A cell has a standard emf of +0.177 V at 298 K. What is the value of the equilibrium constant for the cell reaction a) if n=1? b) if n=2?
nFEo = -RTlnK
a.) K =9.8*10^2
B.) K =9.5*10^5
To determine the value of the equilibrium constant for the cell reaction, we can use the Nernst Equation, which relates the standard emf of a cell to the equilibrium constant.
a) If n = 1:
The Nernst Equation for the cell reaction is given by:
E = E° - (RT / nF) * ln(Q)
Where:
E = Cell potential under non-standard conditions
E° = Standard cell potential
R = Gas constant (8.314 J/mol·K)
T = Temperature in Kelvin
n = Number of moles of electrons transferred in the cell reaction
F = Faraday's constant (96,485 C/mol)
Q = Reaction quotient
In this case, since n = 1, the Nernst Equation simplifies to:
E = E° - (RT / F) * ln(Q)
We are given E° = +0.177 V at 298 K. To calculate the equilibrium constant, we need to determine the reaction quotient, Q.
For a general cell reaction:
aA + bB → cC + dD
The reaction quotient, Q, is defined as:
Q = (C^c * D^d) / (A^a * B^b)
Since the question does not provide the specific reaction, we cannot determine the values of A, B, C, and D. Therefore, without additional information, we cannot calculate the value of the equilibrium constant.
b) If n = 2:
Using the same Nernst Equation:
E = E° - (RT / nF) * ln(Q)
With n = 2, we have:
E = E° - (RT / (2F)) * ln(Q)
Again, since we do not have the reaction equation, we cannot calculate the equilibrium constant without further information.
To determine the equilibrium constant for the cell reaction, you'll need to use the Nernst equation. The Nernst equation allows you to calculate the cell potential or the equilibrium constant of a cell reaction given the standard emf, temperature, and the number of electrons transferred in the cell reaction.
The Nernst equation is given by:
Ecell = E°cell - (RT / nF) * ln(Q)
Where:
Ecell is the cell potential
E°cell is the standard cell potential
R is the gas constant (8.314 J/mol·K)
T is the temperature in Kelvin
n is the number of electrons transferred in the cell reaction
F is the Faraday constant (96,485 C/mol)
ln(Q) is the natural logarithm of the reaction quotient
a) If n=1:
Given that the standard emf (E°cell) is +0.177 V, and assuming room temperature (298 K), you can calculate the equilibrium constant (K) using the Nernst equation.
Ecell = E°cell - (RT / nF) * ln(Q)
When n=1, the equation becomes:
Ecell = E°cell - (RT / F) * ln(Q)
Plugging in the values:
Ecell = 0.177 V - ((8.314 J/mol·K * 298 K) / (1 * 96,485 C/mol)) * ln(Q)
Simplifying:
Ecell = 0.177 V - 0.026 V * ln(Q)
From the Nernst equation, you can equate the cell potential (Ecell) to zero to find the equilibrium condition. At equilibrium, the reaction quotient (Q) equals the equilibrium constant (K). Therefore, you can set Ecell to zero:
0 = 0.177 V - 0.026 V * ln(K)
Solving for K:
0.026 V * ln(K) = 0.177 V
ln(K) = 0.177 V / 0.026 V
ln(K) = 6.808
K = e^(6.808)
Using a calculator:
K ≈ 9036
Therefore, when n=1, the equilibrium constant (K) is approximately 9036.
b) If n=2:
Using the same procedure as above, you can calculate the equilibrium constant (K) when n=2. Plugging the values into the Nernst equation:
Ecell = 0.177 V - ((8.314 J/mol·K * 298 K) / (2 * 96,485 C/mol)) * ln(Q)
Simplifying:
Ecell = 0.177 V - 0.013 V * ln(Q)
Setting Ecell to zero:
0 = 0.177 V - 0.013 V * ln(K)
ln(K) = 0.177 V / 0.013 V
ln(K) = 13.615
K = e^(13.615)
Using a calculator:
K ≈ 75092
Therefore, when n=2, the equilibrium constant (K) is approximately 75092.