The Ideal Gas Law: Density - Online Tutor, Practice Problems & Exam Prep

Density, a fundamental concept in chemistry, measures the mass of a substance per unit volume. For gases, which are less dense than solids and liquids, density is typically expressed in grams per liter. The ideal gas law, represented by the equation:

$PV=nRT$

can be adapted to calculate the density (D) of a gas at specific pressure (P) and temperature (T) conditions. By manipulating the ideal gas law, we derive the formula for density as:

$D=\frac{P\mathrm{MM}}{RT}$

where MM is the molar mass, R is the ideal gas constant, and T is the temperature in Kelvin. This relationship is crucial for understanding the behavior of gases under varying conditions and can be remembered by the mnemonic "Dreams push me over rough times," symbolizing the equation components. This formula is essential for predicting how gases will respond to changes in pressure and temperature, which is critical for applications in various scientific and industrial processes.

The Ideal Gas Law: Density

Recall that density represents the amount of mass per unit of volume. And when it comes to our density formula, its density equals mass over volume. Now gases are much less dense than solids and liquids. So when it comes to their density, we're going to say their density is in grams per liter.

M here is still mass of the gas in grams. And then here volume V will be in liters. So again, when it comes to density, it's massvolume and gas is being much less dense. Don't use grams per milliliter or grams per centimeter, but instead use grams per liter.

The Ideal Gas Law: Density Example 1

Here it says an unknown gas sample has a density of 1.70g per liter. If the sample has a volume of 120 mills, what is its mass? In grams. All right, so here we're given two values what we want to start out with the 120 MLS because it just has one unit by itself, easier to manipulate.

So we're going to start out with 120 milliliters, realize that we need to isolate our grams, which are found here, And in order to isolate those grams, I need to cancel out the liters. So that tells me that I need to convert my 120 MLS into liters first. So remember 1 milli is 10^-3 liters.

Now that I have liters, I can bring in my density, which is 1.7g per one liter. So here liters cancel out and I'll be left with grams at the end. When I punch that in, that gives me, oh point 204 grams of my unknown gas.

Here my number has three sig figs because 1.70 has three sig figs and this has four sig figs. Remember, we want to go with the least number of sig figs when when it's reasonable. Here .204 grams is a reasonable answer for our unknown gas.

The Ideal Gas Law: Density

Now we can say that the ideal gas law can be used to determine the density of a gas under certain pressure and temperature conditions. Here we're going to say that the new derived version of the ideal gas law which we can apply to density is d=P⁢mRT.

And to help us remember this order, just remember dreams push me over rough times here. Dreams D stands for density push. Me is pressure times molar mass over R * T rough times. So if you can remember this phrase, it's a great way to remember this version of the ideal gas law when it relates to density.

Now if you want to look and see how we derived this formula, you can click on to the next video, but only do so if you're a professor really cares on how you derive these different types of formula. If they don't, then just remember this phrase. After that, go to the next series of videos and let's put this formula to practice.

The Ideal Gas Law: Density

So If you've clicked on this video, that means you want to see how we derive this new version of the ideal gas law. So it all begins with our density formula. Remember, density equals mV. Remembering this, we move on to the ideal gas law where we're dealing with molar mass.

Remember, molar mass equals MRT over PV, and we just said that mV represents density. So take them out and put in density. We still have RT and over P we need to isolate density, so just use algebra to isolate it. Multiply both sides by pressure. So molar mass times pressure equals DRT.

Isolate our density by dividing out RT and we see now that density equals PmRT. So that's how we went about an isolated and derived this new formula for density. Now this is how we derived it. But just remember, dreams push me over rough times. Knowing that helps you to write the formula very quickly.

Now that we've seen this formula, click onto the next video and let's put it to work.

