Question

A resistance thermometer is a thermometer in which the electrical resistance changes with temperature. We are free to define temperatures measured by such a thermometer in kelvins (K) to be directly proportional to the resistance \(R\), measured in ohms \((\Omega) .\) A certain resistance thermometer is found to have a resistance \(R\) of \(90.35 \Omega\) when its bulb is placed in water at the triple-point temperature \((273.16 \mathrm{~K}) .\) What temperature is indicated by the thermometer if the bulb is placed in an environment such that its resistance is \(96.28 \Omega ?\)

Short answer

The temperature indicated by the thermometer when its resistance is \(96.28 \Omega\) is approximately \(290.82 K\).

Step-by-step solution

Understand the Direct Proportionality

First, we need to establish the relationship between temperature and resistance. Given that temperature in Kelvin is directly proportional to the resistance in ohms, we can write an equation like this: \(T = k \cdot R\) where \(T\) is the temperature in Kelvin, \(R\) is the resistance in ohms, and \(k\) is the proportionality constant we need to find.

Calculate the Proportionality Constant

To find \(k\), we employ the given values of \(T\) and \(R\) at one known state. The problem provides the values at the triple-point temperature of water which is \(273.16K\) and \(90.35\Omega\). Insert these values into our equation to calculate \(k = T / R = 273.16K / 90.35\Omega = 3.022 K/\Omega\) (approximately).

Calculate the Unknown Temperature

Using the value of \(k\) we just calculated, we can now determine the unknown temperature corresponding to a resistance of \(96.28\Omega\). Substitute \(k\) and \(R\) into the proportion equation to find the temperature \(T = k \cdot R = 3.022 K/\Omega \cdot 96.28\Omega = 290.82K\) (approximately)

Key concepts

Electric Resistance

Electric resistance is a crucial concept for understanding how Resistance Thermometers work. Resistance essentially measures how difficult it is for electric current to flow through a conductor. The unit of resistance is the ohm, symbolized by \(\Omega\). Resistance depends on several factors, like the material's properties, its length, and its temperature.

In a resistance thermometer, as temperature increases, so does the resistance of the thermometric material. This change is because the conductor's atoms vibrate more energetically with temperature, obstructing the flow of electrons more significantly. The relationship between resistance and temperature forms the backbone for using resistance thermometers for temperature measurement.

Kelvin Temperature

The Kelvin scale is widely used in scientific measurements for its simplicity and absolute nature. A kelvin (K) is equivalent to one degree Celsius, but the scale starts at absolute zero, the hypothetical point where all thermal motion ceases. This makes the Kelvin scale appealing for scientific calculations and understanding physical properties, like resistance and temperature relationships.

Using the Kelvin scale for resistance thermometers ensures accurate and consistent readings, as it directly correlates with physical behavior at atomic levels. Given the direct proportionality with resistance, the measured temperature changes can be easily calculated if the constant of proportionality is known.

Proportionality Constant

The proportionality constant, often denoted as \(k\), links two directly proportional quantities—in this case, temperature and resistance. In a resistance thermometer, you utilize a known state, such as the triple-point of water, where temperature is 273.16 K, to determine this constant.

Using the equation \(T = k \cdot R\), where \(T\) is temperature in Kelvin and \(R\) is resistance in ohms, allows for the calculation of the unknown temperature. Once \(k\) is determined, it provides a conversion factor between resistance and temperature, enabling consistent measurements across different conditions.

Ohms

Ohms measure the electrical resistance within a circuit. The symbol for ohms is \(\Omega\). A higher resistance indicates a greater opposition to current flow, which can be due to increased temperature. In the context of resistance thermometers, measuring in ohms is essential because it provides a quantifiable way to determine temperature changes.

By reading the resistance in ohms and knowing the proportionality constant, it's possible to calculate the exact temperature in Kelvin. Therefore, understanding resistance in terms of ohms is a fundamental concept for interpreting thermometric data using resistance thermometers effectively.