To determine the resistance of the platinum-resistance thermometer at 45 degrees Celsius, we can use the formula for linear interpolation based on the known resistances at 0 degrees Celsius and 100 degrees Celsius.
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Establish the known values:
- \( R_0 = 10 , \Omega \) at \( T_0 = 0 , °C \)
- \( R_{100} = 18 , \Omega \) at \( T_{100} = 100 , °C \)
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Calculate the change in resistance per degree Celsius:
- The change in resistance from 0°C to 100°C: \[ \Delta R = R_{100} - R_0 = 18 , \Omega - 10 , \Omega = 8 , \Omega \]
- The change in temperature from 0°C to 100°C: \[ \Delta T = T_{100} - T_0 = 100 , °C - 0 , °C = 100 , °C \]
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Determine the resistance change per degree Celsius: \[ \text{Change in resistance per °C} = \frac{\Delta R}{\Delta T} = \frac{8 , \Omega}{100 , °C} = 0.08 , \Omega/°C \]
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Calculate the resistance at 45°C:
- The temperature change from 0°C to 45°C: \[ \Delta T_{45} = 45 , °C - 0 , °C = 45 , °C \]
- The resistance at 45°C can be calculated as follows: \[ R_{45} = R_0 + (\text{Change in resistance per °C} \times \Delta T_{45}) \] \[ R_{45} = 10 , \Omega + (0.08 , \Omega/°C \times 45 , °C) \] \[ R_{45} = 10 , \Omega + 3.6 , \Omega = 13.6 , \Omega \]
Therefore, the resistance of the thermometer at 45 degrees Celsius is 13.6 Ω.
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