Question

a platinum resistance thermometer has resistance of 52.5 ohms and 9.75 ohms at 0 degree Celsius and 100 degree Celsius respectively.when the resistance is 8.25 ohms, find the temperature?

Answer 1

A platinum resistance thermometer has resistance of 52.5 ohms and 9.75 ohms at 0 degree Celsius and 100 degree Celsius respectively. When the resistance is 8.25 ohms, find the temperature?

Solution

The electrical resistivity of most materials changes with temperature. If the temperature $T$ does not vary too much, a linear approximation is typically used:

$$\rho(T) = \rho_0 \left(1 + \alpha * (T - T_0)\right)$$

where $\alpha$ is called the temperature coefficient of resistivity, $T_0$ is a fixed reference temperature (usually room temperature or $0{}^{\circ}C$), and is the resistivity at temperature. The parameter $\alpha$ is an empirical parameter fitted from measurement data.

(http://en.wikipedia.org/wiki/Electrical_resistivity_and_conductivity#Temperature_dependence)

Thus:

$$\rho(100) = \rho(0) \left(1 + \alpha * (100 - 0)\right)$$

We are given:

$$\rho(100) = 9.75\ \Omega$$ $$\rho(0) = 52.5\ \Omega$$

So:

$$9.75 = 52.5 * (1 + \alpha * (100 - 0))$$ $$1 + 100 * \alpha = \frac{9.75}{52.5}$$ $$\alpha \approx -0.00814\ \frac{\Omega}{K}$$

So, for platinum:

$$\rho(T) = 52.5 * (1 - 0.00814 * T)$$

Where $T$ in Celsius;

We are given $\rho(T_x) = 8.25\ \Omega$

$$8.25 = 52.5 * (1 - 0.00814 * T_x)$$ $$T_x \approx 103.55\ {}^{\circ}C$$

Answer: 103.55 degree Celsius