The Drop of Potential at the Cathode in Flames

Abstract

The theory as given by J. J. Thomson for the drop of potential at plane electrodes has been modified by allowing for recombination in the layer and a similar theory for cylindrical electrodes has been worked out. The equation for plane cathodes is $V_2={\left(\frac{32\pi i}{75k_1}\right)}^{\frac{1}{2}}{x_2}^{\frac{3}{2}},$ and for cylindrical cathodes is $V_2={\left(\frac{2i}{3k_1{({r_2}^2-{r_0}^2)}^2}\right)}^{\frac{1}{2}}\left[{r_2}^{3}log\left(\frac{r_2+{({r_2}^2-{r_0}^2)}^{\frac{1}{2}}}{r_0}\right)-{r_2}^2{({r_2}^2-{r_0}^2)}^{\frac{1}{2}}-\frac{{({r_2}^2-{r_0}^2)}^{\frac{3}{2}}}{3}\right],$ where $V_2$ is the potential drop across the sheath, $i$ the current density, $x_2$ the sheath thickness at the plane cathode, $r_0$ the radius of the cylindrical cathode, $r_2$ the radius of the cylindrical sheath about the cathode, and $k_1$ the velocity of the positive ions for a gradient of one volt per cm. The experimental results for platinum electrodes immersed in pure and NaCl flames agree well with the theoretical equations given. It is found that the drop in potential at the cathode occurs in a sheath of uniform thickness, which completely surrounds the electrode. By plotting the gas potential at various points in the flame against distance from the cathode it is possible to estimate the thickness of the sheath. Over 95% of the potential drop takes place across the sheath at the cathode provided it is of clean platinum. If the cathode is not clean electrons are emitted which partially neutralize the accumulation of positive ions and thus reduce the sheath thickness. By measuring $V_2$, $i$, $x_2$ or $r_0$ and $r_2$, and making the proper substitutions in the above equations, the mobility $k_1$ of the positive ions is found to average 12.4 for a pure flame and 8.1 cms per sec. for one volt per cm for a NaCl flame.