Question

Two absolute scales X and Y assigned numerical values 200 and 450 to the triple point of water. What is the relation between \[{{T}_{X}}\] and \[{{T}_{Y}}\]?
A. \[9{{T}_{X}}=4{{T}_{Y}}\]
B. \[4{{T}_{X}}=9{{T}_{Y}}\]
C. \[{{T}_{X}}=3{{T}_{Y}}\]
D. None of these

Answer 1

Hint: Under a particular temperature and pressure, the three forms of matter will coexist together. In the case of pure water, it is 273.16K. Now equate it with the temperature in the given scale and find the ratio between them. Use \[{{T}_{1}}={{T}_{2}}=273.16K\]

Complete step-by-step answer:
The three states of matter that are found on our planet are solid, liquid and gas. Matter can change from one state to another. We are very familiar with the three states of \[{{H}_{2}}O\]. Those are ice, water and vapour. We have often seen transitions at our surroundings itself. Now let us think: is it possible to find these three states together at a particular temperature? As you know, the freezing and boiling point of water is \[{{0}^{0}}C\]and \[{{100}^{0}}C\]respectively. Then how is it possible to find all the three in a specific temperature?
It is possible in the case of water where we find all the three forms. There is a point where solid, liquid and gas exist in an equilibrium state, the triple point. For pure water, the triple point is at \[0.1{{C}^{0}}\]or \[32.1F\]or \[273.16K\]. i.e. at the triple point of water,
\[0.1{{C}^{0}}=32.1F=273.16K\]……………………(1)
Here in this problem, two new sets of scales are given – X and Y scale.
A triple point, on the X scale,\[{{T}_{1}}=200X\]
At triple point, on the Y scale, \[{{T}_{2}}=450Y\]
But from equation (1) we can write
\[{{T}_{1}}={{T}_{2}}=273.16K\]……………………………..(2)
(Since the SI unit of temperature is in the Kelvin scale, I have taken it.)
\[\therefore \]So we can write as
\[{{T}_{1}}=23.16K\]i.e
\[200X=237.16K\]
Thus the value of X is given in terms of K as
\[X=\dfrac{237.16}{200}K\]…………………………….(3)
Similarly, we can find the value of y in terms of K.
\[Y=\dfrac{237.16}{450}K\]……………………………….(4)
At the triple point of water,
\[{{T}_{X}}\]is the triple point of water on the X scale
\[{{T}_{Y}}\]be the triple point of water on the Y scale.
Therefore we can writ
\[\dfrac{273.16}{200}{{T}_{X}}=\dfrac{237.16}{450}{{T}_{Y}}\] or
\[\dfrac{1}{200}{{T}_{X}}=\dfrac{1}{450}{{T}_{Y}}\]
\[45{{T}_{X}}=20{{T}_{Y}}\]i.e.
\[9{{T}_{X}}=4{{T}_{Y}}\]
From this we got option A is the correct answer.

Note: Like this, we can make our own scale. Only under certain conditions, the triple point exists. It purely depends on temperature and pressure. At this point, it is effortless for the change of state to happen.