viscosity which is bigger than the interaction of liquid medium. Because of the value of viscosity is more increasing, the formation of particle in colloid and sol suspension is almost approaching the gas particle. By using its statement, Helmholtz free energy function F and entropy S can be used as the investigation of nanoparticle as follows. ${F= U-TS (1)}$ ${S=\frac{U}{T}+k\ln(B) (2)}$ By substitute equation (2) into equation (1) and k = nR/N with the state of one mole, then the energy function into equation (3). ${F= -(\frac{RT}{N})\ln(B) (3)}$ Note that, B is the partition function of the state of energy Ei per exponent sum of RT (J). Through this sense, B can be expressed in equation (4). ${B=\sum exp(\frac{-E_i}{RT}) (4)}$ With ${E_i= (\frac{h^2}{(2mV^{2/3}})(n_x^2+n_y^2+n_z^2) (5)}$ So the summation in equation (4) carried out on three variables nx, ny, nz, respectively from -∞ to ∞. This sum takes a little sub symbols qx, qy, and qz are used to replace (h2/(2mV2/3))1/2.nx, (h2/(2mV2/3))1/2.ny, and (h2/(2mV2/3))1/2.nz. By replacing those all, then the sum of equation (4) can be replaced in the integral form of equation (6) with A of ((2mV2/3)/h2)3/2. ${B= A(\int_{\infty}^\infty {dg_x}\int_{\infty}^\infty {dg_y}\int_{\infty}^\infty {dg_z}) {exp(-\frac{q_x^2+q_y^2+q_z^2}{kT})} (6)}$ Now just to integrate of the equation (6) with a technique integration that produces (π/a)(1/2) (a = 1/kT) for variable qxn, qyn, and qzn has n zero [2], so the function B into equation (7). ${B= (\frac{V^{2/3}2\pi{mkT}}{h^2}) (7)}$ It is known that V is the volume of the space for the number of colloidal particles N per mole of particles n. With of substituted N/n into equation (7), then B = JVn obtained with J = (2πn1/3mr.kT/h2)3/2.The next step is substituting the derivative of B into equation (7)as its result can be shown in equation (8). ${F= -(\frac{RT}{N})(\ln(J)+n\ln(V)) (8)}$ From now on equation (1) should be related to the laws of thermodynamics in order to obtain the derivative F of volume with temperature fixed at -P. In the same way, equation (8) derived for a fixed volume with temperature so that the results are comparable to -P in order to obtain the osmotic pressure which can be seen in equation (9). ${P= \frac{RTn}{VN} (9)}$ To consider the state of the particles suspended in a liquid in a state of equilibrium, then the situation will be easier to calculate how big a shift in the osmotic pressure of the particles with Stokes to bring style to any direction of x perpendicular. Thus, the function will be the Helmholtz equation (10). ${\part{F}= \part{U}-(T\part{S})=0 (10)}$ The inner energy in U have the mole of particles per volume with Stokes force (Stoke's law) Kv (K = 6πkr), so that the inner energy ∂U and entropy ∂S can be found by equation (11) and (12). ${\part{U}= -Kv\int_{0}^{l} {\part{x}}{dx} (11)}$ ${\part{U}= -(\frac{R}{N})\int_{0}^{l} {\frac{\part{(\frac{n}{V})}}{dx}}\part{x}dx (12)}$ From now on to integrate equation (11) and (12), so the equation (10) can become the equation (13) in which there is the formulation of diffusion in the Fick's law (Fick's second law). ${\frac{Kv}{K}-D\frac{\part{v}}{\part{x}}= 0 (13)}$ In equation (13), the diffusion D can be calculated for how much, if Kv= (RT/N)(∂v/∂v)(a state of equilibrium). Thus, equation (13) can be equation (14). ${D=\frac{RT}{N6\pi kr}(14)}$ Equation (14) are still not able to determine the diffusion causes the particle radius can not be known. To calculate the particle radius and diffusion, the derivatife of its formulation can be found by equation (15) with a solution for f (function)in equation (16) which were all obtained from the integral form of the series because the time interval and the shift of particle is very small. ${\frac{\part{f}}{\part{t}}=D\frac{\part^2{f}}{\part{x^2}}(15)}$ ${f(x,t)=n\frac{exp(-\frac{x^2}{4Dt})}{\sqrt{4\pi Dt}}(16)}$ By deriving equation (15) with respect to time and distance, then shift the particle can be determined as follows. ${\frac{\part{f}}{\part{t}}=n\frac{exp(-\frac{x^2}{4Dt})}{\sqrt{4\pi D}}(-t^{-\frac{1}{2}}{\frac{x^2}{4Dt^2}+\frac{1}{2}t^{-\frac{3}{2}})} (17)}$ ${\frac{\part^2{f}}{\part{x^2}}=n\frac{exp(-\frac{x^2}{4Dt})}{\sqrt{4\pi Dt}}({\frac{x^2}{4Dt^2}-\frac{1}{2}t^{-1})} (18)}$ Equation (14), (17), and (18) can be substituted into equation (15) and the results can be seen in equation (19), known as the shift of the particle. ${x=\sqrt{\frac{RTt}{N3\pi kr}}(19)}$ In other words, the equation (19), the radius of the particle r can not be determined, because the shift of the particle x is not yet known how its value, therefore, to obtain it can be done by letting the particles are in a state of motion circular uniform (stirred condition by magnetic Stirrer) by applicable Stokes force as sentifugal style. Thus the particle diameter can be determined as follows. ${\sum {F}=ma_s(20)}$ ${Kv=m\frac{v^2}{r_s}(21)}$ ${x=\frac{Kt}{A_r n}r_s(22)}$ Equation (19) and (22) identified as useful for particle radius r is shown in equation (23). ${r=\frac{1}{\pi\eta}\sqrt\frac{{A_r}^2 nRT}{108A_v t{r_s}^2}(23)}$, where Av is Avogardo's number (6.023 × 1023 molecule). By looking at equation (23), the shortest radius r of particle can be determined experimentally (sol-gel method) in the room temperature, then it has mole n whose value is very small, the viscosity η is very large, the time of stirrer t is very long and the radius of the container is also very long (rs). Need to know that, by adjusting those variables as simple as possible, nanoparticle can be reached by sol-gel method. ==References==
1. Manurung, P., Hanes, P., Pardede, I., Fathurhoman, A., dan Simbolon, H (2012). Desain Mikrostruktur Nanotitania dari Bahan Titanium Triklorida. Prosiding : Seminar Nasional & Teknologi-IV.
2. Edwin, J., Purcell., dan Varberg, D., (1996). Kalkulus dan Geometri Analitis (fifth ed.). Erlangga. p. 484.