Problem 19 - Master Resource Book in JEE Main Physics

Question

In an experiment to determine the specific heat of a given liquid by method of mixtures. If room temperature recorded by one thermometer is \(29^{\circ} \mathrm{C}\) and that by second thermometer is \(27.5^{\circ} \mathrm{C}\). If steady temperature of metal in hypsometer is \(62^{\circ} \mathrm{C}\). What will be corrected temperature of metal? (a) \(60^{\circ} \mathrm{C}\) (b) \(60.5^{\circ} \mathrm{C}\) (c) \(61^{\circ} \mathrm{C}\) (d) \(63.5^{\circ} \mathrm{C}\)

Step-by-Step Solution

Answer

The corrected temperature of the metal is closest to 61°C, which is answer (c).

Step 1: Understanding Temperature Readings

We have two thermometers recording different room temperatures: Thermometer 1 records the room temperature as \(29^{\circ} \mathrm{C}\), while Thermometer 2 records it as \(27.5^{\circ} \mathrm{C}\). These readings indicate a calibration difference between the two thermometers.

Step 2: Determine Average Room Temperature

To find an accurate room temperature, we take the average of the two different readings to mitigate their calibration difference: \[ \text{Average Room Temperature} = \frac{29^{\circ} \mathrm{C} + 27.5^{\circ} \mathrm{C}}{2} = 28.25^{\circ} \mathrm{C} \]

Step 3: Determine Average Temperature Difference

Calculate the temperature difference between the two thermometers' room temperature readings: \[ \text{Difference} = 29^{\circ} \mathrm{C} - 27.5^{\circ} \mathrm{C} = 1.5^{\circ} \mathrm{C}. \]

Step 4: Adjust Recorded Temperature of Metal

Assuming similar calibration differences apply to the measured temperature of the metal, we adjust the recorded temperature of \(62^{\circ} \mathrm{C}\) by subtracting the average temperature difference: \[ \text{Corrected Metal Temperature} = 62^{\circ} \mathrm{C} - \left(\frac{1.5^{\circ} \mathrm{C}}{2}\right) = 62^{\circ} \mathrm{C} - 0.75^{\circ} \mathrm{C} = 61.25^{\circ} \mathrm{C}. \]

Step 5: Select Closest Answer

The calculated corrected temperature \(61.25^{\circ} \mathrm{C}\) is closest to answer (c) \(61^{\circ} \mathrm{C}\).

Key Concepts

Temperature Correction

In experiments, precise temperature readings are crucial. Often, different thermometers might give slightly different readings. This leads to the need for temperature correction.
When attempting to measure a temperature accurately, if you have two different temperature readings, first consider taking an average. This helps in balancing out any discrepancies between individual readings. This average can give us an estimate closer to the true room temperature, useful for further calculations.
For example, if one thermometer reads the room at \(29^{\circ} \mathrm{C}\) and another at \(27.5^{\circ} \mathrm{C}\), the calculated average would be \(28.25^{\circ} \mathrm{C}\). This value represents a corrected approximation of the room temperature.
Using this correction method ensures more reliable calculations later, especially when applied to measure further temperatures in experiments.

Calibration Difference

No two thermometers are exactly identical, especially if sourced separately or manufactured in different batches. This is why calibration difference is an important concept.
A calibration difference refers to the deviation in readings between two separate measuring devices. In our example, we saw a discrepancy of \(1.5^{\circ} \mathrm{C}\) between two thermometers measuring the same ambient condition.
To manage this variance, we use the average value of the discrepancy as a correction factor. This factor helps adjust readings to more accurately reflect the true temperature. Therefore, when you calibrate based on differences, any subsequent readings from these devices are adjusted by this calibration gap.

• Helps standardize differing thermometer readings.
• Enables uniformity in experimental results.
• Adjusts for technical manufacturing variations or environmental effects.

Mixtures Method

The mixtures method is a classic experimental technique used to find the specific heat of a substance. It involves mixing two bodies (usually a solid and a liquid) at different temperatures, allowing them to reach a thermal equilibrium.
In this approach, the principle of energy conservation is used. The heat lost by the hotter body is equal to the heat gained by the cooler body.

• Helps in determining specific heat capacity.
• Accounts for energy exchange during mixing.
• Useful for substances that might otherwise be difficult to handle directly.

For example, if a heated metal piece is placed into a liquid, the temperature of the metal will decrease as the liquid's temperature increases until they reach a stable temperature.
Any errors or variances, such as incorrect starting temperatures due to calibration differences, can lead to inaccurate results. Thus, applying the temperature corrections and appropriate calibrations as described is crucial for accurate measurements through the mixtures method.