Question

Be sure to answer all parts.

If a temperature scale were based on the freezing point (5.5°C) and boiling point (80.1°C) of benzene
and the temperature difference between these points was divided into 197 units (called °X), what would
be the freezing and boiling points of water in °X?

Answer 1

To convert temperatures between the Reaumur and Celsius scales, you can use the formula: TR = 80/100 * TC.

Step-by-step explanation:

To answer this question, we must compare the Reaumur scale to the Celsius scale. The difference between the freezing point and boiling point of water on the Reaumur scale is 80°R. On the Celsius scale, it is 100°C. Therefore, 100°C = 80°R. Both scales start at 0° for freezing, so we can create a simple formula to convert between temperatures on the two scales.

Solution:

1. Derive a formula to convert from one scale to the other:

TR = 80/100 * TC

Answer 2

answer: (first one) -3.67 degrees
(second one) 63.3 degrees

Whenever you want to derive a temperature scale or make-up a new entirely, it is easiest to use some basic Calculus. First, the difference of the temperature in Celcius is (80.1 - 5.5 = 74.6), while the difference in the new unit is X is simply 50 because the problem tells you that's how many units of X there'll be. As a result, the derivative of degrees X with respect to degres C is given by:
dX/dC = (50)/(74.6) = 0.67
Moving dC to the right side of the equation and integrating gives you:
Integral(dX) = Integral(0.67dC)
X = 0.67C + constant
Now, you let the the melting point of benzene be 0 degrees X or X = 0 and since the melting point in Celsius is 5.5 or C = 5.5, you can solve for the constant:
0 = 0.67(5.5) + constant
- 3.7 = constant
Thus the equation for converting to degrees Celcius to degrees X is:
X = 0.67C - 3.7
Now, using this equation we can determine the melting and boiling points of water by plugging 0 and 100 for C and solving for X:
X(mp H2O) = 0.67(0) - 3.7 = -3.7
X(bp H2O) = 0.67(100) - 3.7 = 63.3