Lesson Video: Converting between the Celsius and Fahrenheit Temperature Scales Physics • 9th Grade

In this lesson, we will learn how to convert between the Celsius temperature scale and the Fahrenheit temperature scale.

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Video Transcript

In this video, we’re talking about converting between the Celsius and Fahrenheit temperature scales. Although these are the only two temperature scales there are, they are two of the most commonly used. By the end of this lesson, we’ll be able to take a temperature given in one of these two scales and convert it to the other. To understand how to do that, it will help to know where the Fahrenheit and Celsius temperature scales came from in the first place.

Now, when we talk about temperature scales in general, we know that one end of the scale corresponds to hotter temperatures and the other end to cold ones. And any particular temperature scale is defined by the values it gives to specific temperatures. For example, let’s say that we wanted to make up our own temperature scale. We could call it the Nagwa Scale. Let’s say that we define this scale by saying that the temperature of the sun is 1000 and the temperature of a snowflake is zero. So then, if someone asked us, what’s the temperature of the sun? We could tell them that it’s 1000 degrees Nagwa. Or similarly, a snowflake is zero degrees Nagwa. That’s the temperature scale we’ve come up with.

Now, this may seem a little silly, but actually not much more than this goes into defining a temperature scale. Partly for this reason, throughout history, many different scales have been developed. Two of the most common are the ones we’re talking about here, Celsius and Fahrenheit. Of these two, it was the Fahrenheit temperature scale that was developed first, and it’s named after its inventor, the German physicist Daniel Gabriel Fahrenheit. The Fahrenheit scale is oriented around the temperature at which water boils. And for historical reasons, the number on this scale at which that event occurs is 212. Then, if we go down to a colder temperature, the temperature at which water freezes and become a solid, on this scale that’s at 32.

Now, on some temperature scales, and the Fahrenheit scale is one of them, when we name a value on that scale, we include the word degrees in the description of it. For example, we would say that water boils at 212 degrees Fahrenheit. That where degrees points us back to the fact that the original form of this temperature scale was developed by an astronomer. And the idea was that temperatures could be subdivided in the same way that angles could be divided, into degrees. So anyway, water boils at 212 degrees Fahrenheit, and it freezes at 32 degrees Fahrenheit.

Over time, thanks in part to Daniel Fahrenheit’s ability to make very accurate thermometers, the Fahrenheit temperature scale grew in popularity. Sometime later though, a Swedish astronomer named Anders Celsius came up with an idea for a slightly different scale. For his scale, Celsius used the same kinds of physical events, water boiling and water freezing, but he assigned different numbers to them. Celsius wanted to make a temperature scale that could easily be divided into 100 equal parts. In other words, he wanted to make a centigrade scale. In this word, we can see the word cent, which has to do with 100 or one 100th. For example, a century of years or a cent of money.

Anyway, with that goal in mind, Celsius assigned the value of 100 to the temperature at which water boils and zero to the temperature at which it freezes. So then, on the Celsius scale, there’s a 100-degree difference between these two temperatures. But notice that that’s not the case on the Fahrenheit scale. If we subtract 32 from 212, we find the difference between these values is 180 degrees Fahrenheit. This helps us see that there are really two differences between the Fahrenheit and the Celsius scales. One difference, as we’ve already seen, is that they assign different temperature values to the same events.

For example, when water boils in the Fahrenheit world, we say that happens at 212 degrees. But in the Celsius world, we say it happens at 100 degrees. So, different numbers there, that’s one difference between the scales. But another difference is that changing one degree in one scale is not the same as changing one degree in the other scale. Let’s say that we take these two temperature differences on the two different scales, 180 degrees Fahrenheit and 100 degrees Celsius, and we divide them both by 100. If we do that, we find that a temperature difference of 1.8 degrees Fahrenheit is equal to a temperature change of one degree Celsius.

In other words, if we increase the temperature of something by one degree on the Celsius scale, then we’ve increased the temperature of that same object by almost two degrees on the Fahrenheit scale. If we express this difference as a ratio, we would say it’s 1.8 to one or, written as a fraction, 1.8 divided by one. Now, if we multiply top and bottom of this fraction by five, then we get this result of nine-fifths. That’s the ratio of how much a given temperature change in degrees Fahrenheit is to how much that same temperature change is in degrees Celsius. And we’ll keep this fraction in mind because it will show up again later.

