Question

How many minutes are required to deliver \(3.21 \times 10^{6}\) coulombs using a current of 500 A used in the commercial production of chlorine? (a) \(8.3\) (b) \(5.3 \times 10^{4}\) (c) 6420 (d) 107

Short answer

107 minutes are required to deliver the charge.

Step-by-step solution

Understand the Relationship between Charge, Current, and Time

The relationship between charge (Q), current (I), and time (t) is given by the equation Q = I * t, where 't' is the time in seconds. To find the time, rearrange the formula to solve for 't': t = Q / I.

Convert the Charge into Coulombs

The given charge is already in coulombs (C), which is the standard SI unit for electrical charge, so no conversion is necessary.

Substitute the Values into the Formula

Substitute the values of the given charge and current into the rearranged formula. Current (I) is 500 A and charge (Q) is 3.21 * 10^6 C. The calculation for 't' will be: t = (3.21 * 10^6 C) / 500 A.

Calculate the Time in Seconds

Perform the division to calculate the time in seconds: t = (3.21 * 10^6 C) / 500 A which equals 6420 seconds.

Convert the Time from Seconds to Minutes

Since 1 minute equals 60 seconds, divide the time in seconds by 60 to find the time in minutes: time in minutes = 6420 seconds / 60 seconds/minute.

Perform the Final Calculation

Finish the calculation to get the time in minutes: time in minutes = 6420 / 60, which equals 107 minutes.

Key concepts

Understanding the Charge and Current Relationship

When delving into electrochemistry problems, a fundamental concept is the relationship between charge and current. Electric charge, measured in coulombs (C), and electric current, measured in amperes (A), are directly connected through a simple, yet key formula: \( Q = I \times t \). This equation states that the charge (\( Q \)) is the product of the current (\( I \)) and the time (\( t \)) for which the current flows.

For instance, if you have a current of 1 A flowing for 1 second, you'd have transferred 1 C of charge. Inversely, if you know the charge and you need to find out the duration of the current, you simply divide the charge by the current, which gives \( t = \frac{Q}{I} \). It's essential for students to grasp that if the current increases with the charge remaining constant, the time would decrease accordingly, and vice versa.

Practical Applications

In real-life scenarios, like the production of chlorine mentioned in the exercise, understanding this relationship is pivotal for determining operational times and efficiencies. For instance, operators would calculate the time needed to produce a certain quantity of chlorine by applying the correct current, revealing how the charge pertains to time.

Conversion from Coulombs to Minutes

In electrochemistry, and specifically in exercises like the one provided, a frequent task is to convert the amount of electrical charge in coulombs into a particular unit of time—often to minutes, as this unit is more practical for our perception of time. This requires a two-step process: first, determine the time in seconds using the formula \( t = \frac{Q}{I} \), and then convert seconds into minutes since the standard output of the equation is in seconds.

Remember that there are 60 seconds in a minute, making the conversion relatively straightforward: divide the number of seconds by 60 to obtain the time in minutes. Students must not overlook this conversion step because forgetting to convert seconds to minutes can lead to incorrect answers and confusion regarding the scale of the problem.

Why Minutes?

One might wonder why minutes are chosen for such conversions. Minutes provide a good balance between precision and practicality for most electrochemical processes, ensuring the results are easily interpreted and compared in everyday terms.

Calculating Time with Current and Charge

When you are faced with an electrochemistry problem that requires you to find out how long an electrical current needs to run to transfer a certain charge, the process involves plugging values into the definitive formula \( t = \frac{Q}{I} \). If you've accurately determined the charge in coulombs and the current in amperes, calculating time is a matter of simple division.

In practice, you first need to verify that your charge is in the correct unit of coulombs and that your current is in amperes. These SI units are standardized for use in these formulas, which helps to avoid any unit conversion errors. Once the calculation is done and you have the time in seconds, don't forget the last crucial step of converting seconds into your desired unit of time, like minutes in our example problem.

Ensuring Accuracy

Care should be taken to use proper significant figures and units throughout all these calculations to ensure the accuracy of your answer. Consistency in this aspect not only helps in getting the right answer but also in developing good scientific practices.