A platinum resistance thermometer measures temperature on assumption that the change in the electrical resistance of platinum is proportional to the change in temperature. The resistance of the platinum wire at O°c is 10.4π and 10.71π at 100°c. Calculate the temperature when the resistance is 9.61π. What will the resistance of the wire when the temperature is 45°c
A platinum thermometer measures a temperature on the the electrical resistance of a platinum
A platinum
To find the temperature when the resistance is 9.61π, we can use a proportionality equation with the given resistance values at 0°C and 100°C.
Let's denote the resistance at 0°C as R₀ and the resistance at 100°C as R₁.
R₀ = 10.4π
R₁ = 10.71π
The proportionality equation is:
(R - R₀) / (R₁ - R₀) = (T - T₀) / (T₁ - T₀)
Where:
R is the given resistance (9.61π)
T is the temperature we want to find
T₀ is the temperature at 0°C (initial temperature)
T₁ is the temperature at 100°C (final temperature)
Plug in the values we have and solve for T:
(9.61π - 10.4π) / (10.71π - 10.4π) = (T - 0) / (100 - 0)
Simplifying this expression:
-0.79π / 0.31π = T / 100
-0.79 / 0.31 = T / 100
T = (-0.79 / 0.31) * 100
T ≈ -254.84
Therefore, the temperature when the resistance is 9.61π is approximately -254.84°C.
To find the resistance of the wire when the temperature is 45°C, we can use the same proportionality equation with the given resistance values at 0°C and 100°C.
Again, let's denote the resistance at 0°C as R₀ and the resistance at 100°C as R₁.
R₀ = 10.4π
R₁ = 10.71π
(R - R₀) / (R₁ - R₀) = (T - T₀) / (T₁ - T₀)
Plug in the values we have and solve for R:
(R - 10.4π) / (10.71π - 10.4π) = (45 - 0) / (100 - 0)
Simplifying this expression:
(R - 10.4π) / 0.31π = 45 / 100
R - 10.4π = 0.31π * 45 / 100
R - 10.4π = 0.1395π
R = 10.4π + 0.1395π
R ≈ 10.5395π
Therefore, the resistance of the wire when the temperature is 45°C is approximately 10.5395π.
To calculate the temperature based on the resistance of the platinum wire, we need to use the given data and make use of the linear relationship between the resistance and temperature.
Let's first find the constant of proportionality (k) using the given data:
Resistance at 0°C (R₁) = 10.4π
Resistance at 100°C (R₂) = 10.71π
The change in resistance (ΔR) = R₂ - R₁ = (10.71π - 10.4π) = 0.31π
The change in temperature (ΔT) = 100°C - 0°C = 100°C
Now, we can calculate the value of k:
k = ΔR / ΔT = (0.31π) / 100 = 0.0031π/°C
Now, let's solve the first part of the problem:
Resistance = 9.61π
We can use the equation: Resistance = R₁ + k * (Temperature - 0°C)
9.61π = 10.4π + (0.0031π/°C) * (Temperature - 0°C)
9.61π - 10.4π = (0.0031π/°C) * (Temperature - 0°C)
-0.79π = (0.0031π/°C) * Temperature
Divide both sides by (0.0031π):
-0.79 / 0.0031 = Temperature
Temperature = -254.84°C
The temperature when the resistance is 9.61π is approximately -254.84°C.
Now, let's calculate the resistance of the wire when the temperature is 45°C:
We'll use the same equation: Resistance = R₁ + k * (Temperature - 0°C)
Resistance = 10.4π + (0.0031π/°C) * (45°C - 0°C)
Resistance = 10.4π + (0.0031π/°C) * 45
Resistance = 10.4π + 0.1395π
Resistance = 10.5395π
The resistance of the wire when the temperature is 45°C is approximately 10.5395π.
since ∆R/∆T is constant, you want
T such that
(T-0)/(9.61-10.4) = (100-0)/(10.71-10.4)
and R such that
(45-0)/(R-10.4) = (100-0)/(10.71-10.4)