A gas cylinder with a volume of 0.245 cubic meters contains a gas at a temperature of 350 kelvin and a pressure of 120 kilopascals. Find the number of moles of the gas particles in the cylinder. Use 8.31 meters squared kilogram per second squared kelvin mole for the value of the molar gas constant. Give your answer to one decimal place.
In this question, we’re being asked about a gas cylinder, and let’s suppose that this here is that cylinder. We’re told that it has a volume of 0.245 cubic meters, and we’ve labeled this as 𝑉. We’re also told that the gas in the cylinder is at a temperature of 350 kelvin, which we’ve labeled as 𝑇, and a pressure of 120 kilopascals, which we’ve labeled as 𝑃. We’re also given a value of 8.31 meters squared kilogram per second squared kelvin mole for the molar gas constant. And we can recall that this typically has a symbol capital 𝑅.
Given all of this information, the question is asking us to find the number of moles of the gas particles in the cylinder. Let’s label this number of moles as 𝑛. To work out the value of this quantity 𝑛, we can recall a formula known as the ideal gas law. This law says that the pressure of a gas 𝑃 multiplied by its volume 𝑉 is equal to 𝑛, the number of moles of the gas, multiplied by the molar gas constant 𝑅 multiplied by the gas temperature 𝑇. Now, in this equation, we know the value of the pressure 𝑃 and the volume 𝑉. We also know the gas temperature 𝑇, and we’re given a value for the molar gas constant 𝑅. We can see then that there’s just one unknown quantity in this equation, and that’s the number of moles 𝑛 of the gas, which is what we’re asked to find in this question.
What this means is that we can rearrange this equation in order to make 𝑛 the subject. And then, if we substitute in our values for 𝑃, 𝑉, 𝑅, and 𝑇, we’ll be able to calculate the value of 𝑛. To make 𝑛 the subject, we need to divide both sides of the equation by both the molar gas constant 𝑅 and the temperature 𝑇 of the gas. Then, we can see that on the right-hand side, we’ve got an 𝑅 in the numerator, which cancels with the 𝑅 in the denominator. And likewise, the 𝑇 in the numerator and the 𝑇 in the denominator also cancel each other out.
If we then rewrite the equation with the left- and right-hand sides swapped over, we have that the number of moles 𝑛 is equal to the pressure 𝑃 multiplied by the volume 𝑉 divided by the molar gas constant 𝑅 and the gas temperature 𝑇. This quantity 𝑛 is the number of moles of the gas particles, and its SI unit is just units of moles. In order to calculate a value of 𝑛 in moles, we need to ensure that all the quantities on the right-hand side of this equation are given in their own respective SI units. The SI unit for pressure is the pascal, the SI unit for volume is the cubic meter, and the SI unit for temperature is the kelvin. Finally, the SI units for the quantity 𝑅 are meters squared kilogram per second squared kelvin mole.
Notice that though this unit might look a little complicated, the things it’s made up out of — meters, kilograms, seconds, kelvin, and moles — are all SI units. So it makes sense then that this is the SI unit for the quantity 𝑅. Comparing these SI units against the units of the quantities that we’re given, we can see that we have indeed got a molar gas constant 𝑅 in SI units of meters squared kilogram per second squared kelvin mole. We’ve also got a volume 𝑉 in meters cubed, the SI unit for volume, and a temperature 𝑇 in the SI units of kelvin. However, our value for the pressure 𝑃 is in units of kilopascals, while the SI unit for pressure is the pascal. That means that before substituting our values into this equation, we need to convert the pressure 𝑃 from kilopascals into pascals.
To do this, let’s recall that the unit prefix k or kilo- means a factor of 1000, and so one kilopascal is equal to 1000 pascals. That means that to convert a pressure value from kilopascals into pascals, we need to multiply it by a factor of 1000. So then, our pressure 𝑃 of 120 kilopascals is equal to 120 multiplied by 1000 pascals. This works out as a pressure of 120000 pascals.
Now that we’ve got all of these quantities expressed in SI units, we’re ready to substitute those values into this equation in order to calculate 𝑛, the number of moles of the gas particles. Let’s clear some more space on the screen to do this.
We know that 𝑛 is equal to 𝑃 times 𝑉 divided by 𝑅 times 𝑇. And substituting in our values for the quantities on the right-hand side gives us this expression for 𝑛. In the numerator, we have 120000 pascals, which is our value for the pressure 𝑃, multiplied by 0.245 cubic meters, which is our volume 𝑉. Then, in the denominator, we’ve got 8.31 meters squared kilogram per second squared kelvin mole, that’s the molar gas constant 𝑅, multiplied by 350 kelvin, which is our value for the gas temperature 𝑇.
Now, let’s recall that although we’ve got all these various units for the different quantities on the right-hand side of this equation, we know that since they’re all SI units, we’ll get a value for 𝑛 with units of moles. So, rather than writing out all the individual units, we can just write these overall units of moles that we know the value we’ll calculate for 𝑛 will have. We can now evaluate this expression by typing it into a calculator. When we do this, we get a result for 𝑛 of 10.1083 et cetera moles, where the ellipsis here is used to indicate that the result has further decimal places.
We can notice though that the question asks for our answer to one decimal place. To one decimal place of precision, the result runs down to 10.1 moles. Our answer to this question then is that to one decimal place, there are 10.1 moles of the gas particles in the cylinder.