Problem 105 - Chemistry A Molecular Approach

Question

There are exactly 60 seconds in a minute, exactly 60 minutes in an hour, exactly 24 hours in a mean solar day, and 365.24 solar days in a solar year. How many seconds are in a solar year? Give your answer with the correct number of significant figures.

Step-by-Step Solution

Answer

There are 31,557,734 seconds in a solar year, rounded to the correct number of significant figures.

Step 1: Calculate seconds in an hour

Multiply the number of seconds in a minute by the number of minutes in an hour. There are 60 seconds in a minute and 60 minutes in an hour, so there are 60 seconds/minute * 60 minutes/hour = 3600 seconds/hour.

Step 2: Calculate seconds in a day

Multiply the number of seconds in an hour by the number of hours in a day. There are 3600 seconds/hour * 24 hours/day = 86,400 seconds/day.

Step 3: Calculate seconds in a solar year

Multiply the number of seconds in a day by the number of days in a solar year. There are 86,400 seconds/day * 365.24 days/year = 31,557,734.4 seconds/year.

Step 4: Determine the correct number of significant figures

Since the number of days in a year (365.24) has five significant figures, our final answer should be rounded to the same number of significant figures. Therefore, there are 31,557,734.4 seconds in a solar year, which we should round to 31,557,734 to match the significant figures.

Key Concepts

Significant Figures

Grasping the concept of significant figures is essential when dealing with precise measurements and calculations. Significant figures, often called 'sig-figs', are the digits in a number that carry meaning towards its precision. This includes all digits except:

• Leading zeros, which serve as placeholders and don't hold value.
• Trailing zeros in the decimal part, only if they are after a decimal point and after a non-zero digit.
• Any digit that is not considered reliable or well-established.

In the scenario of calculating seconds in a solar year, this concept helps in maintaining the integrity of the calculations. When we multiply the number of days in a solar year (365.24) by the number of seconds in a day, our result should have the same number of significant digits as the number with the least significant figures used in our calculation. This means we have to pay attention to the input data. Here, 365.24 has five significant figures, while other numbers like 60 and 24 are exact values with an infinite number of significant figures. Therefore, our final answer should also reflect five significant figures to remain consistent with the precision of our data, resulting in 31,557,734 seconds in a solar year when rounded properly.

Unit Conversion

Unit conversion is the process of converting a measure from one unit to another. It is a basic skill required in science, engineering, and everyday life to make sense of different measurements. In the context of time calculation, we often convert seconds to minutes, minutes to hours, and so forth.
To correctly convert units, we need to know the conversion factors, such as:

• 1 minute = 60 seconds
• 1 hour = 60 minutes
• 1 day = 24 hours

Using these factors, we convert time upwards or downwards, multiplying or dividing accordingly. It's important to ensure that units cancel out properly, leaving us with the desired unit. For example, to find out how many seconds there are in an hour, we multiply the number of seconds in a minute by the number of minutes in an hour, i.e., \(60 \text{ seconds/minute} \times 60 \text{ minutes/hour} = 3600 \text{ seconds/hour}\). Similarly, we calculate the number of seconds in a day and in a solar year. When chained together, these conversions support each other to achieve the final result.

Time Calculation

Time calculation is fundamental in assessing the duration of events or the span between two points in time. It often involves adding, subtracting, multiplying, or dividing units of time. In our specific exercise, we're tasked with calculating the amount of time contained in a solar year.
The process requires understanding and applying the relationship between seconds, minutes, hours, and days. The calculation begins by multiplying basic units progressively until we reach the desired time frame:

• Start with seconds in a minute.
• Calculate seconds in an hour by multiplying by the number of minutes in an hour.
• Ascertain seconds in a day by multiplying the seconds in an hour by the number of hours in a day.
• Finally, determine the number of seconds in a solar year by multiplying the number of seconds in a day by the number of days in the solar year.

Each step builds upon the previous, accumulating the size of the time unit. It is crucial to work methodically, ensuring accuracy at each stage to avoid compounding errors, and to remember to adjust the final product for significant figures as necessary.