Show how the ratio of two transport numbers \(t\) ' and \(t\) " for two cations in a mixture depends on their concentrations \(\mathrm{c}\) ' and \(\mathrm{c}\) " and their mobilities \(u\) ' and \(u "\).

Electrolysis is a chemical process by which electrical energy is used to drive a non-spontaneous chemical reaction. During electrolysis, an electric current flows through an electrolyte, causing oxidation and reduction reactions to occur at the electrodes. This process decomposes the compounds within the electrolyte, separating the constituent elements or compounds.

For instance, if we consider the electrolysis of aqueous sodium chloride, it results in the production of sodium hydroxide, chlorine gas, and hydrogen gas. Completing an electrolysis reaction requires proper setup, and understanding the movement of ions - 'ion mobility' - within the electrolyte solution is a fundamental part of this process.

Ion mobility refers to the rate at which ions move through a conductor in response to an electric field. In the context of electrolysis, ion mobility is essential because it influences how quickly the ions can travel to the electrodes where they gain or lose electrons.

In simple terms, ions with higher mobility will reach the electrodes faster and will therefore be more likely to participate in the electrochemical reactions. The mobility of an ion is dependent on several factors, including its size, charge, and the viscosity of the medium through which it is moving. Understanding ion mobility is crucial when we look at transport numbers, as it directly impacts the distribution of current among the ions present in the electrolyte.

The transference number, also known as the transport number, is a dimensionless quantity that represents the fraction of the total electric current in a solution that is carried by a specific ion. Each ion in the solution contributes to the total current based on its transport number. The transference number of an ion is influenced by its ion mobility; as such, ions with higher mobility will generally have higher transference numbers because they carry a larger proportion of the current.

For example, in a solution with multiple ions, if one type of ion is much more mobile than the others, it will carry most of the current, resulting in a higher transference number for that ion. Conversely, less mobile ions will contribute less to the total current and therefore have lower transference numbers. This is why transference numbers are crucial to understanding electrical conductivity in electrolyte solutions and the efficiency of electrolysis processes.

The concept of concentration dependence in transport number reflects how the ratio of the transport numbers between different ions in a solution can vary based on their concentrations. Since the transport number is the proportion of the electric current carried by each ion, it intuitively depends on how many of those ions are present (concentration) along with their mobility.

As the concentration of an ion increases, assuming a constant mobility, it is available to carry a larger portion of the electric current. This relationship means that concentration plays a significant role in determining the efficiency and outcome of the electrolytic process. This is particularly important in industrial applications of electrolysis, where precise control over reaction conditions can be directly related to the quality and yield of the desired product.