Now, knowing that we have these two temperature scales, Fahrenheit and Celsius, we want to find a way to convert from one to the other. That is, to take a temperature in one of the scales and change it over to the equivalent temperature on the other scale. Now, there is a mathematical relation that shows us how to do this conversion. And knowing what we know so far about these two scales, we can begin to derive that equation. So far, we have these two temperature correspondences, that 100 degrees Celsius is equal to 212 degrees Fahrenheit and that zero degrees Celsius is equal to 32 degrees Fahrenheit.

Using these correspondences, we want to come up with an equation that relates the temperature in one scale — say the temperature in degrees Celsius, we’ll call it 𝑇 sub C— to an equivalent temperature in degrees Fahrenheit, we’ll call it 𝑇 sub F. Now, the way it’s written right now, we know that this relationship is inaccurate. But we want to figure out what mathematical steps to apply to either side of it so that it is. To start to see what those steps may be, let’s consider this first correspondence, zero degrees Celsius corresponds to 32 degrees Fahrenheit.

This tells us that in a correct equation, if we substitute in zero for our temperature in degrees Celsius, then a correctly defined equation will output 32 in degrees Fahrenheit. That must be the case because we know that zero degrees Celsius corresponds to that temperature in degrees Fahrenheit. So then, what can we do to this equation to make that happen? Well, perhaps the simplest thing is to add 32 to the left-hand side. This way, when we input zero for our temperature in degrees Celsius, then we add 32 to it, we get the result we wanted, 32 in degrees Fahrenheit.

So then, as it’s written, this temperature conversion formula agrees with this first temperature correspondence we have, that zero degrees Celsius corresponds to 32 degrees Fahrenheit. But now, let’s check it for the second correspondence, that 100 degrees Celsius is 212 degrees Fahrenheit. If we substitute in 100 for our temperature in degrees Celsius and then we add 32 to it, we get 132, which is not the 212 we know we should get. This tells us that, in its current form, our conversion formula is still missing something. As it turns out, that something is the conversion ratio we found earlier between degrees Fahrenheit and degrees Celsius. If we multiply that ratio by our input temperature in degrees Celsius and then add 32 to that, we’ll get a different result.

Let’s try it out using 100 degrees Celsius as our input. Nine-fifths times 100 plus 32 is equal to 900 divided by five plus 32, which is equal to 180 plus 32, which adds up to 212. So, we’ve input 100 degrees Celsius and as our output we’ve gotten 212 degrees Fahrenheit. That means that this form of our conversion formula agrees with this correspondence of 100 degrees Celsius corresponding to 212 degrees Fahrenheit. But now, let’s check and see if it still works for the other correspondence. If we insert zero for our temperature in degrees Celsius, then nine-fifths times zero is itself equal to zero, and then when we add 32 to that, we get a result of 32. So, by inputting zero degrees Celsius, as an output we’ve got 32 degrees Fahrenheit, just like we wanted.

What we’ve developed is a general equation for taking a temperature in degrees Celsius and calculating its corresponding value in degrees Fahrenheit. One important point about this equation is that its unitless. Even though there are temperatures in degrees Celsius and degrees Fahrenheit, when we plug them into the equation, we leave off the units. For example, when we substituted in zero degrees Celsius, we did that just plugging in a zero. And then, we calculated out 32, which we knew, as the output was 32 degrees Fahrenheit. This is one of the only cases where we leave off units in the calculation. Now, at this point, given any temperature in degrees Celsius, we could solve for the corresponding temperature in degrees Fahrenheit.

But what about in the opposite direction? To see how to make that kind of conversion, all we’ll need to do is rearrange this equation we already have algebraically. We could think of it this way. Right now, the subject of this equation is 𝑇 sub F, a temperature in degrees Fahrenheit. We want to change this to the new subject, it’s 𝑇 sub C. Our first step we could take towards getting there is to subtract 32 from both sides of this equation. If we do that, the positive 32 and the negative 32 on the left-hand side cancel one another out.