The Ideal Gas Law: Density Example 2

Here we're told that a gaseous compound of nitrogen and hydrogen is found to have a density of OH .977g per liter at .69474 atmospheres and 373.15 Kelvin. What is the molecular formula of the compound? Well, realize here in this question they're giving us density, they're giving us pressure, and they're giving us temperature. With this information, I could find the molar mass of this unknown gas.

So here we're going to say that dreams push me over rough times, and we're going to say that we have density, we have pressure, we have temperature, and we always know what R is. We just need to isolate our molar mass. So multiply both sides by RT, so RT times D = P times molar mass. Divide both sides by pressure so molar mass equals

D R T P

Or take the information given to us. Our density is OH .977g per liter. R is our gas constant. Temperature is already in Kelvin and pressure is already in atmospheres. Doing this we'll see that we isolate. So atmospheres are gone, kelvins are gone, liters are gone, we'll have grams per mole.

So when we plug this into our formula, we're going to get here as our mass, 43.06g per mole. So here's our molar mass. All we do now is we look at the different compounds and we would just calculate their molar masses and see which one comes closest to this 43.06g per mole. Now if you did this correctly, you would see that the answer would have to be option C.

It's the one with the mask closest to our answer. It has a molar mass approximately equal to 43.038g per mole. So option C would be the answer for this particular question. So realize you're giving us density, pressure, and temperature. That means we can use the density form of the ideal gas law to help us find molar mass and then use that information to compare to the molar masses of all these gases.

Consider two containers of gases at the same temperature. One has helium at a pressure of 1.00 atm. The other contains carbon dioxide with the same density as the helium gas. What is the pressure of the carbon dioxide gas sample?

• A. 0.063 atm
• B. 0.091 atm
• C. 0.133 atm
• D. 1.51 atm
• E. 2.71 atm

Determine the molecular formula of a gaseous compound that is 49.48% carbon, 5.19% hydrogen, 28.85% nitrogen, and 16.48% oxygen. At 27°C, the density of the gas is 1.5535 g/L and it exerts a pressure of 0.0985 atm.

• A. C₂₀H₄₄N₈O₄
• B. C₁₆H₂₀N₆O₂
• C. C₁₄H₂₂N₂O₂
• D. C₁₆H₂₀N₈O₄

Here’s what students ask on this topic:

How can density be found using the ideal gas law?

To find the density of an ideal gas using the Ideal Gas Law, you need to start with the law's formula, which is:

$PV=nRT$

Here, P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin.

Density (ρ) is defined as mass (m) per unit volume (V), so:

$\rho =\frac{m}{V}$

To find the mass of the gas, you can use the molar mass (M) of the gas and the number of moles (n):

$m=nM$

Substitute the expression for m into the density equation to get:

$\rho =\frac{nM}{V}$

Now, rearrange the Ideal Gas Law to solve for n/V:

$\frac{n}{V}=\frac{P}{RT}$

Substitute n/V in the density equation with P/RT to get the final expression for density:

$\rho =\frac{MP}{RT}$

So, the density of an ideal gas can be found using the equation:

$\rho =\frac{MP}{RT}$

where M is the molar mass of the gas, P is the pressure, R is the ideal gas constant, and T is the temperature in Kelvin.

How can density be found using the ideal gas law?

Density can be found using the ideal gas law, which is expressed as:

$PV=nRT$

Here, P stands for pressure, V is volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature in Kelvin.

To find the density (ρ), you need to rearrange the ideal gas law to solve for mass (m) over volume (V), since density is mass per unit volume (ρ = m/V). You can replace n (the number of moles) with m/M (mass over molar mass) in the ideal gas law equation:

$PV=\frac{m}{M}RT$

Now, solve for m/V, which is the density (ρ):

$\rho =\frac{PM}{RT}$

This equation shows that the density of an ideal gas is directly proportional to its pressure (P) and molar mass (M), and inversely proportional to the temperature (T) and the ideal gas constant (R). By knowing the pressure, temperature, and molar mass of the gas, you can calculate its density.