Video Transcript
For a gas of a fixed volume and mass, the pressure when divided by its temperature, 𝑃 divided by 𝑇, is 0.16 kilopascals per kelvin. To what temperature must the gas be heated to in order to produce a pressure of 40 kilopascals?
Okay, so in this question, we have a gas, which we are told is at a fixed volume and also a fixed mass. Initially, the gas is at a pressure that we will call 𝑃 and a temperature that we will call 𝑇. Now, we don’t know what these are, but the question does tell us the value of 𝑃 divided by 𝑇, which is equal to 0.16 kilopascals per kelvin. And we’ll keep a note of this down here on the left.
What the question wants us to do is work out what temperature the gas must be heated to in order to produce a pressure of 40 kilopascals. So we must use our knowledge about the relationship between the pressure and temperature of the gas to work out the temperature of the gas when the pressure of the gas is equal to 40 kilopascals. When a gas has fixed volume and mass, the relationship between its pressure and temperature is described by a law known as Gay-Lussac’s law. And Gay-Lussac’s law tells us that the pressure of a gas divided by the temperature of the gas is equal to some constant. This is very useful for us because we’ve been told the pressure of the gas divided by the temperature of the gas at some initial point.
Gay-Lussac’s law tells us that the value of this, 0.16 kilopascals per kelvin, is constant. So this relationship is still valid, even when we’ve heated the gas to the point where the pressure has reached 40 kilopascals. In order to answer this question, we must therefore rearrange this relationship to make 𝑇 the subject of the equation.
Starting with our original relationship of 𝑃 divided by 𝑇 is equal to 0.16 kilopascals per kelvin, we can first multiply both sides by 𝑇. And here, we see that the 𝑇s on the left cancel. Next, we can divide both sides by our constant, 0.16 kilopascals per kelvin. And we see that the 0.16 kilopascals per kelvin in the numerator and denominator of these terms on the right cancel, leaving us with just an expression for 𝑇. Writing this a bit more neatly, we get 𝑇 is equal to 𝑃 divided by 0.16 kilopascals per kelvin. All that’s left for us to do is for us to substitute our known value of 𝑃 is equal to 40 kilopascals into this equation.
Before we go any further, let’s check our units. In the numerator, we have kilopascals, and in the denominator, we have kilopascals per kelvin. We can see that the kilopascals in the numerator and denominator will cancel, leaving us with units of one over one over kelvins, which actually gives us overall units of kelvins, which is what we’d expect for a temperature. So we can rewrite our equation as 𝑇 is equal to 40 divided by 0.16 kelvins.
Now, we could use our calculator to work this out, but we could also work this out by hand. We can write 40 as four multiplied by 10 and 0.16 as 16 multiplied by 0.01. And at this point, we can split the fraction into two parts: four divided by 16 multiplied by 10 divided by 0.01. Now, four divided by 16 is just equal to one divided by four, because 16 is four times as large as four. And as a decimal, this is equal to 0.25. So this part of the fraction is just equal to 0.25.
In the second part of the fraction, 10 divided by 0.01, we can write 0.01 as one divided by 100. So this part of the fraction can actually be written 10 divided by one divided by 100, which is equal to 10 multiplied by 100. And 10 multiplied by 100 is equal to 1000. So 40 divided by 0.16 is equal to 0.25 multiplied by 1000. And finally, putting it all together, 0.25 multiplied by 1000 is just equal to 250. So 𝑇 is equal to 250 kelvins. And this is the answer to our question. The temperature that the gas must be heated to in order to produce a pressure of 40 kilopascals is 250 kelvins.
