Video Transcript
The temperature of four meters cubed of gas is initially 320 kelvin. The gas is heated while being kept at a constant pressure. This causes the gas to expand. The gas is heated until it has a volume of 10 meters cubed. What is the temperature of the gas after it is heated?
In this question, we have a gas which has an initial volume that we will call 𝑉 one. And the question tells us that this initial volume is four meters cubed. We are also told that the gas is at an initial temperature of 320 kelvin and we will call this temperature 𝑇 one. The gas is then heated, which causes it to expand. And we are told that after heating, it has reached a volume of 10 meters cubed, and we will call this volume 𝑉 two. The temperature of the gas will also have changed, and we will call this new temperature 𝑇 two. And this is what the question would like us to calculate.
To answer this question, we will use a gas law that relates the temperature and volume of a gas. The law we will use is Charles’s law. And Charles’s law states that the volume of a gas divided by the temperature of the gas is constant. However, this is only true when two things are satisfied. Number one, the amount of gas does not change, and number two, the pressure of the gas does not change. Now, the question doesn’t say anything about adding or removing gas while it’s being heated. So, we can safely assume that the amount of gas does not change. The question also tells us that the gas is heated while being kept at a constant pressure, so the pressure of the gas does not change. And because both of these conditions are satisfied, we can safely apply Charles’s law to the gas in this question.
Importantly, we can apply Charles’s law to the same gas at two different points in time. Starting with the gas before it is heated, we know that the volume of the gas divided by its temperature is constant. Then, if we look at the gas after it is heated, exactly the same is true. The volume of the gas divided by its temperature is constant. And because this value is a constant, it is the same for any point in time. And in our case, it means that the gas’s volume divided by its temperature before heating is equal to the gas’s volume divided by temperature after heating. And therefore, we have an equation relating the volume and temperature of the gas at two different points in time.
And we know values for 𝑉 one, 𝑇, one, and 𝑉 two. And we’re looking to calculate 𝑇 two. So, all we have to do is rearrange this equation to make 𝑇 two the subject, and we’ll have our answer. Starting with our original equation, we will first take the reciprocal of both sides, where we can remember that for a fraction, the reciprocal just means that the numerator moves to the denominator and the denominator moves to the numerator, which gives us 𝑇 one divided by 𝑉 one is equal to 𝑇 two divided by 𝑉 two. Next, we’ll multiply both sides by 𝑉 two, where we see that the 𝑉 twos on the right cancel, which leaves us with an expression for 𝑇 two.
Writing this a bit more neatly, 𝑇 two is equal to 𝑇 one multiplied by 𝑉 two divided by 𝑉 one. That is to say, the temperature of the gas after it has been heated is equal to the temperature of the gas before it was heated multiplied by the ratio of the volume of the gas after it was heated to the volume of the gas before it was heated.
All that’s left for us to do is to substitute our known values of 𝑇 one, 𝑉 one, and 𝑉 two into this equation. And before we go any further, we must first check our units. First of all, it’s very important that our temperature must be given in kelvin, which in our case is true. Secondly, we see that our units of meters cubed in the numerator of this fraction cancel with the meters cubed in the denominator. And what we can tell from this is actually the units of volume don’t matter as long as both of the volumes given are in the same units. And our overall answer will just be left with units of kelvin, which is what we’d expect for a temperature.
So, the temperature of the gas after it has been heated, 𝑇 two, is equal to 320 kelvin multiplied by 10 meters cubed divided by four meters cubed. Evaluating this expression, we get 𝑇 two is equal to 800 kelvin. So, the temperature of the gas after it is heated is exactly 800 kelvin.
