If a temperature scale were based on the freezing point (5.5^∘ C) and boiling point (80.1^∘ C) o

If a temperature scale were based on the freezing point...

If a temperature scale were based on the freezing point $\left(5.5^{\circ} \mathrm{C}\right)$ and boiling point $\left(80.1^{\circ} \mathrm{C}\right)$ of benzene and the temperature difference between these points was divided into 50 units (called $^{\circ} \mathrm{X} ),$ what would be the freezing and boiling points of water in $^{\circ} \mathrm{X} ?$ (See Figure $1.11, \mathrm{p} .25 . )$

Key Concepts

Unit Division and Scaling

Unit division and scaling refer to the process of dividing a known temperature range into equal intervals on a new scale. This concept is essential when designing a new temperature scale, as it involves determining the size of each unit or degree on the scale based on the difference between the chosen reference points.

Linear Transformation

A linear transformation in temperature conversion implies that the relationship between two temperature scales can be expressed as a straight-line equation. This is done by scaling (multiplying by a constant) and translating (adding or subtracting a constant) the measurements, ensuring that the proportionality between temperatures is maintained.

Temperature Scale Conversion

Temperature scale conversion is the process of mapping temperature measurements from one scale to another, typically through the use of a linear transformation. This involves determining the relationship between two scales based on their fixed reference points and applying an appropriate scaling factor and offset to convert temperatures accurately.

Reference Points

Reference points are the fixed values on a temperature scale, such as the freezing and boiling points of a substance, that define the scale's calibration. These points serve as the anchors from which all other values on the temperature scale are determined, providing a basis for conversion formulas between different scales.

Transcript

00:01 Okay, so this is a pretty involved question that asks us to determine the boiling and freezing points of water, given we invent kind of a new unit of temperature, right? so what they're calling it is this is just degrees x, and they say that zero degrees x is equal to 5 .5 degrees celsius, and 50 degrees x is equal to 5 .5 degrees celsius, and 50 degrees x is equal to, to 80 .1 degrees celsius.

00:35 All right, so those are two just, i think it's the melting boiling point of benzene.

00:42 All right, so the first thing we need to do is we can figure out how many degrees celsius are per each one increase of degrees x.

00:51 So how we're going to do that is we're basically going to do 80 .1 minus 5 .5 over 50.

01:05 So the reason we're using these numbers is the 80 .1 and the 5 .5 are the starting and ending points.

01:10 And the 50s how many steps? so this number should give us the number of steps from 5 .5, or the step width, rather, from 5 .5 to 80 .1.

01:20 We calculate it keeping two sig figs because of the 5 .5, that gives us 1 .5.

01:26 So we can say that one degree x change is the equal to 1 .5 degrees celsius.

01:37 All right.

01:38 So now what some people might do in this situation is they would just say like, oh, well, using this relationship, i can kind of just multiply this by 100 and say that, and like do that kind of multiplication.

01:51 And that's just not actually correct.

01:53 The way you need to go about this is you need to say, how many.

01:57 Steps does it take me to get from the value i have to the value i want? right? so in this case, we're going from 5 .5 to 0.

02:06 All right.

02:08 So how i'm going to do that is the same thing i did last time.

02:10 I'm going to do final, which is zero, minus the initial 5 .5 over the step width, which is going to be 1 .5.

02:19 Right? and this number should give me how many degrees x it takes me to get from 5 .5 degrees celsius to 0 degrees .5...