### Video Transcript

A gas initially has a volume of 0.5 cubic meters and is at a temperature of 20 degrees Celsius. The gas is heated at constant pressure until its temperature is 100 degrees Celsius. What is the volume of the gas after it is heated? Give your answer to two decimal places.

In this example, we have a gas held at constant pressure that is heated so that the gas’s temperature increases. We want to know what effect this has on the volume of the gas. The fact that we’re told the gas is held at a constant pressure can remind us of an experimental gas law known as Charles’s law. This law says that for a gas maintained at constant pressure, at any given instant the volume 𝑉 of the gas divided by the gas temperature 𝑇 is always the same ratio. In other words, if we call our initial gas volume 𝑉 one, 0.5 cubic meters, our initial gas temperature 𝑇 one, which is 20 degrees Celsius, our final gas volume after heating 𝑉 two, and our final gas temperature 𝑇 two of 100 degrees Celsius, then Charles’s law tells us that 𝑉 one divided by 𝑇 one equals 𝑉 two divided by 𝑇 two.

It’s 𝑉 two, the volume of the gas after it’s heated, that we want to solve for. To help us do that, let’s multiply both sides of this equation by the temperature 𝑇 two. This causes that temperature to cancel out on the right. And slightly rearranging our resulting equation, we find that 𝑉 two equals 𝑉 one multiplied by the ratio 𝑇 two to 𝑇 one. Note that we’re given 𝑉 one, 𝑇 two, and 𝑇 one. And so we can substitute them into this equation.

At this point, we must be very careful about something. Charles’s law that 𝑉 divided by 𝑇 is a constant is only true when the temperature 𝑇 is expressed in units of kelvin. Here, our temperatures are in units of degrees Celsius. If we kept these units in calculating 𝑉 two, we would get an incorrect result. What we need to do is to convert each one of our temperatures from a value in degrees Celsius to a value in kelvin. If we go all the way down to absolute zero, a temperature of zero kelvin, that temperature is equivalent in degrees Celsius to negative 273.15 degrees Celsius. It’s also true that a change in temperature of one kelvin is equal to a change in temperature of one degree Celsius.

Therefore, if we considered another temperature, a temperature of zero degrees Celsius at which water freezes, that temperature equals positive 273.15 kelvin. This tells us that to convert a temperature in degrees Celsius to a temperature in kelvin, we’ll add 273.15 to it. In our numerator then, we have our temperature in degrees Celsius and we add 273.15 to it. And that gives us a temperature in kelvin. We get 373.15 kelvin. Likewise, in our denominator we take our temperature in degrees Celsius, add 273.15 to it, and then we get a temperature in kelvin. That simplifies to a temperature of 293.15 kelvin.

We’re now ready to calculate 𝑉 two. And note that as we do, the units of kelvin will cancel out and we’re left just with units of cubic meters, units of volume. The volume we calculate, rounded to two decimal places, is 0.64 cubic meters. This is the volume of the gas after it is heated.