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A constant mass of gas maintained at constant pressure has a volume of 200.0cm³ at the temperature of melting ice, 273.2cm³ at the temperature of water boiling under standard pressure, and 525.1cm³ at the normal boiling point of sulphur. A platinum wire has a resistance of 2.000, 2.778 and 5.28052 at the temperatures. Calculate the values of the boiling point of sulphur given by the two sets of observations, and comment on the results

1 answer

To approach this problem, we can use the concept of thermal expansion, described by Charles's Law, which relates the volume of a gas to its temperature when pressure is held constant.

Given data:

  1. Volume at melting ice (0 °C / 273.2 K): \( V_1 = 200.0 , \text{cm}^3 \)
  2. Volume at water boiling under standard pressure (100 °C / 373.2 K): \( V_2 = 273.0 , \text{cm}^3 \)
  3. Volume at boiling point of sulfur (specifically a temperature we want to determine): \( V_3 = 525.1 , \text{cm}^3 \)

Temperature in Kelvin (to use in thermal expansion)

  • Temperature of melting ice: \( T_1 = 273.2 , K \)
  • Temperature of water boiling: \( T_2 = 373.2 , K \)
  • Resistance of platinum wire at melting ice: \( R_1 = 2.000 , \Omega \)
  • Resistance of platinum wire at boiling water: \( R_2 = 2.778 , \Omega \)
  • Resistance of platinum wire at boiling point of sulfur: \( R_3 = 5.28052 , \Omega \)

Find the temperature corresponding to the boiling point of sulfur using opposition resistances.

Using the ratio of resistances to temperatures, we get:

\[ \frac{R_1 - R_0}{R_2 - R_0} = \frac{T_1 - T_0}{T_2 - T_0} \] Where \( R_0 \) and \( T_0 \) are the resistance and temperature at absolute zero.

From this equation, we can set up the equations for sulfur using the measured \( R_3 \):

\[ \frac{R_1 - R_0}{R_3 - R_0} = \frac{T_1 - T_0}{T_3 - T_0} \]

Let’s find the slopes corresponding to known volumes and temperatures for volumes to temperature adjustments.

Step 1: Calculate the Resistance Ratios

  1. Calculate \( R_0 \) with \( R_1 \) and \( R_2 \): \[ \text{Slope} = \frac{R_2 - R_1}{T_2 - T_1} = \frac{2.778 - 2.000}{373.2 - 273.2} \approx \frac{0.778}{100} = 0.00778 \Omega/K \]

Step 2: Calculate \( (T_3 - T_0) \)

Using \( R_3 \): \[ R_3 - R_1 = \text{Slope} \cdot (T_3 - T_1) \] \[ 5.28052 - 2.000 = 0.00778 \cdot (T_3 - 273.2) \] \[ 3.28052 = 0.00778 \cdot (T_3 - 273.2) \] Now solve for \( T_3 \): \[ T_3 - 273.2 = \frac{3.28052}{0.00778} \approx 421.3 \] \[ T_3 = 421.3 + 273.2 \approx 694.5 , K \]

Step 3: Convert to Celsius

\[ T_3 = 694.5 - 273.15 \approx 421.35 , °C \]

Conclusion

The calculated boiling point of sulfur is approximately \( 421.35 , °C \).

Comments on Results:

The results can show how the resistances of the platinum wire are influenced by changes in temperature under isothermal conditions. The method exhibits consistency provided the functional relation stays linear across the measured ranges. A deviation may also suggest non-ideal behavior or additional calibration needs.

Also, comparing the consistency of different measurements would help assess the reliability of the temperature-resistance correlation, yielding the requisite boiling point more closely related to established scientific data.