A gas with a volume of two cubic meters is at a pressure of 500 pascals. The gas is compressed at a constant temperature to a volume of 0.5 meters cubed. What is the pressure of the gas after it is compressed?
Let’s say that this is our gas with an initial volume 𝑉 one of two cubic meters and an initial pressure 𝑃 one of 500 pascals. The gas is compressed at a constant temperature so that its final volume 𝑉 two is 0.5 cubic meters. Based on all this, we want to solve for the pressure of the gas, we’ll call it 𝑃 two, after it is compressed.
We can recall an experimental gas law called Boyle’s law. This law says that when the pressure 𝑃 or the volume 𝑉 of a gas changes while the temperature of the gas is held constant, then the product 𝑃 times 𝑉 at any given instant is the same value. Boyle’s law means that if we take the product of the initial pressure and volume of our gas, 𝑃 one times 𝑉 one, that will be equal to the product of the pressure and volume of the gas after it’s been compressed, 𝑃 two times 𝑉 two. This is what is indicated when we say that 𝑃 times 𝑉 for a gas at constant temperature is a constant.
Since it’s the pressure 𝑃 two that we want to solve for, let’s isolate 𝑃 two by dividing both sides of this equation by the volume 𝑉 two. That cancels 𝑉 two from the right, and we find that 𝑃 two equals 𝑃 one times the ratio of volumes 𝑉 one to 𝑉 two. Substituting in, 𝑃 one is 500 pascals, 𝑉 one is two cubic meters, and 𝑉 two is 0.5 cubic meters. Note that the units of cubic meters in numerator and denominator will cancel. And then two divided by 0.5 is equal to four. Therefore, the pressure 𝑃 two is 500 pascals times four, or 2000 pascals. This is the pressure of the gas after it is compressed.