Then, as a next step, we can multiply both sides of the equation by the inverse of nine-fifths, in other words five-ninths. When we take this step, looking on the left-hand side, multiplying and dividing by five and multiplying and dividing by nine gives a result of one. And that leaves us with this result, 𝑇 sub C is equal to five-ninths times the quantity 𝑇 sub F minus 32. Using this new form of our equation, we can now take a temperature in degrees Fahrenheit and convert it to its equivalent value in degrees Celsius. So then, here are the two equivalent versions of this temperature conversion. Let’s now get some practice with these relationships through an example.

What is 67 degrees Celsius in degrees Fahrenheit? Give your answer to one decimal place.

All right, so here we want to take this temperature in degrees Celsius, it’s 67 degrees, and convert it to degrees Fahrenheit. That is, given this temperature we can call it 𝑇 sub C in degrees Celsius, we wanna convert it to the Fahrenheit temperature scale. To do this, we can recall the mathematical relationship that connects these two scales, Celsius to Fahrenheit. Given a temperature in degrees Celsius, if we multiply that by nine-fifths and add 32 to that result, we get the corresponding value in degrees Fahrenheit. And by the way, this ratio, nine-fifths, is not a random number. This is equal to the ratio of how much a temperature change on the Fahrenheit scale corresponds to on the Celsius scale.

So then, to solve for 𝑇 sub F, the temperature in Fahrenheit that corresponds to 67 degrees Celsius, we’ll take our input temperature written as a pure number with no units, multiply it by nine-fifths, and then add 32 to that. When we do this, the result we find is 152.6. Now, because everything here has no units, our answer also has no units in its current form. But because we’re solving for a temperature in degrees Fahrenheit, we know that really 152.6 is 152.6 degrees Fahrenheit. In our problem statement, we’re told to give our answer to one decimal place, which it is, 152.6. So then, this is our final answer. 67 degrees Celsius is 152.6 degrees Fahrenheit.

Let’s look now at a second example.

What is negative 40 degrees Celsius in degrees Fahrenheit?

In this exercise then, we’re doing a temperature conversion from a temperature in degrees Celsius to one in degrees Fahrenheit. Whenever we do that, it’s helpful to recall the conversion relationship between these two separate temperature scales. Given a temperature in degrees Celsius, we can call it 𝑇 sub C, if we multiply that temperature by nine-fifths and then add 32 to that result, we get the corresponding temperature in degrees Fahrenheit, we’ll call it 𝑇 sub F.

So, what we want to do is solve for 𝑇 sub F, given an input temperature of negative 40 degrees Celsius. To do that, we’ll use this conversion relationship, and we’ll insert negative 40 for our temperature in degrees Celsius. Multiplying negative 40 by nine-fifths, we get negative 360 divided by five. And that fraction is equal to negative 72. When we add positive 32 to negative 72, we get a result of negative 40. And it’s now that we put the correct units onto this result. That’s degrees Fahrenheit.

Now, at first this may seem strange. We started out with negative 40 in degrees Celsius, and we found that, apparently, that’s negative 40 in degrees Fahrenheit. In fact, there is nothing wrong with what we’ve done. Rather, what we’ve discovered is the one single temperature at which that temperature value in degrees Celsius is equal to the same temperature value in degrees Fahrenheit. There is no other temperature for which this is true, but it is true for negative 40. So, negative 40 degrees Celsius is negative 40 degrees Fahrenheit.

Let’s now summarize what we’ve learned about converting between Celsius and Fahrenheit scales. First off, we saw that many different temperature scales exist. The Celsius and the Fahrenheit scales are two of the most common. Furthermore, these scales are defined by the boiling and freezing points of water. Water boils at 100 degrees Celsius, or 212 degrees Fahrenheit, and it freezes at zero degrees Celsius, or 32 degrees Fahrenheit.

And then, lastly, we learned the conversion formula for going from degrees Celsius to degrees Fahrenheit and back. Written one way, it tells us that a temperature in degrees Fahrenheit is equal to nine-fifths times that equivalent temperature in degrees Celsius plus 32. And written another way, it tells us that a temperature in degrees Celsius is equal to five-ninths times the quantity temperature in degrees Fahrenheit minus 32.