A gas is cooled while being kept at a constant pressure. If its temperature in kelvin changes by a factor of 0.25, by what factor does the volume of the gas change?
Okay, so in this question, we’re being asked about a volume of gas, which is being kept at a constant pressure. To visualise this, we could imagine a box filled with gas where the top face of the box is a movable section with a weight resting on top. Because there’s a constant weight resting on the top of the box, this surface experiences a constant downward force. And because the rest of the container is rigid, this means that the gas experiences a constant pressure on all sides even if the volume of the gas changes. In this question, we’re told that the temperature of the gas in kelvin changes by a factor of 0.25. And we’re asked to find the factor that the volume changes by.
If the temperature is changing by a factor of 0.25, this means it started at some initial temperature, which we can call 𝑇 i, which then changed to a value of 0.25 times its initial temperature. At the same time, the volume changed from some initial value, which we can call 𝑉 i, to a final value, which we can call 𝑉 f. Our job is to find the factor by which the volume changed. This is the same as asking, what number do we need to multiply 𝑉 i by in order to get to 𝑉 f? Because this question is asking us to figure out how a change in temperature will change the volume of a gas that’s kept at constant pressure, we can use Charles’s law to help us.
Charles’s law tells us that, at a constant pressure, the volume of a gas is proportional to its temperature. In other words, 𝑉 is directly proportional to 𝑇. And proportionality works both ways, so that means we can also say 𝑇 is directly proportional to 𝑉. When we say the two variables are proportional to one another, what we mean is that a change in one variable will be accompanied by an equal proportional change in the other variable. In other words, both variables always change by the same proportion.
So, for example, if we were to double the size of 𝑉, which is the same as saying we change it by a factor of two or a proportion of two, then we would find that 𝑇 changes by the same proportion. So it doubles as well. Or say, we multiply 𝑇 by a half. That’s the same as saying we change it by a factor of a half or a proportion of a half. Then 𝑉 would change by the same proportion, so it halves as well. Now, in this question, we’re told that the temperature in kelvin changes by a factor of 0.25, which means to get from the initial temperature to the final temperature, we multiply the initial temperature by 0.25.
Because Charles’s law tells us that 𝑉 and 𝑇 are proportional to one another, that means that 𝑉 must change by the same proportion as 𝑇. So in other words, to get from the initial volume to the final volume, we also need to multiply it by 0.25. This change can be expressed algebraically. If we multiply 𝑉 i by 0.25, we get 𝑉 f. So we can write 𝑉 f equals 0.25 times 𝑉 i. So because the final volume is 0.25 times the initial volume, we can say that the volume changes by a factor of 0.25. So this is our answer. If the temperature of a gas kept at constant pressure changes by a factor of 0.25, then the volume changes by a factor of 0.25 as well.