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Exercises · 10.2

Q.Two absolute scales AA and BB have triple points of water defined to be 200 A200\ A and 350 B350\ B. What is the relation between TAT_A and TBT_B?

Himachal HpboseTextbookSubjective· 2mImportance★★★★★est
18% · 10/55 Questions
✓ Free question

The key idea is that the triple point of water is a fixed physical temperature (273.16 K on the Kelvin scale), so the numerical values assigned on scales AA and BB are proportional to the absolute temperature. This gives TA=47TBT_A = \frac{4}{7} T_B.

Why This Works

Temperature scales are arbitrary, but absolute scales have a fixed reference: the triple point of water. On the Kelvin scale, this is exactly 273.16 K. On any other absolute scale, the triple point is assigned some other number — but it represents the same physical temperature. This means the ratio of any temperature to the triple-point value on that scale is the same as the ratio on any other absolute scale.

In other words, if you know the triple-point assignments on two absolute scales, you can convert between them by simple proportion.

Step-by-Step

  1. Identify the fixed point.

    The triple point of water is a universal constant. On scale AA, it is defined as 200 A200\ A. On scale BB, it is 350 B350\ B. Both represent the same physical temperature.

  2. Set up the proportion.

    For any absolute scale, temperature is proportional to the thermodynamic temperature. So if TAT_A is a temperature on scale AA and TBT_B is the same physical temperature on scale BB, then:

TA200=TB350\frac{T_A}{200} = \frac{T_B}{350}

Why? Because the ratio of a temperature to the triple-point value must be the same on both scales — they measure the same thing.

  1. Solve for the relation. Multiply both sides by 200×350200 \times 350:

350 TA=200 TB350\, T_A = 200\, T_B

Divide through by 350:

TA=200350TB=47TBT_A = \frac{200}{350} T_B = \frac{4}{7} T_B

That’s the direct relation.

Tip

You can also think of it as a conversion factor: one degree on scale BB equals 47\frac{4}{7} of a degree on scale AA. So scale AA has larger degrees (fewer of them to reach the same temperature).

Watch out

A common mistake is to write TA=350200TBT_A = \frac{350}{200} T_B — that would mean scale AA gives a larger number for the same temperature, but here the triple point is lower on AA (200 vs 350), so AA's degrees are actually bigger. Always check: if the triple-point number is smaller, each degree is larger.

✓Final answer

The relation is TA=47TBT_A = \frac{4}{7} T_B.

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