Given:
EMF of battery connected to copper voltameter = 12 V
Mass of copper deposited in 30 minutes = 2 g
EMF of battery when the same voltameter is connected = 6 V
Time for deposition of copper in this case = 45 minutes

To find: Mass of copper deposited in 45 minutes

Formula Used:
The amount of substance liberated at an electrode is directly proportional to the quantity of electricity passed through the cell.

Explanation:
1. Calculation of amount of electricity passed through the cell in first case:
From the formula, we know that the amount of substance liberated at an electrode is directly proportional to the quantity of electricity passed through the cell.
So, the amount of electricity passed through the cell can be calculated using the formula:

Q = I * t

where,
Q = Quantity of electricity passed in Coulombs (C)
I = Current in Ampere (A)
t = Time in seconds (s)

We know that the charge on 1 mole of electrons is 1 Faraday (1 F = 96500 C).
The molar mass of copper is 63.5 g/mol.
So, the amount of electricity required to deposit 1 gram of copper is given by:

Q = (1/63.5) * 96500 = 1519.68 C

Now, the amount of electricity required to deposit 2 g of copper is given by:

Q = 2 * 1519.68 = 3039.36 C

We also know that the voltage of the battery is 12 V.
So, the current flowing through the circuit is given by:

I = V/R

where,
V = Voltage of the battery = 12 V
R = Resistance of the circuit (which includes the resistance of the voltameter and the solution)

Since we do not know the resistance of the circuit, we cannot calculate the exact value of current. However, we know that the resistance of the circuit will remain the same in both cases (when the battery is of 12 V and when it is of 6 V). So, we can assume that the current flowing through the circuit is constant in both cases.

Now, we know the amount of electricity passed in the first case, and we can assume the current to be constant. So, we can calculate the time required to deposit 2 g of copper using the formula:

Q = I * t

=> t = Q/I

=> t = (3039.36 C) / I

=> t = (3039.36 C) / (V/R) (since I = V/R)

=> t = (3039.36 C) * (R/V)

2. Calculation of amount of electricity passed through the cell in second case:
Now, when the same voltameter is connected across a 6 V battery, the current flowing through the circuit will be given by:

I = V/R

where,
V = Voltage of the battery = 6 V
R = Resistance of the circuit (which includes the resistance of the voltameter and the solution)

We can assume that the resistance of the circuit remains the same as before. So, the current flowing through the circuit can be calculated as:

I = 6/R (since V/R is constant for both cases)

Now, the time required to deposit the same amount of copper (2 g) using a 6 V