Question

Scientists from the European Space Agency have been studying geysers located around the southern pole of one of Saturn's moons. These geysers send \(250.0 \mathrm{~kg}\) of water into the atmosphere every second. Scientists trying to discover what happens to this water have found that it forms a very thin vapor ring around Saturn. How many moles of water are shot into the atmosphere in a Saturn day? (One Saturn day is equal to 10 hours and 45 minutes.)

Short answer

537,737,513.87 moles of water are shot into the atmosphere in a Saturn day.

Step-by-step solution

- Convert time to seconds

First, convert the duration of one Saturn day into seconds. One Saturn day is 10 hours and 45 minutes. Calculate the total time in minutes first:\[10 \text{ hours} \times 60 \text{ minutes/hour} + 45 \text{ minutes} = 645 \text{ minutes}\]Now convert minutes to seconds:\[645 \text{ minutes} \times 60 \text{ seconds/minute} = 38700 \text{ seconds}\]

- Calculate total mass of water

Calculate the total mass of water ejected into the atmosphere in one Saturn day. Given the eruption rate is 250.0 kg per second:\[250.0 \text{ kg/second} \times 38700 \text{ seconds} = 9,675,000 \text{ kg}\]

- Calculate moles of water

To find the number of moles, use the formula: \[ \text{moles} = \frac{\text{mass}}{\text{molar mass}} \]The molar mass of water (\text{H}_2 \text{O}) is: \[2 \times 1.01 \text{ g/mol (H)} + 16.00 \text{ g/mol (O)} = 18.02 \text{ g/mol}\]Convert the molar mass to kg/mol (since the mass is in kg): \[18.02 \text{ g/mol} \times \frac{1 \text{ kg}}{1000 \text{ g}} = 0.01802 \text{ kg/mol}\]Now calculate the number of moles:\[ \text{moles} = \frac{9,675,000 \text{ kg}}{0.01802 \text{ kg/mol}} \ \text{moles} = 537,737,513.87 \text{ mol}\]

Key concepts

molar mass of water

Water, chemically represented as \text{H}_2 \text{O}, consists of two hydrogen atoms and one oxygen atom. The molar mass of an element is the mass of one mole of that element and is usually measured in grams per mole (g/mol). For water, the molar mass can be calculated by summing the masses of its constituent atoms. The atomic mass of hydrogen (H) is approximately 1.01 g/mol, and oxygen (O) is roughly 16.00 g/mol. Therefore, for water:
\[2 \times 1.01\text{ g/mol (H)} + 16.00\text{ g/mol (O)} = 18.02\text{ g/mol}\text{ (H}_2 \text{O)}\]
This means one mole of water has a mass of 18.02 grams.

unit conversion in chemistry

Unit conversion is an essential skill in chemistry for transforming measurements into different units to solve problems. For example, converting time from hours and minutes to seconds:
1. Start by determining the total minutes in 10 hours and 45 minutes:
\[10 \text{ hours} \times 60 \text{ minutes/hour} + 45 \text{ minutes} = 645 \text{ minutes}\]
2. Then, convert the minutes to seconds:
\[645 \text{ minutes} \times 60 \text{ seconds/minute} = 38700 \text{ seconds}\]

• Understanding the factors to multiply by, such as 60 minutes per hour or 60 seconds per minute, is crucial for accurate conversions.

Similarly, converting mass from grams to kilograms involves recognizing that 1 kg = 1000 grams. So, when converting the molar mass of water from g/mol to kg/mol:
\[18.02 \text{ g/mol} \times \frac{1 \text{ kg}}{1000 \text{ g}} = 0.01802 \text{ kg/mol}\]

mass to moles conversion

Mass to moles conversion is a foundational concept in chemistry, where you use the relationship between mass, molar mass, and moles. The basic formula is:
\[\text{moles} = \frac{\text{mass}}{\text{molar mass}}\]
Here's how you can apply it:
Suppose you need to calculate the number of moles for water being ejected from Saturn's moon. Given the total mass of the water ejected in a day is 9,675,000 kg and you've previously calculated the molar mass of water as 0.01802 kg/mol:
\[\text{moles} = \frac{9,675,000 \text{ kg}}{0.01802 \text{ kg/mol}} = 537,737,513.87 \text{ mol}\]

• Always ensure that the units of mass and molar mass match (both should be either in kg or g).
• This conversion helps quantify the amount of substance using the concept of moles, which correlates directly with particle count using Avogadro's number.

This understanding aids in various calculations in chemical reactions, stoichiometry, and material science.