A faulty thermometer reads $$5°$$C in melting ice and $$95°$$C in steam. The correct temperature on absolute scale will be ______ K when the faulty thermometer reads $$41°$$C.

Solution & Explanation

1. Understand the Linear Scaling of Temperature Scales

Temperature scales are constructed linearly based on two fixed reference points: the lower fixed point (melting point of ice) and the upper fixed point (boiling point of water/steam).

For any linear thermometer scale, the ratio of the difference between a reading and the ice point to the total interval between the steam point and the ice point remains perfectly constant:

$$\frac{\text{Reading} - \text{LFP}}{\text{UFP} - \text{LFP}} = \text{Constant}$$

Where:

• LFP: Lower Fixed Point (Ice point)
• UFP: Upper Fixed Point (Steam point)

2. Set Up the Equivalence Relation

Let us compare the faulty thermometer scale ($$W$$) with the standard Celsius scale ($$C$$):

• For the Faulty Thermometer ($$W$$): $$\text{LFP}_W = 5^\circ\text{C}$$ and $$\text{UFP}_W = 95^\circ\text{C}$$
• For the Standard Celsius Scale ($$C$$): $$\text{LFP}_C = 0^\circ\text{C}$$ and $$\text{UFP}_C = 100^\circ\text{C}$$

Equating their linear scale ratios:

$$\frac{W - \text{LFP}_W}{\text{UFP}_W - \text{LFP}_W} = \frac{C - \text{LFP}_C}{\text{UFP}_C - \text{LFP}_C}$$

$$\frac{W - 5}{95 - 5} = \frac{C - 0}{100 - 0}$$

$$\frac{W - 5}{90} = \frac{C}{100}$$

3. Calculate the True Temperature in Celsius ($$C$$)

We are given that the faulty thermometer reads $$W = 41^\circ\text{C}$$. Substituting this value into our equation:

$$\frac{41 - 5}{90} = \frac{C}{100}$$

$$\frac{36}{90} = \frac{C}{100}$$

Simplify the fraction on the left side:

$$\frac{2}{5} = \frac{C}{100}$$

$$C = \frac{2}{5} \times 100 = 40^\circ\text{C}$$

4. Convert to the Absolute (Kelvin) Scale

The relationship between the standard Celsius scale and the absolute thermodynamic Kelvin scale ($$T$$) is defined by:

$$T = C + 273$$

Substituting the true calculated value of $$C = 40^\circ\text{C}$$:

$$T = 40 + 273 = 313 \,\, \text{K}$$

Correct Numerical Answer: 313