The emf of a Daniell cell at \(298 \mathrm{~K}\) is \(\mathrm{E}_{-1}\) \(\mathrm{Zn}\left|\mathrm{ZnSO}_{4} \| \mathrm{CuSO}_{4}\right| \mathrm{Cu}\) \((0.01 \mathrm{M}) \quad(1.0 \mathrm{M})\) when the concentration of \(\mathrm{ZnSO}_{4}\) is \(1.0 \mathrm{M}\) and that of \(\mathrm{CuSO}_{4}\) is \(0.01 \mathrm{M}\), the emf changed to \(\mathrm{E}_{2}\). What is the relationship between \(\mathrm{E}_{1}\) and \(\mathrm{E}_{2} ?\) (a) \(\mathrm{E}_{1}=\mathrm{E}_{2}\) (b) \(\mathrm{E}_{2}=\mathrm{O} \neq \mathrm{E}_{1}\) (c) \(\mathrm{E}_{1}>\mathrm{E}_{2}\) (d) \(\mathrm{E}_{1}<\mathrm{E}_{2}\)

Electrochemical cells, such as the Daniell cell in our exercise, are devices that convert chemical energy into electrical energy. They consist of two half-cells, each containing an electrode immersed in an electrolyte. In the case of the Daniell cell, one half-cell has zinc metal in a zinc sulfate solution, while the other has copper metal in a copper sulfate solution.

Electrons flow from the zinc electrode to the copper electrode through an external circuit, while ions move through the electrolyte in each half-cell and through a salt bridge that connects the two half-cells. The electrochemical reactions that occur at the electrodes are responsible for the cell's electromotive force (emf), which can be measured in volts. The emf is influenced by a number of factors, including the intrinsic properties of the metal electrodes and the concentration of solutions in each half-cell.

The Nernst equation is vital for understanding how the emf of an electrochemical cell changes with varying conditions. Mathematically, it is expressed as:
\[ E = E^{\circ} - \frac{RT}{nF} \ln Q \]
where \( E \) is the cell emf, \( E^{\circ} \) is the standard emf, \( R \) is the ideal gas constant, \( T \) is the temperature in Kelvin, \( n \) is the number of moles of electrons transferred, \( F \) is Faraday's constant, and \( Q \) is the reaction quotient.

This equation essentially tells us that the cell's emf will vary based on the concentrations of reactants and products, according to the reaction quotient \( Q \). In our example with the Daniell cell, altering the concentrations of \( \mathrm{ZnSO}_4 \) and \( \mathrm{CuSO}_4 \) changes the emf value, which can be calculated using the Nernst equation.

Le Chatelier's principle is a qualitative tool that predicts the response of a system in equilibrium when an external change occurs. It states that if a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium moves to counteract the change.

In terms of electrochemical cells, if the concentration of reactants is increased (for example, more \( \mathrm{ZnSO}_4 \) in the Daniell cell), the principle indicates that the equilibrium will shift to produce more products, hence generating a higher emf. Similarly, an increase in the concentration of products (more \( \mathrm{CuSO}_4 \)) would shift the equilibrium to favour the reactants, resulting in a lower emf. Le Chatelier's principle helps us to understand why the emf of our Daniell cell will change when the concentrations of electrolytes in the half-cells are altered.

The reaction quotient \( Q \) plays a critical role in the Nernst equation and is a snapshot of the ratio of products to reactants at a given moment. For a general cell reaction of the form \( aA + bB \rightarrow cC + dD \), the reaction quotient is expressed as:
\[ Q = \frac{[C]^c[D]^d}{[A]^a[B]^b} \]
where the concentrations in square brackets correspond to the molarity of each species. If \( Q \) is less than the equilibrium constant \( K \), the forward reaction is favored to reach equilibrium. Conversely, if \( Q \) is greater than \( K \), the reverse reaction is favored.

For the Daniell cell, the reaction quotient is based on the concentrations of \( \mathrm{ZnSO}_4 \) and \( \mathrm{CuSO}_4 \). When the concentrations of reactants and products change, as in the exercise, \( Q \) changes, and so does the emf of the cell. Understanding how to calculate and interpret \( Q \) is crucial when evaluating the direction in which an electrochemical cell reaction is likely to proceed.