The resistances of the platinum wire of a platinum resistance thermometer at the ice point and steam point are \( 8 \, \Omega \) and \( 10 \, \Omega \) respectively. After inserting in a hot bath of temperature \( 400^\circ \text{C} \), the resistance of the platinum wire is:
- \( 2 \, \Omega \)
- \( 16 \, \Omega \)
- \( 8 \, \Omega \)
- \( 10 \, \Omega \)
The Correct Option is B
Approach Solution - 1
Use the Temperature-Resistance Relationship for a Platinum Resistance Thermometer:
The resistance \( R \) of a platinum resistance thermometer at a temperature \( T \) (in °C) is given by:
\[ R_T = R_0(1 + \alpha \Delta T) \] where:
\( R_0 \) is the resistance at 0°C (ice point),
\( \alpha \) is the temperature coefficient of resistance of platinum,
\( \Delta T \) is the temperature difference from the ice point.
Given Data:
\( R_0 = 8 \, \Omega \) (resistance at 0°C)
\( R_{100} = 10 \, \Omega \) (resistance at 100°C)
Temperature of the hot bath: \( T = 400°C \)
Calculate the Temperature Coefficient \( \alpha \):
Using the resistance values at 0°C and 100°C:
\[ R_{100} = R_0(1 + 100\alpha) \]
Substitute \( R_{100} = 10 \, \Omega \) and \( R_0 = 8 \, \Omega \):
\[ 10 = 8(1 + 100\alpha) \] \[ 1 + 100\alpha = \frac{10}{8} = 1.25 \]
\[ 100\alpha = 0.25 \] \[ \alpha = \frac{0.25}{100} = 0.0025 \, °C^{-1} \]
Calculate the Resistance at 400°C:
Now, using \( T = 400°C \):
\[ R_{400} = R_0(1 + \alpha \times 400) \]
Substitute \( R_0 = 8 \, \Omega \) and \( \alpha = 0.0025 \):
\[ R_{400} = 8 \times (1 + 0.0025 \times 400) \]
\[ R_{400} = 8 \times (1 + 1) = 8 \times 2 = 16 \, \Omega \]
Conclusion:
The resistance of the platinum wire at 400°C is \( 16 \, \Omega \).
Approach Solution -2
At ice point (\( 0^\circ \text{C} \)): \( R_0 = 8 \, \Omega \)
At steam point (\( 100^\circ \text{C} \)): \( R_{100} = 10 \, \Omega \)
We need to find the resistance \( R_{400} \) when the temperature \( t = 400^\circ \text{C} \).
Step 2: Formula for resistance-temperature relation.
For a platinum resistance thermometer, resistance varies linearly with temperature:
\[ R_t = R_0 (1 + \alpha t) \] where \( \alpha \) is the temperature coefficient of resistance.
Step 3: Find \( \alpha \).
At \( t = 100^\circ \text{C} \):
\[ R_{100} = R_0 (1 + 100\alpha) \] \[ 10 = 8(1 + 100\alpha) \] \[ \frac{10}{8} = 1 + 100\alpha \Rightarrow 1.25 = 1 + 100\alpha \] \[ \Rightarrow \alpha = \frac{0.25}{100} = 0.0025 \]
Step 4: Calculate resistance at \( 400^\circ \text{C} \).
\[ R_{400} = R_0 (1 + \alpha \times 400) \] \[ R_{400} = 8(1 + 0.0025 \times 400) = 8(1 + 1) = 8 \times 2 = 16 \, \Omega. \]
Final Answer:
\[ \boxed{16 \, \Omega} \]
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