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## Worked Examples: Energy of Batteries

**Question 1:** 96,500 C of electricity (charge) was delivered by a 1.3 volt mercury button cell.

How much energy, in joules, was produced by this cell?

**Solution:**

(Based on the StoPGoPS approach to problem solving.)

*What is the question asking you to do?*
Calculate energy in joules

E = ? J

*What data (information) have you been given in the question?*
Extract the data from the question:

Q = quantity of charge delivered = 96,500 C

V = voltage produced (EMF) = 1.3 V

*What is the relationship between what you know and what you need to find out?*
Write the equation for calculating electrical energy:

E = Q × V

*Substitute the values into the equation and solve for E:*

E = 96 500 × 1.3 = 125,450 J
*Is your answer plausible?*
Perform a "rough" calculation by "rounding off" data to see if your answer is in the same "ball park".

Let Q = 100,000 C

Let V = 1 V

Then, E = Q × V = 100,000 × 1 = 100,000 J

Our calculated value of E = 125,450 J is in the same "ball park" as the rough calculation so we are reasonably confident our answer is plausible.

*State your solution to the problem:*
E = 125,450 J

**Question 2:** A battery is made up of a series of 2.0 volt galvanic (voltaic) cells.

The battery delivers 48,250 C of electricity (charge) which provides 579 kJ of energy.

How many galvanic cells made up the battery?

**Solution:**

(Based on the StoPGoPS approach to problem solving.)

*What is the question asking you to do?*
Calculate the number of galvanic cells in the battery.

*What data (information) have you been given in the question?*
Extract the data from the question:

E = energy = 579 kJ

Convert kJ to J by multiplying by 1000 J/kJ

E = 579 × 10^{3} J

Q = quantity of charge = 48,250 C

V_{(cell)} = 2 V

*What is the relationship between what you know and what you need to find out?*
(i) Write the equation to calculate the voltage produced by the battery:

E = Q × V_{(battery)}

(ii) Rearrange this equation to calculate V:

V_{(battery)} = E ÷ Q

(iii) Substitute the values for E and Q into this equation to find the voltage for the battery, E_{(battery)}:

V_{(battery)} = E ÷ Q

V_{(battery)} = 579 × 10^{3} ÷ 48,250 = 12 V

*Calculate how many cells make up the battery:*
1 galvanic cell delivers 2 V

Let *x* = number of cells required to produce a 12 volt battery

*x* × 2 V = 12 V

*x* = 12 ÷ 2 = 6 cells

*Is your answer plausible?*
Work backwards: If you place 6 lots of 2 V galvanic cells in series to produce 48,250 C of charge, how much energy is provided?

Voltage of battery = 2 × 6 V = 12 V

E = Q × V = 48,250 × 12 = 579,000 J = 579 kJ

Since this amount of energy agrees with that given in the question, we are reasonably confident that our calculated value for the number of 2 V cells in series is correct.

*State your solution to the problem:*
There are 6 galvanic cells connected in series to make up the battery.

**Question 3.** A 2.0 volt cell delivers a current of 300 mA, generating 2100 J of energy in the process.

For how long did the cell operate?

**Solution:**

(Based on the StoPGoPS approach to problem solving.)

*What is the question asking you to do?*
Calculate time

t = ? seconds

*What data (information) have you been given in the question?*
Extract the data from the question:

V = voltage = 2.0 V

I = current = 300 mA

Convert milliamps to amps by dividing by 1000 mA/A

I = 300 mA ÷ 1000 mA/A = 0.300 A

E = energy = 2100 J

*What is the relationship between what you know and what you need to find out?*
(i) Calculate quantity of charge, Q

E = Q × V

Rearrange this equation to find Q:

Q = E ÷ V

Substitute in the values and solve for Q:

Q = 2100 ÷ 2.0 = 1050 C

(ii) Calculate time taken, t

Q = I × t

rearrange this to find t

t = Q ÷ I

*Substitute the values into the equation and solve for t:*
t = Q ÷ I

t = 1050 ÷ 0.300

= 3500 seconds

*Is your answer plausible?*
Work backwards: if you run a 2 V cell with a current of 300 mA for 3500 seconds, how much energy is delivered?

(i) E = Q × V,

(ii) Q = I × t,

so E = (I × t) × V = (300/100 × 3500) × 2 = 2100 J

Since this is the same amount of energy given in the question, we are reasonably confident that our calculated value for time is correct.
*State your solution to the problem:*
time = 3500 seconds

## Worked Example: Electrolysis and Energy Calculations

**Question: ** An EMF of 4.5 V produces 1 kg of sodium metal by the electrolysis of Na^{+}.

Calculate the minimum number of kilowatt-hours of electricity needed to produce the sodium metal.

**Solution:**

(Based on the StoPGoPS approach to problem solving.)

*What is the question asking you to do?*
Calculate energy in kilowatt-hours

E = ? kWh

*What data (information) have you been given in the question?*
Extract the data from the question:

V = EMF (voltage) = 4.5 V

m(Na) = mass of sodium metal produced = 1 kg

Convert mass in kg to g by multiplying by 1000

m(Na) = 1 kg × 1000 g/kg = 1000 g

*What is the relationship between what you know and what you need to find out?*
(i) Calculate moles of sodium metal produced:

moles(Na) = mass(Na) ÷ molar mass(Na)

n(Na) = 1000 ~~g~~ ÷ 22.99 ~~g~~ mol^{-1} = 43.5 mol

(ii) Calculate moles of electrons required to produce 43.5 moles of sodium metal from sodium ions:

Na^{+} + e^{-} → Na_{(s)}

mole (stoichiometric) ratio e^{-} : Na_{(s)} is 1:1

Therefore 43.5 moles electrons needed to produce 43.5 moles of sodium metal.

(iii) Calculate the quantity of charge, Q, required:

Q = n(e^{-})F

n(e^{-}) = 43.5 mol

F = 96,500 C mol^{-1} (from data sheet)

Q = 43.5 × 96,500 = 4,197,750 C

(iv) Calculate the required energy, E, in joules

E = Q × V

Q = 4,197,750 C

V = 4.5 V

E = 4,197,750 × 4.5 = 18,889,875 J

*Convert energy in joules to energy in kilowatt-hours*
1 kilowatt-hour = 3.6 × 10^{6} J

1 J = 1 ÷ (3.6 × 10^{6}) = 2.78 × 10^{-7} kWh

18,889,875 J = 18,889,875 × 2.78 × 10^{-7} = 5.25 kWh

*Is your answer plausible?*
Work backwards: Calculate the EMF needed to produce 1 kg of Na_{(s)} using 5.25 kWh of energy.

E = Q × V

So, V = E ÷ Q

E = 5.25 kWk = 5.25 kWh × 3.6 × 10^{6} J ~~kWh~~^{-1} = 1.89 × 10^{7} J

so, V = 1.89 × 10^{7} ÷ Q

Since Q = n(e)F and F = 96,500, Q = 96,500n(e)

so, V = 1.89 × 10^{7} ÷ (96,500n(e))

1 mole e produced 1 mole Na^{+},

n(e) = n(Na^{+}) = mass/molar mass = 1000/22.99 = 43.5

so, V = 1.89 × 10^{7} ÷ (96,500 × 43.5) = 4.5 V

Since this voltage agrees with that given in the question, we are reasonably confident that our calculated value for energy is correct.
*State your solution to the problem:*
E = 5.25 kWh