How many years are there in 1 mole of seconds? Use conversion factors, and show which units cancel.

To convert from seconds to years, we'll need to go through some intermediate units: minutes, hours, and days. The necessary conversion factors are: - 1 minute = 60 seconds - 1 hour = 60 minutes - 1 day = 24 hours - 1 year = 365.25 days (accounting for leap years)

A mole is simply a very large number used in chemistry, defined as Avogadro's number, which is approximately \(6.022 \times 10^{23}\) units in one mole. So, we are given that the time is 1 mole of seconds, or \(t = 6.022 \times 10^{23}\) seconds.

Now, let's start converting seconds to minutes using the conversion factors. $$ t \,(\text{minutes}) = 6.022 \times 10^{23}\ \text{seconds} \times \frac{1 \, \text{minute}}{60\, \text{seconds}} $$

Next, we'll convert minutes to hours using the conversion factors. $$ t \,(\text{hours}) = 6.022 \times 10^{23}\ \text{seconds} \times \frac{1 \, \text{minute}}{60\, \text{seconds}} \times \frac{1\, \text{hour}}{60\,\text{minutes}} $$

Continuing with the conversion process, we will now convert hours to days. $$ t \,(\text{days}) = 6.022 \times 10^{23}\ \text{seconds} \times \frac{1 \, \text{minute}}{60\, \text{seconds}} \times \frac{1\, \text{hour}}{60\,\text{minutes}} \times \frac{1\, \text{day}}{24\, \text{hours}} $$

Finally, we'll convert days to years, taking into account leap years by using an average of 365.25 days per year. $$ t \,(\text{years}) = 6.022 \times 10^{23}\ \text{seconds} \times \frac{1 \, \text{minute}}{60\, \text{seconds}} \times \frac{1\, \text{hour}}{60\,\text{minutes}} \times \frac{1\, \text{day}}{24\, \text{hours}} \times \frac{1 \,\text{year}}{365.25\,\text{days}} $$

Now that all the conversion factors are in place, calculate the final expression to find the number of years. \[ t \,(\text{years}) = 6.022 \times 10^{23}\ \text{seconds} \times \frac{1 \, \text{minute}}{60\, \text{seconds}} \times \frac{1\, \text{hour}}{60\,\text{minutes}} \times \frac{1\, \text{day}}{24\, \text{hours}} \times \frac{1 \,\text{year}}{365.25\,\text{days}} \] Notice that the units of seconds, minutes, hours, and days all cancel out, leaving us only with years. \[ t \, (years) = 6.022 \times 10^{23} \times \frac{1}{(60)(60)(24)(365.25)} \, \text{years} \] Using a calculator, we find: \[ t \,(\text{years}) \approx 1.91 \times 10^{16} \, \text{years} \] So there are approximately \(1.91 \times 10^{16}\) years in one mole of seconds.