Problem 24 - Chemistry The Molecular Science

Question

You light a \(25-W\) light bulb with the current from a \(12-V\) lead-acid storage battery. Calculate how much energy the light bulb utilized after \(1.0 \mathrm{~h}\) of operation. Calculate how many coulombs passed through the bulb. Assume \(100 \%\) efficiency. \((1 \mathrm{~W}=1 \mathrm{~J} / \mathrm{s} .)\)

Step-by-Step Solution

Answer

Energy used: 90000 J; Charge passed: 7488 C.

Step 1: Understanding Energy Consumption

Since power is given by the formula \( P = \frac{E}{t} \), where \( E \) is energy in joules and \( t \) is time in seconds, we can rearrange it to find \( E = P \times t \).

Step 2: Converting Time to Seconds

The time given is \(1.0 \h\), which needs to be converted into seconds. Thus, \( t = 1.0 \times 3600 \text{ seconds} = 3600 \text{ seconds} \).

Step 3: Calculating Energy Used by the Light Bulb

Substitute the values into the energy formula: \( E = 25 \text{ W} \times 3600 \text{ s} = 90000 \text{ J} \). So, the light bulb used \( 90000 \text{ joules} \) of energy.

Step 4: Understanding Charge Flow

The current \( I \) can be calculated using \( P = VI \), where \( V \) is the voltage. Rearrange to find \( I = \frac{P}{V} \).

Step 5: Calculating Current Through the Bulb

Substitute the known values into the formula: \( I = \frac{25 \text{ W}}{12 \text{ V}} = 2.08 \text{ A} \).

Step 6: Calculating Total Charge

Charge \( Q \) can be calculated using \( Q = I \times t \). Substitute \( I = 2.08 \text{ A} \) and \( t = 3600 \text{ s} \): \( Q = 2.08 \times 3600 = 7488 \text{ C} \). So, \( 7488 \text{ coulombs} \) passed through the bulb.

Key Concepts

Power Calculation

In the realm of physics, power refers to the rate at which energy is used or transferred, and it is usually measured in watts (W). To calculate the power consumed by a device, we use the formula:

• \( P = \frac{E}{t} \)

where \( P \) is power, \( E \) is energy in joules, and \( t \) is time in seconds.
In this scenario, a 25-watt light bulb is the focus, which implies it consumes 25 joules of energy per second. Applying this understanding allows us to deduce how much energy is consumed over a more extended period of time, by rearranging the formula to find \( E \) if needed.

Electric Charge

Electric charge is a fundamental property of matter that interacts with electric fields to produce current. Charges are generally denoted by \( Q \) and measured in coulombs. In electricity, charge relates to the flow of electrons through a conductor, creating what we observe as electric current.
Within a circuit, understanding electric charge is crucial as it affects how devices power up and function. When a current flows through a bulb, it carries a flow of charges that illuminate the bulb and is a key factor in power calculations.

Coulombs

The coulomb (named after Charles-Augustin de Coulomb) is the SI unit of electric charge. It's the amount of charge transported by a constant current of one ampere in one second. Coulombs provide a tangible way to quantify electric charge in energy-related calculations.
In our example, after one hour of operation, 7488 coulombs of charge pass through the light bulb, highlighting the volume of charge movement in that time frame.
Formula used:

• \( Q = I \times t \), where \( I \) is current and \( t \) is time.

Lead-Acid Battery

A lead-acid battery is a type of rechargeable battery commonly used in vehicles and backup power systems due to its reliability and cost-effectiveness. It consists of lead dioxide and sponge lead with an electrolyte solution of sulfuric acid.
For this exercise, the battery provides 12 volts to the light bulb. This voltage influences how much current flows through the bulb according to Ohm's Law (\( V = IR \)). In this manner, it's capable of powering devices such as our light bulb continuously for the required duration.

Electric Current

Electric current is the flow of electric charge along a conductor, measured in amperes (A). It quantifies how much charge passes a point in a circuit per second.
Current generation relies on the movement of electrons and is driven by voltage from a power source like a battery. In this exercise, we calculate the current flowing through a 25-Watt bulb using the relationship:

• \( I = \frac{P}{V} \)

With the bulb's power and voltage known, calculating the resultant current is straightforward.

Energy Conversion

Energy conversion entails changing one form of energy to another. In electrical applications, it's often the transformation of electrical energy into light, heat, or motion.
Here, the energy stored in the lead-acid battery is converted to light energy by the light bulb. This transformation is typically governed by the principle of efficiency, although the exercise assumes a 100% conversion rate for simplicity.
Understanding this helps in designing systems and choosing components that maximize energy use effectively to power different loads efficiently.