Question Video: Understanding the Relationship between Volume and Number of Moles of an Ideal Gas Physics

Video Transcript

For an ideal gas, for the pressure and temperature of the gas to remain constant, if the number of moles of the gas is increased by a factor of three, by what factor must the volume of the gas change?

Here, we’re considering an ideal gas. That tells us the gas can be described by the ideal gas law, which says that the pressure of an ideal gas multiplied by its volume equals the number of moles of the gas times a constant multiplied by the gas’s temperature. In this equation, the quantity 𝑅 is a constant; it’s called the molar gas constant. In our particular case, the pressure 𝑃 and the temperature 𝑇 of the gas are also held constant.

As a next step, let’s take our application of the ideal gas law and separate it so that constant values are on one side and nonconstant values are on the other. We know that the pressure 𝑃, the molar gas constant 𝑅, and the temperature 𝑇 are constant values. If we divide both sides of our equation by the number of moles of the gas 𝑛 times the pressure 𝑃, then on the left, the pressure 𝑃 cancels out, and on the right, the number of moles of gas cancels out. We get then that 𝑉 divided by 𝑛 is equal to 𝑅 times 𝑇 over 𝑃. We now have all of our constant values on the right side of this expression and all of the variables on the left.

Since 𝑅 and 𝑇 and 𝑃 are all constants individually, the ratio 𝑅 times 𝑇 divided by 𝑃 is also a constant. If we call this overall constant 𝑘, then we can write 𝑉 divided by 𝑛 is equal to 𝑘.

In our situation, we’re told that the number of moles of the gas, that’s 𝑛, is increased by a factor of three. In other words, what was 𝑛 now has become three times 𝑛. We want to know by what factor the volume 𝑉 must change along with the change in 𝑛 so that the ratio of these quantities is still equal to the constant 𝑘. If 𝑉 divided by 𝑛 is equal to 𝑘, then if 𝑛 is multiplied by a factor of three, 𝑉 must be as well. That way the new ratio three 𝑉 divided by three 𝑛 is still equal to 𝑉 divided by 𝑛, which is equal to 𝑘. We know then by what factor 𝑉 must correspondingly be multiplied.

For an ideal gas, where the pressure and temperature of the gas are held constant, if the number of moles of the gas is multiplied by a factor of three, then the volume must also be multiplied by a factor of three.